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. MPDE12 Models in Population Dynamics and Ecology UFSM Federal University of Santa Maria Santa Maria - RS - Brazil 10-13 September 2012 Supported by CAPES, CNPq, FAPERGS and SBMAC . WELCOME The MPDE12 meeting continues on from the successful series of MPDE meetings (MPDE’07, MPDE’08, MDBE’09, MPDE’10) recently hosted at the University of Leicester and (MATE 2011) hosted at the University of Essex in Colchester, all of them in the UK. This year the meeting left the North Hemisphere and crossed the Atlantic to be held in Brazil. It is an honor for us to host the meeting at UFSM and we have the pleasure to great and welcome all of you who are participating in it. Considered the ”Heart of the State”, Santa Maria is located in the central area of the state of Rio Grande do Sul, ”Great River of the South”, the southernmost state in Brazil. Despite the European influence, the ”gauchos” - the inhabitants of Rio Grande do Sul,- strongly cultivate the traditions of the ”Pampas” region around the border with Uruguay and Argentina such as drinking ”chimarrão” (a tea drunk in special gourd cups), eating ”churrasco” (the typical barbecue) and wearing the traditional clothes which are the ”bombachas” (baggy trousers), boots and large hats for men and long dresses for women. Although different from what you may have seen about Brazil, we believe you will find and hopefully enjoy our ”Brazilianess”. Established in 1961, UFSM is a governmental, regionally oriented University which has as its main purpose to perform teaching, research and extension activities. UFSMs mission is the professional formation, seeking to generate and cultivate science and technology in the interest of regional development and the transmission of universal knowledge. Many students and young researchers from Brazil and South America will find here an extraordinary opportunity to contact the state-of-the-art in Population Dynamics and we are confident it will bring enthusiasm to lead the local Biomathematician population to its carrying capacity. We acknowledge the support from the UFSM, the Brazilian Society for Applied Mathematics and Computing (SBMAC) and the Brazilian Government Funding Agencies (FAPERGS, CNPq and CAPES). BEM-VINDOS Diomar Cristina Mistro Luiz Alberto Dı́az Rodrigues . . Organization and Structure Organising Committee Diomar C. Mistro (UFSM, Brazil) Luiz Alberto D. Rodrigues (UFSM, Brazil) Sergei Petrovskii (Leicester, UK) Scientific Committee Edward Codling (Essex, UK) Horst Malchow (Osnabrück, Germany) Nicholas Britton (Bath, UK) Nitant Kenkre (Albuquerque, USA) Wilson C. Ferreira Jr. (UNICAMP, Brazil) Plenary Speakers Alan Hastings (Davis, USA) Bernd Blasius (Oldenburg, Germany) Horst Malchow (Osnabrück, Germany) Mark Lewis (Edmonton, Canada) Nicholas Britton (Bath, UK) Sergei Petrovskii (Leicester, UK) Ulrike Feudel (Oldenburg, Germany) Vitaly Volpert (Lyon, France) Wilson Castro Ferreira Jr. (UNICAMP, Brazil) 5 . Plenary Talks . A simple approach to questions of subsidies and time in ecology Alan Hastings Distinguished Professor Environmental Science and Policy University of California Davis CA 95616, USA I will develop simple models describing the interactions among age structured population dynamics, time dependent resource availability and the role of subsidies. These models will show that all these features need to be explicitly included to understand the factors that permit species to persist or coexist in simple systems. Comparisons will be made to data previously collected by Polis and colleagues for islands in the Gulf of California. 9 Marine bioinvasion in the network of global shipping connections Bernd Blasius Institute for Chemistry & Biology of the Marine Environment Carl von Ossietzky University of Oldenburg, Germany Transportation networks play a crucial role in human mobility, the exchange of goods, and the spread of invasive species. With 90world trade carried by sea, global shipping provides one of the most important modes of transportation. Shipping also constitutes the world largest transportation vector for marine bioinvasion, transferring accidentally numerous species around the world. Here, we use information about the itineraries of 16,363 cargo ships during the year 2007 to construct a network of shipping connections between ports. We perform a statistical analysis of the network topology and reveal marked differences to standard gravity models. Coupling the shipping network with port environmental conditions and biogeography, we develop a model for marine bioinvasion by world-wide ballast water exchange. The model allows to identify high risk invasion routes, hot spots of bioinvasion, and major source regions from which bioinvasion is likely to occur, and it can be used to classify coastal ecoregions with respect to total invasion risk and risk composition from other regions. Our model predictions agree with observations in the field and reveal a pattern of maximal invasion risk at intermediate geographic distances. Finally, we apply the model to investigate strategies for risk reduction by ballast water treatment. Motivated from the invasion process, we present a conceptional model for the spread of a binary variable (here: invaded or non-invaded port) on a complex network. Despite its simplicity, the model exhibits complex dynamics and shows many properties that set it apart from similar models of epidemic spread or cascading failures. 10 Infection and control of a competitive invader Horst Malchow Department of Mathematics and Computer Science Institute of Environmental Systems Research University of Osnabrueck, Germany A plant competition-flow model of Lotka-Volterra type is considered for conditions of invasibility of a certain model area occupied by a native species. Shortdistance invasion is assumed as diffusion whereas long-distance seed dispersal can be stratified diffusive or advective. The variability of the environment due to contingent landslides and artificial causes such as deforestation or weed control leads to the temporary extinction of one or both species at a randomly chosen time and spatial range. The spatiotemporal dimension of these extreme fragmentation events as well as a possible selected harvesting or targeted infection of the invading weed turn out to be the crucial driving forces of the system dynamics whereas annual periodicities of certain system parameters are less influential (Malchow et al. 2011, 2012). Furthermore, the mathematical model allows for differential competitiveness amongst and between infected and uninfected invaders. It is found that pathogen-induced modifications of competition can tremendously alter the stability and persistence of host–pathogen systems (Sieber et al. 2012). 11 The Mathematics Behind Biological Invasion Processes Mark Lewis Department of Mathematical Sciences University of Alberta Edmonton, Canada Models for invasions track the front of an expanding wave of population density. They take the form of parabolic partial differential equations and related integral formulations. These models can be used to address questions ranging from the rate of spread of introduced invaders and diseases to the ability of vegetation to shift in response to climate change. In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions, for accelerating invasion waves, and for nonlinear stochastic interactions that can determine spread rates. 12 Interspecific kleptoparasitism Nicholas F. Britton Department of Mathematical Sciences & Centre for Mathematical Biology, University of Bath, Claverton Down, Bath, BA2 7AY, UK, e-mail [email protected] Kleptoparasitism is parasitism by theft. In a typical kleptoparasitic host– parasite interaction the parasite steals some resource, such as a food item, from the host. The host and parasite belong to the same species in intraspecific and to different species in interspecific kleptoparasitism. An interaction may be considered as an asymmetric game, with the parasite (or intruder) deciding whether to challenge the host (or owner) for the resource and the host deciding whether to resist the challenge. In the intraspecific but not the interspecific case a single animal may play the role of owner at one time and that of intruder at another. We review the intraspecific case, originally discussed by Ruxton and Moody [1] and others, and then go on to analyse the interspecific case, considering both the adaptive and the replicator dynamics. Depending on parameter values, the replicator dynamics system may not settle to a steady state but to oscillatory behaviour in strategy space. [1] G D Ruxton and A L Moody, The ideal free distribution with kleptoparasitism, Journal of Theoretical Biology, 186, 449–458, 1997. 13 Biological Invasion: Observations, Theory, Models, Simulations Sergei Petrovskii Department of Mathematics University of Leicester, UK Biological invasion admittedly consists of a few distinctly different stages such as exotic species introduction, establishment and geographical spread. Each of the stages has its own specific mechanisms and implications, which require application of specific research approaches. In my talk, I focus on the challenges arising during the stage of the geographical spread. A well-developed theory based on diffusion-reaction equations predicts a simple pattern of alien species spread consisting of a continuous traveling boundary or ’population front’ separating the invaded and non-invaded regions. A propagating population front has been a paradigm of the invasive species spread for several decades. However, it also appears to be at odds with some observations. In some cases, the spread takes place through formation of a distinct patchy spatial structure without any continuous boundary. Perhaps the most well known and well studied example of this ’patchy invasion’ is the gypsy moth spread in the USA. In order to address this problem theoretically, I first re-examine the current views on possible mechanisms of the patchy spread and argue that the importance of the stratified diffusion may be significantly overestimated. Second, I will revisit the traditional diffusion-reaction framework and show that the patchy spread is, in fact, its inherent property in case the invasive species is affected by predation or an infectious disease and its growth is damped by the strong Allee effect. The patchy spread described by a diffusion-reaction model appears to be a scenario of alien species invasion ”at the edge of extinction” and this can have important implications for the management and control of the invasive species. Finally, I will show that patchy spread is not an exclusive property of the diffusion-reaction systems but can be observed as well in a completely different type of model such as a coupled map lattice which is capable of taking into account environmental heterogeneity. 14 Biodiversity of plankton: non-equilibrium coexistence of competing species Ulrike Feudel University of Oldenburg, Germany Several solutions have been offered in the literature to solve the paradox of planton which states that in equilibrium the number of coexisting species can not exceed the number of ressources. We study a competition model similar to the one introduced by Huisman and Weissing who showed that coexistence of more species than ressources becomes possible in non-equilibrium states such as periodic or chaotic states. They called this phenomenon supersaturation. In many studies of competition models Liebig,s law of the minimum is used to account for the fact that the least available nutrient will determine the growth rate of the plankton species. However this would require that the organisms can instantaneously switch their physiological regulation system, which is problematic. It is more natural to assume that there is a co-limitation for all ressources, so that all ressources contribute to the growth rate. Therefore our model is based on the dynamic energy budget theory [2] which uses the concept of a synthezising unit. This concept is based on the mechanisms of enzyme kinetics and considers all ressources as complementary. Using this model we study the dynamics of the competing species which can exhibit competitive exclusion, heteroclinic cycles, stable coexistence in a fixed point and periodic solutions. Moreover, we find the coexistence of more species than ressources in parameter regions where periodic and chaotic solutions are possible. Hence we can show that supersaturation is possible in a model with a more realistic approach to the uptake of ressources. It is important to note that this model exhibits supersaturation in parameter ranges which are realistic. Our study reveals the dynamical mechanism how supersaturation can occur: it is due to a transcritical bifurcation of limit cycles. 15 Multi-scale modelling in cell population dynamics Vitaly Volpert University of Lyon, France Multi-scale modeling in biology takes into account cells in their interaction with each other and with the surrounding medium, intracellular (molecular) level, various feedbacks and regulations from organs and tissues. We will discuss possible approaches to the description of these complex biological phenomena with continuous and with hybrid discrete-continuous models. They will be illustrated with various examples, such as erythropoiesis, tumor growth or blood coagulation. 16 Revisiting the 1879 Model of Evolutionary Mimicry by Frederico Müller, a GermanBrazilian Biomathematician Extraordinaire Wilson Castro Ferreira Jr. Department of Applied Mathematics Universidade de Campinas - UNICAMP, Brazil In this talk we will present some mathematical approaches to describe an extraordinary phenomenon widely cited in the biological literature as ”Müllerian Mimicry”. Mimicry in general is an evolutionary phenomena which was observed and registered a long time ago by Henry W. Bates in a form named ”Batesian” today . In such cases, a species, called the model, develops toxic or, at least, very distasteful substances in their bodies, accompanied by highly visible signals for their identification. Predators are bound to learn of this toxicity in a hard way, i.e., by eventually tasting one of the prey individual, usually by killing it, and subsequently keeping in memory their ”(Non)-Search Image” as a result of the unfortunate episode. Of course, there is no advantage to the prey population if the hapless predator dies after the lesson, since this knowledge is a personal one and not, in general, transferrable to other individuals by teaching. However, and on the other hand, a good lesson must be suitably incisive to make it remembered for a long time. What is most interesting is that sometimes other species, called mimic, oftenly quite unrelated to the model except for sharing with it a visually acute predators, develop a similar exterior visual signaling, which does not take too much demand on their internal metabolism. By using such ”cheap” strategy the mimic population learn excellent protection from predators without much expenditure of metabolic resources. Observations of mimicry phenomena among many different species in nature happened during the second half of the 19th century while a fierce battle for the acceptance of Darwin’s evolutionary theory was still undecided and they contributed decisively as one of the foremost argument in Darwin’s favor. Müllerian Mimicry is a subtle phenomenon from an evolutionary point of view and happens when two different species, both of them toxic as well, and under pressure from the same predator, develop a similar strong visual signal in such a way that ”teaching casualties” become a shared onus. The 17 tasting of any individual, no matter from which prey population, will turn the hapless predator into a ”learned” one which will treat both prey populations with due respect afterwards. This type of mimicry was first described by Frederico Müller, a Brazilian-German naturalist living in Desterro (Florianópolis, today), in a paper published in 1879. In this article Müller who was also a mathematician by training, developed a non trivial model to better describe the related evolutionary process, a work which can no doubt be regarded as the first Brazilian Biomathematical piece. Frederico Müller, as he was known in Brazil, was born in Germany in 1822, baptized as Johann Friedrich Theodor Müller, graduated in Natural History and Mathematics from the University of Berlin in 1844 and emigrated to Brazil in 1852 where he worked as a teacher, and a naturalist for the Museo de Historia Natural do Brazil until his death in 1897. His work appeared in more than 240 papers published in the European scientific literature under the name of Fritz Müller, and was highly regarded by luminary biologists of his time, such as Ernst Haeckel, Alfred Wallace and Charles Darwin, with whom he corresponded extensively and enthusiastically after reading the ”On the Origin of Species” in 1861. In this talk we plan to present some mathematical approaches to Müllerian Mimicry starting from Müller’s original up to recent developments. 18 Mini-Symposium From movement to foraging: the latest modelling advances in animal ecology Organizer: Luca Giuggioli University of Bristol, UK Quantifying animal movement and their foraging behaviour is key to advance our understanding of a variety of application areas including species invasion, conservation biology, epidemic disease spread and optimal search strategies. The mini-symposium aims to bring the audience at the forefront of research concerning the latest advances in modelling the movement and foraging behaviour of animals. One talk will be about Gaussian statistics, the central limit theorem, and normal diffusion, as well as how and why the movement of animals may sometimes be better described by anomalous diffusion rather than by normal diffusion. The second talk will be on how different landscapes and resources distribution may influence the best strategies of foraging. And the third will describe how to model animal searching for food in confined domain. Gaussian statistics, normal diffusion and when these fail Gandhi Viswanathan Federal University of Rio Grande do Norte - UFRN, Brazil We will review Gaussian statistics, the central limit theorem, and normal diffusion, as well as how and why the movement of animals and dispersal processes etc. may sometimes be better described by anomalous diffusion rather than by normal diffusion. 20 Epidemic spread in an animal population with overlapping home ranges Luca Giuggioli University of Bristol, UK We develop a framework to calculate encounter times of two random walkers in one dimension when each individual is segregated in its own spatial compartment and shares with its neighbor only a fraction of the available space. Encounter times are used to study the spatial propagation of an infectious disease in a population of susceptible and infected territorial individuals with overlapping home ranges, and which may transmit an epidemic when they meet. We determine analytically the macroscopic propagation speed of the epidemic as a function of the microscopic characteristics: the confining geometry, the animal diffusion constant, and the infection transmission probability. 21 How different landscapes and resources distribution may influence the best strategies of foraging Marcos Gomes Eleuterio da Luz Department of Physics Federal University of Paraná UFPR, Brazil Quantifying animal movement and their foraging behaviour is key to advance our understanding of a variety of application areas including species invasion, conservation biology, epidemic disease spread and optimal search strategies. But when characterizing such dynamics, a fundamental aspect to be considered is how a certain response to the features of the environment will influence attempts to improve the process, like minimizing costs in finding a prey or a mate. In this talk, we discuss the different aspects associated to the environment, like the diversity or source targets, their distribution, the forager power of detection, etc, which may determine the best strategies during foraging. 22 Mini-Symposium Evolutionary Dynamics Organizers: Wilson Castro Ferreira Jr.a and Fábio Chalubb a b UNICAMP, Brazil Universidade Nova de Lisboa, Portugal Evolutionary Dynamics is the study of the mathematical principles behind biological evolution. And, as was stated by one of the most important scientist of the XXth century, Theodosius Dobzhansky, ”nothing in biology can be understood except in light of evolution”. The correct understanding of the evolution was only possible after its merge with Mendelian genetics, in the beginning of the XXth century. Furthermore, the combination of these two fundamental theories was only made possible by a new generation of scientists with basic training in physics and mathematics in what is know as the ”modern synthesis of evolution”. This mini-symposium shall be understood in this framework, i.e., the boundary between mathematics, physics and biology. It will consist of three 40min talks with the common factor of using mathematics to provide a correct understanding of diverse biological phenomena. In José Fontanari’s talk information theory will be used to investigate the formation of chromosomes in a primordial soup of free genes; Fabio Chalub and Max Souza will present a joint work, where they show how detailed models of evolution can be simplified and how the simplified model provides a correct understanding of the detailed model. Continuous approximations of discrete evolutionary processes Fabio Chalub Universidade Nova de Lisboa, Portugal Joint work with Max O. Souza (UFF, Brazil) We consider simple discrete evolutionary processes (e.g., the Moran or the Wright-Fisher process) and obtain continuous approximation to the forward and to the backward evolutions. The continuous approximation (in its most general form) consists of a degenerated partial differential equation of drift-diffusion type. The forward equation must be supplemented by a set of conservation laws, while the backward evolution must be supplemented by incomplete information on the boundaries. In both cases, these extra conditions can be obtained directly from the discrete evolutionary process. We will also show how simple expressions for the fixation probability of a given type and the time to fixation of any type, in a population of two types, can be obtained from the continuous approximation and we will compare these expressions with numerical simulations for the discrete evolution. 24 Non-zero-sumness and the origin of complexity in prebiotic evolution José Fernando Fontanari Instituto de Fı́sica de São Carlos Universidade de São Paulo, Brazil The co-existence of distinct selfish genes or templates seems to be a prerequisite for the evolution of complex cellular life. In fact, this co-existence has been advanced as a solution to the information crisis of prebiotic evolution and has served as motivation for the proposal of two competing information integration models - the hypercycle and the gene-package models. However, a recent analysis [Silvestre & Fontanari, J. Theor. Biol. 252, 326-337 (2008)] has shown that both models suffer from the same deficiency, namely, in presence of mutants the total amount of information is constant, regardless of the number of co-existing templates. The main hindrance to achieving coexistence in the package model framework is the competition between template types within the packages (i.e., vesicles or primitive cells) which play a zero-sum game. Apparently, this problem can be solved by a single artifact - the template linkage in chromosomes, which would then play a non-zero sum game, since they are essentially in the same boat. Here we investigate the viability of this solution by studying the fate of a primordial chromosome in a population of free genes. Analysis of the probability of fixation of the chromosome in the population indicates that the chromosome lineage prospers only for unrealistically small vesicles, which would be doomed due to the lack of genetic redundancy well before the emergence of the ancestral chromosome. 25 Multiscaling Modelling in Evolutionary Dynamics Max O. Souza Universidade Federal Fluminense, Brazil Joint work with Fabio Chalub (Universidade Nova de Lisboa, Portugal) We start from a family of continuous approximations to the probability density of a frequency dependent Moran process studied by Chalub & Souza in [1]. These approximation, depending on the scalings, can be of diffusive or nondiffusive type, the latter being equivalent to the Replicator Dynamics. We then study the small diffusion limit, and show how the Replicator Dynamics can be consistenly fitted in a diffusive approximation. Some additional results concerning the fixation probabilites in this limit are also presented. [1] Fabio A. C. C. Chalub & Max O. Souza, From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics, Theoretical Population Biology, 76 (4) 268-277, 2009. 26 Mini-Symposium Dynamics of Infectious Desease Organizer: Jorge X. Velasco-Hernández IMP, México A mathematical model for Toxoplasmosis dynamics within-between-host Brenda Tapia Santos Facultad de Matemáticas, Universidad Veracruzana, México In any infectious disease there are two key processes in the host-parasite interaction. One is the epidemic process associated with disease transmission, and the other is the immunological process at the level of the individual host. When we consider coupling within-between-host dynamics we have some questions: How does the within-host dynamics influence the transmission of a pathogen from individual to individual? Will the model predictions in terms of the virulence and basic reproduction number of the pathogen be altered if the two process are dynamically linked? In this work we propose a framework that explicitly links the epidemiological and immunological dynamics through an environmental compartment. We use as a model system the infection by Toxoplasma gondii. 28 Spatial patterns in the spread of Dengue in Mexico Jorge X. Velasco-Hernández IMP, México In this talk we describe spatial patterns present in the spread of Dengue in for Mexican states. We describes the detected relationships between different municipalities using data mining techniques. We also present a mathematical model for the dynamics of Dengue and some partial results related to the comparison between its predictions and field data. 29 Stochastic amplification in vector-borne epidemics Marcos Aurelio Capistrán Ocampo CIMAT, México Let us consider a SIR-SI model formulated as a continuous-time Markov jump process. Through standard results, an approximate Langevin equation can be derived. Further, the Wiener-Khinchin theorem, allows to derive an analytical form for the Power Spectral Density (PSD) of the fluctuations of the state variables. We offer an analysis of the PSD in the regime of parameters corresponding to dengue and malaria. 30 Contributed Talks . A mathematical model for the interbreeding with Neanderthals Armando G. M. Neves and Maurizio Serva UFMG, Brazil Until 2010 most researchers believed that, although both groups had coexisted at the same places, humans and Neanderthals had not interbred. This result was supported mainly by evidences coming from mitochondrial DNA. In 2010, Green et al. experimentally proved by direct sequencing of nuclear DNA from Neanderthal fossils that some interbreeding between humans and Neanderthals occurred. As a consequence, it is estimated that living non-Africans have 1 to 4% their nuclear DNA being of Neanderthal origin. I will describe solvable models for the interbreeding process leading to the above situation. 33 Basins of attraction and global analysis of a three-dimensional population dynamics system Artur César Fassoni and Lucy Tiemi Takahashi UNICAMP In this work, we propose a model that describes the interaction between two populations of plants, in a context of competition and allelopathy. The natural habitat of one is invaded by the other, which competes with the native by the natural resources and produces a phytotoxin that inhibits their growth and spread. This phenomenon is known as allelopathy [1]. Although it is a phenomenon common to most plants and studied long ago by agronomists and biologists [2], there are few results with mathematical treatment about it [3]. The proposed model consists of an autonomous system of three ODE’s and eight parameters. Via theoretical tools which characterize the basin of attraction of the attractors equilibria, very used in the analysis of power systems [4], we made a full qualitative analysis of the system, characterizing the omega-limit set of all solutions and obtaining the description of each equilibrium stability throughout all the parameter space. We also studied the influence of each parameter on the size of basins of attraction, which allows control strategies aimed at the survival or extinction of certain specie. [1] H.P. Bais et al, Allelopathy and exotic plant invasion: from molecules and genes to species interactions, Science, 301:1377-1380,2003. [2] Inderjit, Allelopathy and plant invasions: traditional, congeneric, and biogeographical approaches, Biological Invasions, 10:875-890,2008. [3] D.R. Souza et al, A multiscale model for plant invasion through allelopathic suppression, Biological Invasions, 12:1543-1555,2010. [4] H.D. Chiang et al, Stability region of nonlinear autonomous dynamical systems, IEEE Trans. on Automatic Control, 33(1):16-27,1988. 34 Experiments and theory on dispersive models of population continuous ageing Bernardo A. Mello, Lucielli S. Leolato and Regis S. S. dos Santos Universidade de Brası́lia Age structure is a relevant property of biological populations. They are particularly important when describing insects, since these animals go throw several phenological phases during their development. Among the aspects affected by the phenology are reproduction, mobility, mortality, resistance to adverse situations, and susceptibility to plague control methods. However, the description of these populations must consider the biological, not the chronological, age. We present three models for the dynamics of the biological age of populations. One is the Fokker-Plank equation and the other two were specially designed to include dispersion in the development rate while forbidding rejuvenation (B. Mello, Phys. Rev. E, 82, 21918, 2010). We describe the dynamic properties of the models and compare the population distribution generated by each of then. We show that the distributions of the three models are Gaussian if we wait long enough. The experimental part of that work was performed at Embrapa Wine and Grape, in the city of Vacaria-RS. We studied one of the most important plagues of apple orchards, the Grapholita molesta, as know as oriental moth. The individuals were isolated and reared at the controlled temperatures of 15 oC, 20 oC, 25 oC, and 30 oC. The instants when each of them went through the phenological events were recorded. We used the results of the experiments to determine the models’ parameters of maximum likelihood. We also found the maximum likelihood of the Gaussian distribution and compared it with the likelihood of the biological age models. 35 Modeling pathogen adaptation to perfect and imperfect treatments Bourget Romaina,b, Loı̈c Chaumonta and Natalia Sapoukhinab a LAREMA Université d’Angers, UFR Sciences 2 Boulevard Lavoisier, 49045 Angers b INRA, UMR1345, Institut de Recherche en Horticulture et Semences, QUASAV, F-49071 Beaucouzé, France Humans use various treatments inducing resistance to hosts in agronomy (i.e., crop resistance, chemicals) or in medicine (i.e., vaccine, antibiotics) in order to limit or prevent pathogens development. However, pathogens are capable to adapt to the treatments making them ineffective. Thus, a current question in epidemiology is how sustainably manage extant treatments. We built two stochastic models of pathogen adaptive dynamics describing adaptation process to two distinct types of host resistance: total resistance which prevents pathogens development and partial resistance which only limits pathogens development. The goal of this study was to estimate the effect of the treatment parameters and pathogen/host characteristics on the treatment durability. Our models were based on birth and death stochastic processes which allowed us to model the dynamics of rare events such as mutations and migrations, but also the dynamic of small emergent populations. Using simulations, we found that the migration rate of a pathogen population needs to be taken into account in designing durable treatment strategies; since it can alter current criteria for the critical proportion of the host treated that could impede pathogen adaptation. Moreover, we identified the conditions under which multicomponent treatment inducing total host resistance can be durable. In the case of the partial resistance, we showed that the speed and shape of the pathogen adaptation curve was determined by the mutation probability law. Our results allow us to increase our understanding of interactions between deployed treatments strategies and the treatment durability. 36 Mathematics Equivalent Representation for Model-Based Individuals Carlos Manuel Viriato Neto, Érica Keith de Morais and Erivelton Geraldo Nepomuceno Federal University of São João del Rei The need to understand the dynamics of the spread of disease resulted in the emergence of a new area of science: mathematical epidemiology, and this is aiming the development of models that can help trace policies to control these diseases. With this study we obtained an accurate mathematical representation for Individuals Based Model (IBM) through stochastic equations incorporating aspects that allow to take into account the particularities of the IBM. This equivalence will contribute notably to advance the possibility of determining control for multi-agent systems. 37 Economical epidemic model for a controlled system Chakib Jerry Moulay Ismail University, Morocco In this contribution, we consider a SIR model as in Tridane ( Impulsive optimal control by vaccination for influenza with post-contact prophylaxis, Proceeding of Syst. Theo. Mod. Anal. Con., Fes 2009, pp. 579-586.) but here the system is controlled by the treatment rate. Considering the problem of determining optimal controls to minimize the total outbreak size over the course of the epidemic and using necessary condition of optimality. Our goal is to draw conclusions about the effect of the shortage of the drug treatment on the management of strategies of control policy. 38 Species traits and density dependence; unraveling patterns in butterfly population dynamics Claire Dooleya, Michael Bonsalla, Tom Oliverb a b University Oxford, UK Centre for Ecology and Hydrology Different forms of density dependence are observed in British butterfly populations. As particular forms of density dependence may cause extinction events it is important to identify and understand the mechanisms responsible for these observations. First, we investigate which species are more susceptible to certain forms of density dependence to explore the role of density dependence in terms of a species’ conservation status. Second, we investigate the effect of species traits on susceptibility to different forms of density dependence. Analysis was carried out on the UK Butterfly Monitoring Scheme data and species traits were tested whilst controlling for phylogenetic relatedness using the Markov chain Monte Carlo Sampler for Multivariate Generalised Linear Mixed Models in R. We hope that our results will advance knowledge of how British butterfly populations behave and will allow us to assess the occurrence of density dependence at a species level. 39 Landscape ecology: a mathematical overview Claudia Pio Ferreira IBB/Unesp, Brazil We present a mathematical model applied to agricultural pest control through landscape ecology. In particular, we are interested in studying the control of Diabrotica speciosa, a pest of many crops throughout Central America and South America, and vector of viral and bacterial diseases. Using the formalism of cellular automata (CA) a mosaic of landscapes will be constructed to acess the pest temporal evolution, and the influence of the mosaic spatial struct, over the size of the insect population. The results will be compare with those given by the mean field theory. 40 Consequences of an alternative food for predator in the Holling-Tanner predation model Claudio Arancibia-Ibarra and Eduardo González-Olivares Pontifı́cia Universidad Católica de Valparaı́so, Chile A model derived from the well-known Holling–Tanner (or May-Holling-Tanner predator-prey model is analyzed, in which the growth equation for predators is of logistic type and the functional response is Holling type II. We assume the environmental carrying capacity of predators is given by the function Ky = K(x) = nx + c, obtaining a modified Holling-Tanner, a particular Leslie-Gower model. The new parameter c > 0 represents the population size of an alternative food for predators, that is, when the amount is small, they can consume other resource; then, is a generalist predator. The model is described by the twodimensional autonomous nonlinear differential equations system of Kolmogorov type. We will identify key properties of system, setting the bifurcation diagram, trying to establish the quantity of limit cycles surrounding a positive equilibrium point. The results will be compared with the May-Holling-Tanner model. 41 Bitstring model to study the Dengue’s spread Crysttian Arantes Paixão, Iraziet da Cunha Charret and Renato Ribeiro de Lima FGV/RJ, Brazil Dengue is a disease transmitted by mosquitoes in tropical and subtropical regions of the world. It is considered one of the most important viruses for the human population. One of the ways to combat this virus is through the implementation of vector control. In this work, we propose a computational model that simulates the spread of the virus, including the life cycle of the vector, Aedes aegypti, of the human population and the serotypes of the virus of genus Flavivirus. The model is based on a modification of the bit-string technique. With this model, we attempted to capture the main features of the epidemiological cycle and the infection process. To each individual a strip of bits is assigned, which contains all the information that will be used during the simulation. In the winged stage the mosquitoes may visit other areas and reproduce, ovopositing in breeding sites scattered through the modeled area, restarting the vector’s life cycle. Regarding the process of infection, the mosquitoes can be infected by four types of viruses when they meet with an already infected human. With the contact, the mosquito becomes infected and begins to infect the human population, spreading the disease. The performance of the model also was evaluated by studying the simulation time and memory resources used. It is noteworthy that the main advantage of the computational model proposed lies in its ability to optimize the use of processing and memory resources around 80%, on average, when compared with individual-based models or cellular automata. 42 Modelling the evolution of individual trap counts over time: Numerical techniques for mean-field and individual based models Daniel Bearup and Sergei Petrovskii University of Leicester, UK Insect trapping is commonly used in ecological studies and pest monitoring programs to estimate relative population levels. However estimating absolute population sizes from such data remains challenging. Experimental practice typically requires that a number of traps are placed within a habitat. Trap counts are then collected at regular intervals over a period of time. Each trap induces a perturbation in the population distribution within the field which is reflected in the dynamics of the trap counts observed. Modelling of this evolving interaction between a trap and the population in its vicinity provides a way to directly link trap counts to population density. Insect movement is treated as stochastic motion either directly, through individual based modelling, or approximately through a mean-field model. Depending on the type of insect movement the mean-field model may be tractable to analytical approaches in one dimension. However in general these models can only be approximated numerically. The development and optimisation of explicit numerical algorithms to handle this problem is the primary focus of this talk. Individual based models are developed in parallel to these mean-field approaches. Such models more directly represent the underlying processes but are less tractable to analysis. Thus these models are used primarily as a means to validate the results obtained from mean-field approximations, but they also display interest dynamics of their own. 43 Modelling directed flow across fragmented habitats using electrical network theory Daniel Cowleya, O. Johnsona and M. Pocockb a b University of Bristol, UK Centre for Ecology and Hydrology and University of Bristol Range shift is a potentially important strategy for species to respond to anthropogenic climate change. However, species need to disperse across landscapes in which suitable habitat can be sparse and fragmented. How does this habitat loss and fragmentation affect the spread of species? Previous models of habitat connectivity consider undirected flow, but this is not appropriate when considering flow over the large temporal and spatial scales involved in range shift. Here we develop a model for directed flow across networks of habitat patches based on the analogy between random walks and electric networks, proposing metrics with useful ecological interpretations. We use ’current’ to assess the relative importance of each patch within the network and identify the most important patches in real landscapes. ’Effective resistance’ is given as an overall measure of network resistance for the comparison of different networks and landscapes. We also demonstrate that the model is robust across a range of ecologically realistic input parameters. Strikingly, the pattern of patch importance varies according to the direction of flow, demonstrating the importance of our modelling approach. These results can be used to inform conservation policy in order to facilitate species movement in the face of climate change or inhibit the spread of invasive species. 44 A modified Leslie-Gower type predation model with a sigmoid functional response and weak Allee effect on prey Sebastián Valenzuela-Figueroa and Eduardo Gonzales-Olivares Pontificia Universidad Católica de Valparaı́so, Chile In this work, a continuous-time predator-prey model is analyzed, in which: i) The Allee effect affect prey population, ii) the functional response is Holling type III or sigmoid, and iii) the predator growth function is of logistic type (Turchin, 2003). This latter assumption characterizes the Leslie-Gower type models, where the environmental carrying capacity of predators Ky is proportional to the prey population size; we assume that Ky = K(x) = nx + c [2], obtaining a modified Leslie-Gower model (González-Olivares et al., 2011). The new parameter c > 0 represents an alternative food for predators, that is, when the amount is small dams, they can consume other resource; then, we have a generalist predator (Turchin, 2003). The Allee effect is described by the simpler known form (González-Olivares and Rojas-Palma, 2011), obtaining an autonomous nonlinear differential equations system of Kolmogorov type. The analysis must be made separately for the strong Allee effect and weak Allee effect, due the number of limit cycles can change with respect to this parameter (González-Olivares and Rojas-Palma, 2011). The results will be compared with the model studied in Tintinago-Ruiz and E. González-Olivares (2012), in which the Allee effect is absent and with those in which other mathematical forms are used to describe the Allee effect. E. González-Olivares, J. Mena-Lorca A. Rojas-Palma and J. D. Flores, Dynamical complexities in the Leslie.Gower predator.prey model as consequences of the Allee eect on prey, Applied Mathematical Modelling 35 (2011) 366.381. E. González-Olivares and A. Rojas-Palma, Multiple Limit Cycles in a Gause Type Predator.Prey Model with Holling Type III Functional Response and Allee Eect on Prey, , Bulletin of Mathematical Biology 73 (2011) 1378- 1397. P. Tintinago-Ruiz and E. González-Olivares, Dynamics of a Leslie-Gower type predator-prey model with sigmoid functional response, (2012) in preparation. P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Mongraphs in Population Biology 35 Princeton University Press 2003. 45 The influence of trait characteristic as weak force dictating qualitative changes in population dynamics and direct consumption as a strong force determining quantitative changes Emanuelle Arantes Paixãoa, Lucas Del Bianco Fariaa and Michel Skin da Silveira Costab a b Universidade Federal de Lavras, Brazil Laboratório Nacional de Computação Cientı́fica - LNCC Most models that describe food chains and food webs are composed by interactions mediated only by the density. However, many empirical studies report the existence of trait characteristics and its significant influence on food webs and natural communities. Thus, the main objective of this study was to analyze the influence of the trait characteristic in two models of tri-trophic chains, with and without allochthonous input. For these analysis, computer simulations and numerical analysis were carried out. In all the simulations, was considered a set of parameters, which results in populational chaotical dynamics. The term trait characteristic used in this study refers to the foraging behavior of consumers and therefore is related to the tradeoff between its foraging time and predation risk by a top predador. The results observed lead the conclusion that there is a balance between weak and strong interaction forces in nature. Here, the trait characteristic may be considered as the weak interaction and the strong interaction is the force of the direct consumption. Thus, the incorporation of the trait characteristic promoted stabilization to the chaotical systems, suggesting its qualitative effects on the population dynamics. On the other hand, the forces of direct consumption might have a strong influence on quantitative variation of population densities. As all biological systems, the food chains and food webs may be characterized as complex system. Thus, the trait characteristic may be intrinsic mechanisms of the system which promotes food web stability and species coexistence. 46 Pattern-Non-Pattern Transition for a Nonlocal Population Dynamics Jefferson A. R. da Cunhaa, Andre L. A. Pennab and Fernando Albuquerque de Oliveirab a b University of Goiás, Brazil University of Brası́lia, Brazil In this article, we study pattern formation for one-species population in nonlocal domains. The nonlocal growth and competition terms are defined from the parameters alpha and beta ranging in length L. In this space (α; β) we have analyzed a coexistence curve α∗ = α(β ∗ ) which delimits domains for the existence (or not) of pattern formation in population dynamics systems. Patternno-pattern transition emerges from this model when nonlocal interaction among the individuals are present. 47 Sustainable fisheries management through close seasons with variable lengths Fernando Córdova-Lepea, Rodrigo Del Vallea and Gonzalo Robledob a Universidad Católica del Maule, Chile b Universidad de Chile, Chile A model of regulation by fishing closures of varying duration will be presented. The closures are limiting the productive effort and they are alternated with comparatively short periods of openings. In this framework, it is defined a regulation that determines the length of the next period of closure as a function of the stock not captured in the current access. It is shown that under certain threshold of fishing effort, there is convergence to steady long closures that ensure the ecological sustainability of the resource. 48 On the evolutionary ecology of cyclically asexual plant parasites Castel M, Mailleret L, Ravigné V, and Frédéric Hamelin Agrocampus Rennes, France In many plant parasites, sexually produced forms are often the only way to overwinter. From an ecological perspective, the need to find a mate prior to overwintering may generate an Allee effect and lead the population to extinction. Nevertheless, many plant parasites also reproduce asexually within the season, which tends to compensate for the sexual Allee effect. From an evolutionary standpoint, investing into asexual reproduction is likely to have a detrimental effect on the sexual forms’ survivorship. This type of trade-off is actually commonly considered in studies about parasites that infect either directly or via free-living forms. Yet, the case of sexual free-living forms remains overlooked so far. Thus, it remains unclear whether investing into asexual transmission should be selected for in the long run. This raises a series of issues: can a mixed sexual-asexual investment be selected for? Can evolution drive the species to extinction? Can asexuality be lost? To investigate these issues, we built a simple ”semi-discrete” epidemiological model which includes recurrent episodes of sexual reproduction prior to winter interruption. Using the Adaptive Dynamics framework, we showed that various evolutionary outcomes are possible, ranging from a simple monomorphic evolutionary endpoint to evolutionary branching and the extinction of one morph. Sexual reproduction can promote disruptive selection, which is impossible in the analogous asexual model. We discuss how these results can help interpreting recent population genetics studies on e.g. the phytopathogenic fungus Leptosphaeria maculans, whose European populations are quasi-purely sexual, whereas Canadian populations invest more into asexual reproduction. 49 The paradoxical affinity between modularity and dependence asymmetry Gilberto Corso UFRN, Brazil We discuss the relationship between two patterns from interaction networks of ecology of communities: modularity and asymmetric specialization. Indeed, asymmetric specialization and modularity express two opposite features: the first suggests an interplay of generalists and specialists forming an entangled web of interconnected species while the second brings the idea of groups of species interacting in isolated cliques. We perform the analysis using the Dependence Asymmetry DA which is the simplest way to quantify asymmetric specialization. We construct an algorithm that find the pattern of maximal DA and we perform an analytic estimations for the upper bound of DA. Finally, we study the symmetric modular structure that has null DA; moreover we forced an asymmetric mismatch in this pattern which generates a high DA as we compare with a random pattern and with the maximal possible value. We conclude that, despite the opposite notions suggested by the studied patterns, if a modular pattern has enough asymmetry it resembles a specialized asymmetric pattern. 50 An individual energy balance model for Greater Rhea (Rhea americana) and its implication on recruitment Marı́a Verónica Simoya, Graciela Ana Canziania and Gustavo J. Fernándezb a Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina b Universidad de Buenos Aires, and CONICET, Argentina In Grater Rhea populations (Rhea americana) inhabiting the Humid Pampas the number of males breeding each year is low. Given that the male takes care of incubation, this low number could be due to the high cost of reproduction for males. We propose an individual-based model for estimating energy budgets of rhea to analyze the association between the population recruitment and the individual characteristics of adult males. The individual model was based on a system of equations calculating the weight of an individual as a function of its ingestion rate and the energetic cost of its activity pattern with daily step. The ingestion rate was calculated from field experiments. Daily energetic cost was estimated from observed activity patterns at the field, taking into account gender and factors that influence behavior (e.g., photoperiod, season). Concatenating the daily model output, the weight dynamics over any period can be obtained. We use simulations for analyzing issues related to reproduction such as the minimum weight that the male needs to attain to be able to face the cost associated to breeding and incubation, and the proportion of males that could reach it. 51 Partial Synchronization in a heterogeneous metapopulation model Jacques A. L. Silva Federal University of Rio Grande do Sul, Brazil Metapolation models are known to display a wide variety of emergent collective behavior, in special, synchronization phenomena. A metapopulation model of n equal patches can exhibit total synchronization. The local stability of the synchronized attractor is described by its transversal Lyapunov number expressed as a product of the Lyapunov number of the map describing the single patch dynamics and a constant depending on the dispersal fraction and on the spectrum of the interaction matrix. Moreover the dynamics on the synchronized attractor is precisely the local dynamics on each patch. Here we present a metapopulation model introducing some heterogeneity, maybe due to differences in patch quality or size. This heterogeneity induces differences in survival and reproduction in each patch thereby leading to different local maps. We consider a model with n patches, k of them with local dynamics given by a map f while the remaining n − k have local dynamics described by a map g. Under appropriate conditions on the interaction matrix C, the network can display partial synchronization phenomena where a clusters of size k and another cluster of size n − k are formed and interact as if they were two patches. The dynamics of the two clusters does not resemble any of the two isolate patch models. An expression for the transversal Lyapunov number is obtained based on a decomposition of the phase space as a direct sum of the invariant manifold of dimension 2 where the partial synchronized attractor is confined and its orthogonal complement of dimension n − 2. 52 The Rosenzweig-MacArthur predation model considering double Allee effects on prey Jaime Huincahue-Arcos and Eduardo González-Olivares Pontifı́cia Universidad Católica de Valparaı́so, Chile This work deals with a model derived from the well-known RosenzweigMacArthur predator-prey model, a particular case of Gause type model, in which a double Allee effect affecting the prey is considered. To describe this ecological phenomenon we use the mathematical form proposed in The obtained results show significant differences with the Rosenzweig-MacArthur model in which the Allee effect is absent. Also, the quantity of limit cycles differs with the numbers obtained in other model studied in which the Allee effect and described by a simpler form, which is topologically equivalent to that used in this work. 53 Mathematical modelling and numerical simulation of the temporal-spatial population dynamics in the presence of environmental impact: a case study João Frederico C. A. Meyer and Paulo C. Carmona Tabares University of Campinas, Brazil Argentine and German antarctic researchers have been observing the behavior of interacting species in Potter Bay, along the King George’s Island coast. The need for numerical simulation of scenarios arose form the need to test different hypotheses as to what was causing a difference in the expected behavior of said species. The authors, following previous work by members of the Mathematical Ecology research sub-group of the Campinas Biomathematics team, proposed a nonlinear system of Partial Differential Equations in which, besides the interacting species, the presence of the sediment was considered, as well as its effects on the interacting populations. The resulting system includes the equations of Diffusion-advection-reaction for the sediment, as well as Lotka-Volterra-type Diffusive equations for the two interacting species in which the negative influence of the sediment is included. A second-order finite difference scheme is presented, with which numerical results, albeit with estimated parameters, computational results were obtained in agreement with observed characteristics. The qualitative output showed, quite surprisingly (for the authors, at least), that the randomness in the spatial occupation of the bay was seriously affected creating a different occupational setup, a result which has been observed in the study area, by local research groups. In the presentation, the system is presented, justified and discussed, as well as the numerical approximation tools and the results presented in graphical form, in a MATLAB software environment. 54 Continuum Three-Zone Model for Swarms João Plı́nio Juchem Netoa, J.M. Millerb, A. Kolpasb and L. Rossib a Federal University of Rio Grande do Sul, Brazil b University of Delaware, USA We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popular biological models of aggregations. In continuum models, individuals are replaced with density and velocity functions. Individual behavior is modeled with convolutions acting within three interaction zones corresponding to repulsion, orientation, and attraction, respectively. We begin with a variable-speed first-order model in which the velocity depends directly on the interactions. Next, we present a variable speed second-order model. Finally, we present a constant-speed second-order model that is coordinated with popular individual-based models. For all three models, linear stability analysis shows that the growth or decay of perturbations in an infinite, uniform swarm depends on the strength of attraction relative to repulsion and orientation. We verify that the continuum models predict the behavior of a swarm of individuals by comparing the linear stability results with an individual-based model that uses the same social interaction kernels. In some unstable regimes, we observe that the uniform state will evolve toward a radially symmetric attractor with a variable density. In other unstable regimes, we observe an incoherent swarming state. 55 Dynamics of a predator-prey model with Allee effect on prey and ratiodependent functional response José D. Floresa and Eduardo González-Olivaresb a b University of South Dakota, USA Pontifı́cia Universidad Católica de Valparaı́so, Chile We proposed a ratio-dependent predator-prey model with Allee effect on the prey. We present a parametric analysis of the stability properties of the dynamics of the system in which the functional response is a function of the ratio of prey and predator. An important mathematical feature of these type of models is that while the functional response is undefined at the origin, the origin is singular equilibrium. We present the different types of system behaviors for various parameter values, showing the existence of separatrix curves in the phase plane determining that the long-term system’s dynamic is dependent on the initial conditions. The model is study analytically as well as numerically, including stability and bifurcation analysis. We also discuss the biological relevance of the method regarding both coexistence (conservation) and extinction (biological control) issues. 56 Wavelet Clustering a tool for analyzing of spatial and temporal patterns of epidemics based on their dynamical properties: Application to dengue in Thailand Kévin Cazellesa and Bernard Cazellesb a Université Montpellier 2, France b France Over the past few decades, our world has experienced the emergence, or the re-emergence, of several infectious diseases in connection with our fast-changing environment. The understanding of the spatial and temporal patterns of the transmission of these diseases is of fundamental importance for predicting the patterns of emerging epidemics in our changing world. Adapted tools that account the non-stationary nature of the phenomena underlying to disease transmission are needed to analysis these observed patterns. We propose to use cluster analysis that quantifies the dissimilarities between wavelet spectra and classifies these results using hierarchical clustering. Wavelet spectra are compared one with each other using a multivariate method defining an orthonormal basis that maximizes the mutual covariance for each pair of spectra. Dissimilarity value is obtained from comparing the decomposition of the both spectra onto this basis. The hierarchical clustering is then applied on the obtained set of dissimilarities and groups the spectrum according to their time-frequency similarities. An example of these emerging diseases is the resurgence of dengue. We have previously showed that the dengue epidemics can be influenced by large climatic oscillations but the associations between dengue cases and climatic factors appear transient underlying the importance of the nonstationary aspect of these epidemics. Here we will extend our previous results by incorporating the spatial dimension of the disease propagation using wavelet clustering. These preliminary analyses will facilitate the development of models that describe the interactions of different factors on the transmission of the dengue virus. 57 Evaluating the risk of hemorrhagic dengue disease Lourdes Estevaa and Hyung Mo Yangb a Universidad Nacional Autónoma de México, México b UNICAMP, Brazil In countries where dengue disease is endemic, its severe manifestations, dengue hemorrhagic fever (DHF) and its associated dengue shock syndrome (DSS), have been increasing in alarming proportions during the past years. The causes determining the occurrence of DHF are not yet fully understood, and two hipotheses have been proposed, the first one assumes that is the quantity of virus inoculated by mosquito (virulence) the factor that determines progression to DHF; the second one says that those persons experiencing a second infection with an heterologous dengue serotype are in major risk to have DHF/DSS (secondary infection or immune enhancement hypothesis). In this work we formulate a mathematical model to evaluate the risk of dengue hemorrhagic disease assuming that this manifestation of the disease is related to the titers of the mosquito-infecting virus dose. We assess the effect of temperature variation on the entomological and epidemiological parameters of the transmitter vector, Aedes aegypty, and the consequences in the proportion of DHF cases. 58 Mathematical modeling of immune response in co-infection with Trypanosoma cruzi and HIV Luiz Fernando de Souza Freitas and Hyung Mo Yang UNICAMP, Brazil The human body has a complex system of defense: the immune system. Such a system has different answers for different attacks to the body. The co-infection with parasites such as Trypanosoma cruzi and HIV virus triggers a important defense mechanism: humoral immunity. Due to the chronic phase of Chagas disease, in most cases asymptomatic, that is reactivated when the main body’s defense cells, T CD4 is not active, succumb by the action of the HIV virus. In order to study the dynamics of co-infection by the disease, Chagas disease and acquired immunodeficiency syndrome, the response of the human immune system, a mathematical model of the system autonomous nonlinear ordinary differential equations is prepared. This model presents a simplified form of the dynamics among immune system, protozoan T. cruzi HIV virus and target cells. 59 On dynamical behavior of the sugarcane borer - parasitoid agro-ecosystem Marat Rafikov and Jean Carlos Silveira Federal University of ABC, Brazil In this paper, we propose a mathematical model of interactions between the sugarcane borer (Diatraea saccharalis) and its larvae parasitoid (Cotesia flavipes). The steady states of the system are determined. And the dynamical behavior of the larvae, parasitized larvae and parasitoid populations is examined. Linear feedback control strategy is proposed to indicate how the natural enemies should be introduced in the environment. Some numerical simulations for supporting the theoretical results are also included. 60 Analysis of the effect of fecundity and survival parameters on an adequate management of the endangered Blue-throated Macaw (Ara glaucogularis) populations Marı́a Laura Maestria , Igor Berkunskyb,c and Rosana Ferratib a Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina b Instituto Multidisciplinario sobre Ecosistemas y Desarrollo Sustentable, Universidad Nacional del Centro de la Provincia de Buenos Aires, and c CONICET The Blue-throated Macaw (Ara glaucogularis) is a rare, critically endangered and endemic macaw of savannahs in northern Bolivia. During the last decade intensive conservation work was carried out in order to identify and address the critical parameters delaying the population’s recovery. Field conservation actions were aimed at increasing nest site availability, protecting active nests against predators and increasing nestling survival. The application of these actions resulted in higher nest success and higher nestling survival. Despite this reproductive improvement, the breeding population is not yet recovering. The number of macaws at each breeding site has been stable or decreased and at a few sites, local extinction has taken place. We propose a stage-structured matrix population model for simulating its dynamics. Fecundity and survival parameters were determined from field data and permanence in a class parameters were computed assuming a stable stage distribution. Sensitivity and elasticity of each parameter were analyzed. The simulation of different management scenarios involved the quantiles of fecundity and survivorship parameters for the first class. Simulation results show a larger incidence of the fecundity parameter, in agreement with field observations indicating that the reproductive population has not yet recovered in spite of the current conservation efforts. As the Blue-throated Macaw is currently a conservation dependent species, we consider the value of increasing the wild population through the release of confiscated and captive-bred individuals. The model validates the possibility of increasing the wild populations in such a way, as well as by incubating eggs fallen from nests. 61 Analysis of sustainable harvest of guanaco population in Patagonia using a stage structured matrix model Mario Ignacio Simoy and Mauro Andrés Nardı́n Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina Guanaco (Lama guanicoe) is a herbivore belonging to the Camelidae spp. It has the widest spatial distribution among the South American camelids and inhabits the most diverse environments. Guanaco populations can be found from the Peruvian Andes to Tierra del Fuego and Navarino Island, from sea level plains to 4500 m high mountain valleys. Here we present a matrix population model for analyzing the dynamics of guanaco populations in the province of Neuquén, Argentina. It was parameterized using vital rates data from publications. The model is structured in four stages: Chulengos, Juveniles, Young Adults and Adults. Sensitivity and elasticity analyses allow understanding the effect of vital parameters on the population dynamics. The population growth rate indicates an increasing population. Thus, harvest was included in order to analyze scenarios of sustainable exploitation. Four alternative harvest scenarios, in which only one stage is being harvested, were evaluated. Conditions for sustainability were established. 62 Small initial contagion approximation (SIC) in stochastic epidemic networks Mark Kelbert, Igor Sazonov and Michael B. Gravenor Swansea University, UK A useful tool in epidemic network analysis is the approximation of small initial contagion (SIC) in which it is assumed that a small share of initially infected triggers an outbreak in each network node (epidemic centre). This approximation is very natural when individuals are mainly concentrated in urban centers slightly interacting between each other due to migration of infectives. However, when the number of initially infected is small, the discreteness of population can essentially affect the dynamics of the outbreak making it stochastic. In the initial stage the model of interaction between individuals and their contamination is a random process, and the deterministic approach works well only if the number of infectives is large all the time. In the proposed method, we account for the discreteness of population when the number of infectives is small and derive equations for evolution of the probability distribution. At the time of developed outbreak, we solve deterministic equations with the random initial conditions with the distribution computed in framework of the randomized model. To confirm the resulted a large number of computer simulations have been conducted. The results of direct simulation are in a good agreement with the approximate models. Sazonov, Kelbert, Gravenor 2011: Travelling waves in a network of SIR epidemic nodes with an approximation of weak coupling, Mathematical Medicine and Biology, 28 Sazonov, Kelbert, Gravenor 2011: A two-stage model for the SIR outbreak: Accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage, Mathematical Biosciences, 234 63 Fractional calculus approach to dynamical systems with memory Matheus Jatkoske Lazo FURG, Brazil The calculus with fractional derivatives and integrations of non-integers orders started more than three centuries ago with l’Hôpital and Leibniz when the derivative of order was suggested. This subject was also considered by several mathematicians as Euler, Fourier, Liouville, Grunwald, Letnikov, Riemann and others up to nowadays. Although the fractional calculus is almost as old as the usual integer order calculus, only in the last three decades it has gained more attention due to its applications in various fields of science, engineering, economics, biomechanics, etc (see [1-4] for a review). Fractional derivatives are in general nonlocal operators and are historically applied in the study of nonlocal or time dependent processes. In this context, fractional calculus can be useful to investigate memory effects in dynamics of populations. The main purpose of this work is to review some recent results and propose some new potential applications of Fractional Calculus on dynamical systems with memory. [1] Sabatier, J., Agrawal, O. P. and Tenreiro Machado, J. A. (eds), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Netherlands, 2007. [2] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [3] Hilfer, R. (ed), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [4] Magin, R. L., Fractional Calculus in Bioengineering, Begell House Publisher, 2006 64 When can we trust our model predictions? Unearthing structural sensitivity in biological systems Matthew Adamson and A. Yu. Morozov University of Leicester, UK It is well recognized that models in the life sciences can be sensitive to small variations in their model functions, a phenomenon known as ’structural sensitivity’. Conventionally, modellers test for sensitivity by varying parameters for a specific formulation of the model functions, but models can show structural sensitivity to the choice of functional representations used: a particularly concerning problem when system processes are too complex, or insufficiently understood, to theoretically justify specific parameterisations. Here we propose a rigorous test to detect structural sensitivity in a system - with respect to the local stability of equilibria - the main idea being to project infinite dimensional function space onto a finite dimensional space by considering local properties of the model functions. As an illustrative example, we use our test to demonstrate structural sensitivity in the seminal Rosenzweig-MacArthur predator-prey model, and show that the conventional parameter-based approach can fail to do so. We also consider some implications structural sensitivity has for ecological modelling: we argue that trying to find an ’optimal’ parametrisation of a model function may simply be an ill posed problem when the model exhibits structural sensitivity, and we suggest that structural sensitivity in biological models may explain irregular oscillations often observed in nature. 65 Stable Periodic Orbits and Chaos in the Biotic Iron Cycle Miguel A. Dumetta and James P. Keenerb a UFPR, Brazil b University of Utah,USA We model the Biotic Iron Cycle for Pyrite (with bacteria Acidithiobacillus Ferrooxidans) by a system of nonlinear ODEs. This investigation was motivated by reports of periodic solution in pH and bacteria population in pyrite dump sites acidic environments . Chemical reactions, reaction rates, and bacteria population models are taken from the literature. For certain parameter values, the model contains up to four non-trivial steady states, two of which are stable. There are two subcritical Hopf bifurcations, associated to a parameter that measures the capacity of the bacteria to metabolize the limiting step reaction of the iron cycle. At one of this values, there is a sequence of unstable periodic orbits that converge to a homoclinic bifurcation with saddle quantity negative and hence by Shil’nikov theorem there are periodic stable solutions in a neighborhood of the homoclinic orbit. Numerical confirmation of these facts are found as well as the existence SNP bifurcation, doubling period bifurcation and stable chaos for a range of values of the bifurcation parameter. It can be shown that the rich structure of the system is a generic behavior. 66 Effect of the diffusion on the adaptive process in spatially structured populations Elder S. Claudino, Iram Gleria, M. L. Lyra and Paulo Roberto de Araujo Campos UFRPE, Brazil It is well established that adaptation occurs through the occurrence and the subsequent fixation of beneficial mutations in natural populations. Here we investigate the role of the diffusion process on the dynamics of fixation of advantageous mutations. Our model assumes a finite population which is spread over a continuous two-dimensional lattice. The individuals are subject to the natural selection and mutations occur at a constant rate. Competition is local, and its extent is tuned by the radius r. Moreover, the model assumes that the individuals can diffuse over space at a constant velocity v. The direction of the displacement at each time step is completely random. Though it is not observed any strong effect of the diffusion process on the fixation probability of beneficial mutations, the rate at which the mutations reach fixation is strongly influenced by the diffusion velocity v. Particularly, we find that these effects are more prominent in the regime of low density of particles (individuals), and when the radius r is not sufficiently large. We hope that this kind of modelling can shed new light into our understanding about adaptation in spatially structured populations such as those found in biofilms. 67 Epidemiological model of a network of closed patches with border contagion. An epidemic threshold result Fernando Córdova-Lepea, Rodrigo Del Vallea and Gonzalo Robledob a Universidad Católica del Maule, Chile b Universidad de Chile, Chile Conditions for the persistence of a SIS type disease in a meta-population of n- neighbors patches (spatially distributed in a directed graph) are given. We will assume that: a) The migration rates among patches are null. b) In each patch, the inner contagion rate and the recovery rate determine a basic reproductive number less than one. c) In the border of two connected patches, there exist contagions between susceptible and infectious individuals of different patches. The paper novelty is to find, in this framework, a threshold for an endemic disease level. 68 Backward Bifurcation in a model for bacteriocin production regulated by quorum sensing Roberta Regina Delboni and Hyun Mo Yang UNICAMP, Brazil Bacteriocins produced by lactic acid bacteria are defined as extracellularly primary or modified products of bacterial ribosomal synthesis, and that can inhibit or kill pathogenic bacteria such as Listeria monocytogenes, Staphylococcus aureus and Clostridium botulinum. The cell-density-dependent regulation of bacteriocin production is a phenomenon, called quorum sensing, that involves specific molecules that are directly sensed by membrane-located histidine kinases, after which the signal is transmitted to an intracellular response regulator that activates transcription of target genes. These molecules that act as signals, accumulate in the environment as the cell density increases and activate signal transduction cascades that result in production of the auto-inducer by the stimulated bacteria cell. Besides their function as antimicrobial peptide, the nisin (a specific bacteriocin) exhibit a peptide pheromone function that plays an essential role in quorum sensing control of it biosynthesis. The logistic growth curve is usually appropriate to describe the growth of lactic acid bacteria, since it takes into account self-inhibition caused by the production of lactic acid and the depletion of nutrientes. The bacteria have a maximum rate of growth, but when the amount of bacteriocin produced is low, the bacteria multiply more slowly because they are adapting to the environment. From an ecological point of view it may be suggested that as cells of lactic acid bacteria grow to high numbers, the need to produce defensive metabolites like bacteriocins diminishes. When the population senses that a certain cell concentration is reached, the production of bacteriocin may consequently decrease. With this assumptions, we develop a mathematical model of non-linear ordinary differential equations to understanding the bacteriocin production by lactic acid bacteria and the quorum sensing mechanism. By analyzing the model it is possible to verify conditions for the existence of multiple equilibria and backward bifurcation. 69 Predicting the occurrence of an epidemic using the SIS epidemiological model Robin N. Thompson University of Cambridge, UK One of the key questions in epidemiology is whether or not, given the arrival of a pathogen in a system, an epidemic is going to occur. We consider a stochastic version of the SIS model, showing analytically that traditional measures of the probability of an epidemic rely on various assumptions - including a ”large” population and a basic reproduction number much larger than unity. We will then consider the exact definition of an epidemic. In particular, we define an epidemic to be hitting a threshold number of infected individuals, and consider the probability of an epidemic occurring given initial disease incidence data. Our results have clear implications in many areas of epidemiology. For example, important diseases such as meningitis may be modelled using the SIS model. Our method provides a framework to estimate whether the number of available beds in a hospital will be exhausted, from data at the start of the epidemic. Another example is that of Huanglongbing citrus disease. One control strategy that is used is roguing (the removal of infected plants, replanting with new susceptible plants). Since legislation in Brazil states that, if the number of infected individuals reaches twenty-eight percent of the grove, then the grove must be removed, our methods can be used to predict whether grove removal is going to be necessary. 70 A Leslie-Gower type predation model with sigmoid functional response and double Allee effect on prey Ruth Becerra-Klix and Eduardo González-Olivares Pontificia Universidad Católica de Valparaı́so, Chile This work deals with a continuous time predator-prey model of Leslie-Gower type (Gonzalez, 2011; Turchin, 2003) which main characteristic is the logistic type of growth predator equation (Turchin, 2003). In this type of models, the environmental carrying capacity of predators is proportional to the amount of available prey (Turchin, 2003). We assume the functional response is of sigmoid type and prey are affected by double Allee effect whose mathematical form is proposed in above papers. Although there are other mathematical ways to describe the multiple Allee effects,it can be shown that they are topologically equivalent to that used in this work The results here obtained will be compared with the Leslie-Gower, in which the Allee effect is absent. E. González-Olivares, J. Mena-Lorca A. Rojas-Palma and J. D. Flores. Dynamical complexities in the Leslie.Gower predator.prey model as consequences of the Allee effect on prey, Applied Mathematical Modelling 35 (2011) 366.381. P. Turchin, Complex population dynamics. A theoretical/empirical synthesis, Mongraphs in Population Biology 35 Princeton University Press, 2003. 71 A study of the effect of the Allee effect and landscape heterogeneity on an animal dispersed plant Salvador Lou Vega and Wilson Castro Ferreira Jr. UNICAMP, Brazil The Allee effect can alter the dynamics of an invasion, it can reduce the expansion speeds and be responsible for the lag phases observed in some invasions, and can influence the the dynamics of the colonization process. Petrovskii et al. (2002) suggest that Allee effect may be responsible for a patchy invasion pattern. Deterministic models for the spread of organisms, have predicted a smooth wave expansion, which may be reasonably for homogeneous environments or when considered at large distance scale where the environmental heterogeinity is averaged. Nonetheless, at smaller scales the expansion may be not a smooth front wave, but a patchy pattern expansion, which indeed occurs in nature. These pattern formation has been attributed to environmental stochasticity. Petrovskii et al. (2002) argue that patchy invasion can arise through deterministic models, as they show through a predator-pray model with an Allee effect. In this framework we will study the colonization process of an animal dispersed plant subject to Allee effect through an integrodifference model for the growth and dispersal of a plant in heterogeneous environments. Environmental heteogenity will be introduced in the dispersal process. We suspect that the seed dispersal kernel generated by its animal vector togehter with an Allee effect in plants may induce a patchy invasion. Bibliography. Petrovskii, S.V, A. Morozov, E, Venturino. 2002. Allee effect makes possible patchy invasion in a predator-prey system. Ecology Letters, 5: 345-352. 72 Evolution of population stability: theory meets experiment Sudipta Tung, Amitabh Joshi and Sutirth Dey Indian Institute of Science Education and Research, India Although simple growth models predict population stability due to reduced intrinsic growth rate (r) the underlying biology remains unclear. Here we describe an individual based model (IBM) for laboratory populations of Drosophila melanogaster that explicitly incorporates major life-history features that are likely to be relevant to the dynamics. We find good agreement between model predictions and observations from a laboratory experiment. We also demonstrate that the critical minimum size requirement for successful pupation and basal female fecundity of the flies are the important determinants of population stability. 73 Fitting models for memory-induced animal movement to empirical movement trajectories Ulrike Schlaegel and Mark Lewis University of Alberta, Canada In this talk we present mechanistic models for complex animal movement patterns and fit them to movement data. This is built on the framework of random walks, with interactions of movement decisions and dynamic information from memory. Movement decisions of animals are influenced by a complex interplay of internal goals (such as food acquisition or territorial defense), static information about the environment (such as landscape features) and dynamic information obtained through experience (such as the state of static food sources or locations of mobile food sources). At any given time, information can be derived both from perceptual cues and from memory. In our model information is stored in a map-like representation of the home range. This map effects movement and is itself dynamically updated based on the movement. To test our understanding of the behavioural processes underlying movement, we confront the models with movement data. We demonstrate how Markov Chain Monte Carlo techniques can be used to fit the models to empirical movement trajectories and to select between competing models. To account for measurement errors, the models are embedded into the hierarchical framework of state-space dynamics. 74 . Posters Is the Cattle Farming Intensification the better choice to Reduce Environmental Pressure on Brazilian Areas? Adriano Gomes Garcia and Magda da Silva Peixoto UNESP/Botucatu, Brazil Bovine cattle farming in Brazil are predominantly extensive, converting large natural areas into grazing. An alternative to reduce the occupied area is to intensify cattle breeding by increasing productivity, but it increases the use of water and animal supplements, mainly soybean. Intensification of pasture management has been subsidized by the Brazilian government as a means of reducing deforestation. We used a mathematical model in fuzzy language to check if an intensification policy of the Brazilian government could reduce cattle farming impact on these areas. Our results indicate that cattle farming intensification could strongly decrease grazing areas, which theoretically reduces deforestation. However the intensification does not solve the deforestation problem in the Amazon and the Brazilian Pantanal. Expanding pastures revolves around the estate speculation, not linked to cattle breeding in the Amazon. In the Pantanal, the extensive cattle farming have caused less environmental impact than intensive, since it is based on small farmers who do not exceed ecosystem capacity. Furthermore, simulations indicate that a cattle farming intensification puts a strong pressure on hydric sources. 77 Boundary Induced Phase Transitions in the bacterial colonies model Anderson A. Ferreira UFPel, Brazil We present a one dimensional stochastic automata model inspired by reaction diffusion fisher equation in one dimensional. Through simulation of Monte Carlo and Mean Field Analysis we determine the critical exponents and the critical parameters that define the extinction point of the population. A Free boundary problem for Ecology model Oscar Alexander Ramirez Cespedes and Deccy Yaneth Trejos Angel UFG, Brazil This poster will present a mathematical model that includes free boundary. This model describes the population dynamics between two groups of animals of the same species defined in a finite region. Numerical simulations illustrate different dynamics model based on the free boundary and shows the importance of the spatial distribution of the two groups of individuals. 78 Müllerian mimicry: his inventor and historical origins Divane Marcon and Wilson Castro Ferreira Jr. UNICAMP, Brazil In the year of 2012 we (should) commemorate the 190th anniversary of Johannes Friedrich Theodor Müller, a German born Naturalist, physician and teacher, who immigrated to Brazil in his thirties, where he was known and naturalized as Frederico Müller, and lived most of his life in Desterro (now Florianópolis), Santa Catarina, until his death in 1897. He arrived in Brazil in 1852 as part of a German immigrant group and helped decisively in the establishment of a community, which later became the city of Blumenau, but his scientific mind was immediately caught by the tropical exuberance of his new country. Müller made major contributions to Natural Science, having published extensively in the European scientific literature and kept in touch with developments of his time through a large correspondence with some of the main European scientists like Ernst Haeckel and Charles Darwin. His observations on crustaceans helped in affirming the evolutionary theory of Charles Darwin, with whom he exchanged letters for 17 years. Darwin used to refer to Fritz Müller, the name he used to sign in his papers, as the ”prince of observers.” Among his many contributions to population dynamics, the most remarkable was the field observation of a mimicry phenomenon between different species of non-palatable butterflies, now widely known as Müllerian Mimicry. What is even more remarkable was the fact that, besides perceiving this subtle phenomenon in Nature, he also proposed a theory based on a Mathematical Model. His observations and theory on this subject was published in 1879 in the German journal Kosmos and received wide coverage in the scientific literature elsewhere, as for example reviews in American Naturalist and the Proceedings of British Society of Zoology as well as by an detailed commentary in the later editions of Darwin’s ”Origins”. Although, he was named a naturalist of the Museo de História Natural do Rio de Janeiro by the Emperor D. Pedro II for many years and was recently bestowed posthumous honors of Doctor Honoris Causa, by the University of Santa Catarina, (UFSC), his scientific legacy is unfortunately scarcely known in the biomathematics milieu today. With this poster we intend to present some biographical notes about the singular life of this remarkable man and also outline the importance of his original model as a precursor of the modern approach to evolutionary population dynamics. 79 The Influence of Control by Segregation on the Dynamics of Equine Infectious Anemia Evandro Estevão Marquesonea and Norberto A. Maidanab a Universidade Tecnológica Federal do Paraná, Brazil b Universidade Federal do ABC, Brazil The equine infectious anemia (EIA) is a disease determined by a virus, exclusive of equine, that causes great economic losses. It is a disease that has no treatment and its main form of control is the elimination of infected animals. The virus is transmitted mechanically by bloodsucking insects, especially species of Tabanus tanamus (horse ies and deer y) and Stomoxys (Stable Fly). He survives only for short periods in the mouthparts of the ies. [2],[4]. In this work we propose a model for study the EIA dynamic. We consider the population of horses and insects. The horse population is divided into three subpopulations: susceptible horses, infected horses and asymptomatic horses. In the population of insects are considered two subpopulations: non-disease-carrying and diseasecarrying. The transmission is modeled by the law of mass action. According to a federal law, each state should execute the control of the disease and sacrifice infected animals. Although, in regions such as the Pantanal, where the disease is widespread, is allowed segregation of animals in paddocks, obeying some rules. In this work we examine the effects of control by sacrifice or segregation in the dynamics of the disease. The simulations were performed in subpopulations of infected and asymptomatic horses. The results show that the daily control done in the subpopulation of infected, is effective to prevent the disease. References [1] J.D. Murray, ”Mathematical Biology”, Springer, Berlin (2002). [2] R. A. M. S. Silva, U. G. P. Abreu, A.T. M. Barros, ”Anemia infecciosa equina: epizootiologia, prevenção e controle no pantanal”,Embrapa Pantanal. Circular Técnica, 29 (2001) 7-16. [3] R.O. Vargas, ”Anemia Infecciosa Equina”, Monografia de Especialização, Universidade Castelo Branco, Campo Grande-MS (2008). 80 Mathematical Methods for Bacterial Population Dynamics Francisco Quevedo Camargo USP, Brazil Our main goal in this project is to study phenomena that emerge in bacterial populations by using differential equations and comparing theoretical previsions with experimental data. At first, we approached the effects of different growth laws on population size structure. We have also conducted some theoretical work on the problem of critical patch area for populations moving through chemotaxis instead of usual diffusion. 81 Model for vaccination against polio Irene Duarte Gandica,Lina Marcela Ocampo, Maria Mercedes González and Edwin Fernando Duque Marı́n Universidad del Quindı́o, Colombia Poliomyelitis is an acute infection caused by the polio virus, which affects the human central nervous system. It is transmitted through fecal-oral and respiratory contact. There are two types of vaccine: live attenuated virus/oral polio vaccine (OPV) and inactivated polio vaccine (IPV). Nowadays there is a vaccination scheme in 5 years old children with OPV. Vaccinated children spread the virus in the environment (derived from the vaccine) and some people who are in contact with the virus become vaccinated by herding behavior. This work presents a mathematical model describing the dynamics of this infection in a population where both types of vaccination are carried out. The population is divided into two age groups and it is used Michaelis-Menten interaction. Different vaccination strategies are simulated and analyzed. 82 A seed dispersion and growth vegetation model Jaqueline Maria da Silvaa, Cezar Weltera and Mauricio V. Kritzb a Universidade Federal dos Vales do Jequitinhonha e Mucuri, Brazil b LNCC, Brazil We present a model for the tree growth process in flooded areas of the Amazon Forest that contemplates seed dispersion processes and influence of light in the model. These processes, strongly influenced by the annual food, occur early in the plant’s life cycle and affect the distribution, structure and dynamics of tree populations. The study of these processes is very important to understand the dynamics of the flooded ecosystems and their sustainable management. To study the spatial distribution of seeds in a region and the growth of germinated seeds, we define an influence region for a mature tree in the growth of the seeds, the growth neighborhood of young trees and the probability of a seed to fall at any point of this neighborhood. We make the hypothesis that every year each mature tree produces affixed number of seeds. The seeds are distributed in accordance with a probability that varies with the distance from a mature tree. Seed-germination and the growth process depend on access to the solar light. In particular, since the canopies of mature-trees in a neighborhood of a youngtree contain sufficient dense leaves and twigs structure, the light shall vary and influence the growth of surrounding trees. Indeed, for most species, the nearer a young-tree is from a mature-tree, smaller will be its possibility to survive. In some cases it may occur the death of these individuals by shading. We thank for FAPEMIG for financial support to participate of this meeting. 83 A Solution through the Generalized Integral Transform Technique for an SIR Epidemic Model with Spatial Diffusion João Bosco Soares, Emanuel Negrão Macêdo and João Nazareno Nonato Quaresma UFPA, Brazil A hybrid solution based on the Generalized Integral Transform Technique (GITT) is obtained for an SIR epidemic model with spatial diffusion, which may describe the spread of diseases such as whooping cough. The GITT approach is a hybrid methodology based on eigenfunction expansions for solving linear or nonlinear in multiphysics problems. An extensive parametric analysis is done in order to investigate the influence of typical governing parameters for such physical situation. Comparisons with results from the literature for typical situations are performed to demonstrate the consistency of the final results and to show the ability of the GITT approach in handling problems dealing with SIR epidemic models. 84 Finite Element model for the dissemination of infection toxoplasma gondii John Alexander León Marı́n and Irene Duarte Gandica Universidad del Quindı́o, Colombia Toxoplasmosis is a parasitic zoonosis of wide world distribution, that infects a large proportion of human and animal populations, it produced for Toxoplasma gondii parasit. Epidemiological studies have shown that most of the wordl the presence of cats is critical for parasite transmission to different intermediate host (human, domestic animals). Moreover, In Vancuver Canada, an epidemic outbreak was associated to contaminated water reservoir of the city by a wild cat and Brasil, an epidemiological survey also associated consumption of unfiltered water with disadvantaged socioeconomic infection. In this paper, the spread by Felis catus of Toxoplasma gondii parasite is modeled. The dynamics is described with a system of partial differential equations defined on a irregular region, including initials and boundary conditions, that combines a model of type SIR with an equation of diffusion for the parasite. This paper aims to apply an algorithmic scheme, which consists of the approximation space by the finite element and the temporal, the crank-Nicolson method, and to approximate the solution of the resulting nonlinear system, uses a successive linearization at each time step, a model describing the spread of toxoplasmosis in a regin with irregular borde, which takes into account the parasite spread by mechanical transport (birds, rodents, insects, etc.) through a velocity field. 85 A stochastic study for small populations using Gillespie algorithm Josemeri A. Jamieniak and Claudia Pio Ferreira UNESP, Brazil All biological populations present random oscillations that are significant when we study small populations. In this context, stochastic population models consider natural oscillations in the population dynamics by using random variables. Therefore each simulation results in different outputs even when the initial values are kept, increasing transient-time and requiring extensive computational efforts. In probability theory, stochastic process is the opposite of deterministic process because it can evolve through countless ways by keeping the same initial condition. Each way is possible, however they have different probabilities. This indetermination is introduced by probability distributions that rule process evolution. In this work, stochastic simulations were run for small populations using Gillespie algorithm. We consider that each population component was distributed in a finite set of discrete states and their interactions result in population oscillations. Our goal here is to compare stochastic and deterministic simulations outputs. 86 On the studies of a Matrix model of Aedes aegypti dynamics Luciano Medina Peres FURG, Brazil During the last several years, many models for the study and interpretation of population dynamics has been proposed [1, 2, 5]. More recently, the introduction of mathematical theory in the analysis of population dynamics has improved significantly the knowledge of many interesting and relevant aspects related with the dynamics population models. In this contribution we are interested in the matrix models of Leslie and Lefkovitch [3, 4] for describe the dynamics of the Aedes aegypti population and the relation of this with human iterations. Leslie’s model are quite used for human population dynamics and Lefkovitch’s model can be easily adapted to the Aedes aegypti population. In order to address the iteration between the Aedes aegypti vector and the human population, we couple the models in a Malthus type dynamics [1, 2, 5]. The potential advantages of the models of Leslie and Lefkovitch is that once established a matrix ”M” formed by values of fertility, survival probability and phase shift, the ability to change parameters in ”M” can be done without necessarily knowledge of, for example, size or probability of any age group. On the other hand, these models have limitations in predict the behavior for populations with migration (as a form of growth), and the differences between individuals of the same class (as infected, not infected and recovered) [7]. The main tool of our analysis is based on the concept of linear algebra [6] and differential equations [1, 2] that provide the rates of growth and extinction of the Aedes aegypti vector with the human iterations. Moreover, we intend to test the proposed model with real data. 87 Basins of attraction on a classic model of competition between two species Artur César Fassoni, Lucy Tiemi Takahashi and Laércio José dos Santos UFJF, Brazil In this paper we present a global qualitative analysis of a classic model of competition between two species. A global energy function was determined, which provides a new demonstration that all system trajectories converge for one of its equilibrium points. Then we studied the influence of parameters on the size and shape of the basins of attraction. In the case of strong competition, the system has two asymptotically stable equilibrium and a saddle, and we verify that the higher is the rate of interspecific competition of a species, the greater is the basin of attraction of asymptotically stable equilibrium that corresponds to the predominance of the same. In addition, the relative birth rate of species influences the curvature of the boundary of the basins of attraction, which in this case is the stable manifold of saddle point: the analysis of influence of this parameter shows that, for a species that begins with a few individuals in habitat, it is advantageous that its birth rate is greater than that of other species; conversely, if a species begins with a large number of individuals, it is advantageous to its birth rate is less than that of other species. We showed this result analytically, by obtaining the expression of curvature of the separatrix between the basins of attraction, depending on the parameter that represents the ratio of birth rates. Finally, for the special case in which birth rates are equal, we get the exact formula of the separatrix between the basins. Research partially supported by FAPEMIG - Demanda Universal grant no. CEX APQ-00149-08. 88 Dynamics of chickenpox: Hopf bifurcation Ailton Luiz Vieira, Lucy Tiemi Takahashi and Laércio José dos Santos UFJF, Brazil In this work, we propose a model based on the results obtained by Vieira and Takahashi, in 2009, aiming at a more realistic formulation for the dynamics of varicella zoster herpes coupled to, common diseases in Brazil. The qualitative analysis of equilibrium points of the model we see the emergence of a Hopf bifurcation. And by analysis of Hopf, non-degeneracy and transversality, guarantee the emergence of periodic orbits, which was corroborated by the results of numerical simulations. [1] SOTOMAYOR TELLO, J. M.; MELLO, L. F.; SANTOS, D. B.; BRAGA, D. C. Bifurcation analysis of a model for biological control, Mathematical and Computer Modelling, v. 48, p. 375-387, 2008. [2] VIEIRA, A. L.; TAKAHASHI, L. T., A sobrevivência do vı́rus varicelazoster (The survival of the varicella-zoster virus), Biomatemática, v. 19, p. 109-124, 2009. Research partially supported by FAPEMIG - Demanda Universal grant no. CEX APQ-00149-08. 89 Dynamics of populations with fractional order derivatives Luverci do Nascimento Ferreira FURG, Brazil Differential equations involving fractional order derivatives have recently gained attention in various fields of science, engineering, economics, biomechanics, etc (see [1-4] for a review). Fractional derivatives are nonlocal operators that can be useful to study complex systems, and is special, to investigate systems with temporal memory effects. In this context, the fractional-order Lotka-Volterra equations was recently considered by some authors [5-7]. They found that the most evident consequence of the fractional order derivatives is to drive the population to the equilibrium. However, a more detailed study of the effect of fractional-order derivatives is lacking. In order to investigate the effect of fractional order derivatives on population modeling, in the present work we consider the fractional-order generalization of Logistic, Nurgaliev and Lotka-Volterra equations. [1] Sabatier, Agrawal and Tenreiro Machado (eds), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Netherlands, 2007. [2] Kilbas, Srivastava and Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [3] Hilfer (ed), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [4] Magin, Fractional Calculus in Bioengineering, Begell House Publisher, 2006. [5] Rivero, Trujillo, Vasquez and Velasco, Applied Mathematics and Computation, http://dx.doi.org/10.1016/j.amc.2011.03.017. [6] Das, Gupta, Journal of Theoretical Biology 277 1-6, 2011. [8] Camargo, Calculo Fracionario e Aplicações, Tese (doutorado), Unicamp, Campinas, 2009. 90 Evolution of virulence driven by predator-prey interaction: Possible consequences for population dynamics A.Yu. Morozov and Matthew Adamson University of Leicester, UK The evolution of pathogen virulence in natural populations has conventionally been considered as a result of selection caused by the interactions of the host with its pathogen(s). The host population, however, is generally embedded in complex trophic interactions with other populations in the community, in particular, intensive predation on the infected host can increase its mortality, and this can affect the course of virulence evolution. Reciprocally, the evolution of virulence within an infected host can affect the patterns of population dynamics of a predator consuming the host. Surprisingly, neither the effect of predation on the evolution of virulence within a host, nor the influence of the evolution of virulence upon the consumer’s dynamics has 91 Fuzzy Logic applied as alternative method to estimate age and growth of freshwater stingray Paratrygon aiereba Maria Lúcia Góes de Araújo, Rodney Bassanezi, Karla Tribuzy, Rosânegla Lessa, Francisco Santana DIMAR/UFRPE, Brazil Many age and growth studies of elasmobranch utilize growth patterns in calcified structures as vertebral centra to determinate the age of these fishes. The routine of this method involve vertebra manual preparation and reading, verification and validation of estimated age, and in all these steps can be introduce subjective errors in analysis. Uncertainty is part of fish growth analysis, and the most traditional growth models used are deterministic models, which have the accuracy as a major feature. To incorporate uncertainty in growth parameter estimation of freshwater stingray Paratrygon aiereba, we have applied Fuzzy logic theory in generalized von Bertalanffy growth model and compared the results with those by human expert. The weight-age model used has the conjecture that the greatest variation in weight occurs when the individual reaches the first maturation (inflection point). Each model variable (α - anabolism, β -catabolism and γ- alometric parameter) was divided into three triangular fuzzy sets reflecting low, medium, and high values. Three growth curves were generated related to low (inflection point =7kg; α=2.2230, β=0.1022, γ=0.2127); medium (inflection point =12 kg; α=1.2974, β=0.1619, γ=0.4681) and high (inflection point =15.76 kg; α=1.0027, β=0.3695, γ=0.7448). Von Bertalanffy generalized model parameters estimated by age rings readings (W = 54880.85, k = 0.097041, t0 = −0.594) reflect the current data set. The prediction of age compared to that by an expert agrees to the extent of 70%. The accuracy can be improved with more data about the species. These results show that fuzzy logic is a useful approach to describing Paratrygon aiereba growth. 92 The jaguar’s patches: how many habitat remnants are needed to keep jaguars in a fragmented landscape? Marina Zanin Gregorinia, Francisco Palomaresb and Daniel Britoa a b Universidade Federal de Goiás, Brazil 2Estación Biológica de Doñana, Espain Habitat loss and fragmentation (HLF) are the main threat to biodiversity, acting synergistically in the environment. However, to discriminate the effects of habitat loss from fragmentation on species is important due to the influence on population dynamics. Thus, we isolated the effects of these processes on jaguar (Panthera onca) population viability, finding critical thresholds for habitat loss and for habitat fragmentation (HCT and FCT). The population viability was analyzed by using the VORTEX software. To estimate HCT, we modeled population dynamics in a landscape of 10,000km2 and reduced the total area gradually. To calculate FCT we maintained total area of suitable habitat constant throughout scenarios, which were subdivided progressively. HCT and FCT were calculated by piecewise linear regression. The sensitivity analysis was done by a regression tree through 200 hypothetical populations with random values of mortality and proportion of male breeders, because these were life history characteristics poorly estimated in the published literature. Jaguar HCT varies largely (1,933 to 7,535 km2), but it may be used as an indicator of protected areas size. We observed jaguar metapopulation viability on restricted environmental conditions, with large areas and high densities. In fragmented landscapes, the total area needed to maintain a viable metapopulation increase compared with in areas of a single patch. However, a large part of the current jaguar distribution is composed by highly fragmented landscapes, making viable the strategy of landscape with patches distributed in order to maintain a viable metapopulation. These results are generalization that may be better estimated through higher accurate values of adult female mortality, since the high sensitivity of this parameter on the viability calculated. 93 Herbivory-taxis and Non-local Aggregation in a Plant Herbivore System Luiz Alberto D. Rodrigues, Otonio Dutra da Silva and Diomar C. Mistro Universidade Federal de Santa Maria, Brazil The dispersal of herbivorous insects in large plantations must be modelled by taking into account many behavioral aspects of the individual movement. The insect ability to search for (and find) what is the best for his nutrition needs and the corresponding mechanism responsible for it are of fundamental importance to the development of a mathematical model. Also, many insects present an aggregation tendency which seems to be a population strategy to optimize harvesting and survival. In the present work, both kinds of behaviour are microscopically described by a Coupled Map Lattice model which includes a short range taxis for plant quality to a long range taxis with respect to their own population density. A very short range random search, which is indispensable for any taxis, is implicitly included. Numerical simulations are used to show the spatio-temporal distribution of the herbivore density and plant quality. 94 1/f noise in a SIS model for dengue epidemics Romuel F. Machado, Sérvio P. Ribeiro, Michelle C. Pedrosa and Everaldo Arashiro a Universidade Federal de Ouro Preto, Brazil We have devised a simplified discrete mathematical model for the life cycle of Aedes aegypti coupled to a SIS model in order to investigate the epidemic dynamics. Using a SIS model is justified by the fact that individuals infected by certain dengue serotype can be infected by another one. The environmental influence is taken into account by expressing crucial quantities such as transition rates and adults mortality as functions of temperature. The dependence of these quantities on temperature were obtained from empirical data and the temperature time series that feeds the model is a real one taken from a meteorological station located at Ipatinga (MG). By adjusting the average temperature of this series we can model different scenarios for the mosquito development from unfavorable (very low and very average values) to favorable ones (medium values). We show that there is a transition from an epidemic to endemic regime as the average temperature changes from very low/high values to medium values. The epidemics for high average temperatures, where the power spectrum of adults mosquitoes time series exhibit the typical power-law behavior of 1/f noise for small frequencies, is more severe than for low ones. The present findings highlight the existence of real possibilities of further invasion of A. aegypti in originally cold regions in South America due to global warming effects on average temperature. 95 Population density and home range size of Formicarius colma (Birds, Formicariidae) in a primary forest plot in the Central Amazon Tatiana Straatmanna,b and Gonçalo Ferrazb a b Instituto Nacional de Pesquisas da Amazônia, Brazil Biological Dynamics of Forest Fragments Project, Smithsonian Tropical Research Institute, Manaus, AM, Brazil. Spatiotemporal studies of site occupancy have proven useful in wildlife population analysis. Yet, the biological meaning of occupancy often depends on information about the density and movements of organisms, especially when sites lack well-defined boundaries, as in vast unbroken regions of tropical forest. In this study, we provide population density (d), home range size (from parameter σ, which represents the home range standard deviation) and detection probability (ρ) estimates for a population of the Rufous-capped Antthrush (Formicarius colma) in a primary forest plot in Central Amazonia. We captured and color banded 14 individuals of F. colma during June 2011. In the five following months we carried out field surveys to search for the banded birds using playback. We analyzed the data using a SECR model for search-encounter data in a Bayesian framework using MCMC and compared the results with preexisting assessments of F. colma density in the same area. Our posterior estimate for σ was 0.20 Km (95% credible bounds: 0.11 - 0.27). The σ mean was used as the home range radius, leading to a home range area of 12.56 ha, higher than the territory size estimates from previous studies (6.58 and 7.3 ha). Density estimate was 5.7 ind. per 100 ha (95% c.i.: 3.1 - 8.9), lower than the results previously found (21 and 11 individuals). For the detection probability ρ we got a maximum detection probability at zero distance of 0.40 (95% c.i.: 0.06 - 0.86). The differences between estimates may result from the distinct analytical and datacollection methods used. Differently from the previous works, we used locations only from marked birds, during a longer sampling period, while accounting for imperfect detection. 96 Modeling Latitudinal Gradient in Ecological Network Robustness using Bayesian Analysis Vinicius Augusto Galvão Bastazini and Valério De Patta Pillar Universidade Federal do Rio Grande do Sul, Brazil. The latitude-niche breadth hypothesis has been a central and long-standing topic in ecology, and, one of its many predictions is that ecological networks at higher latitudes should be less specialized. Consequently, it is expected that network robustness (i.e., the system’s tolerance to species extinctions; NR) should increase with an increase in latitude, making ecological networks less prone to secondary extinctions at higher latitudes. However, insofar this prediction has never been addressed. We tested the association between latitude and NR using Bayesian analysis to estimate model parameters and effect size. We based our analysis on 13 bipartite networks, available in databases. Our dataset encompassed different continents and types of ecological interactions. We ran two sets of analyses simulating replicated random extinctions: the first based on removal of species from lower trophic levels, and the second on removal of species from higher trophic levels. As predicted, NR was positively associated with latitude. When species are randomly removed from the higher trophic level, NR increases at a rate of 6% with latitude (95%CrI: 0.02 ≤ β1(std) ≤ 0.09; Median = 0.06), compared to an increment of 3%, when species are removed from the lower trophic level (95%CrI: 0.02 ≤ β1(std) ≤ 0.06; Median = 0.03). Besides the obvious implications for theoretical ecology, our findings also provide essential information for applied conservation, as they indicate that tropical ecological networks are more likely to undergo processes of secondary extinctions. 97 Dynamics of host and parasitoids with integrated pest management policy: theory and experimentation R. A. Moral, A. P. M. B. Battel, J. A. Neves, E. N. Lopes, and Wesley Augusto Conde Godoy ESALQ - USP, Brazil. The population dynamics in a host parasitoid system is investigated using a combination of integrated pest management policies with predator prey theory using data obtained in laboratory. A mathematical model is proposed to investigate effects of management strategies on the population dynamics of the insects. The sensitivity of model parameters is investigated by employing the bifurcation theory in order to show the parameter space associated with the possible dynamic behaviors. The results are discussed in the context of integrated pest management. 98 Plenary Speakers Alan Hastings Bernd Blasius Univ. Davis, USA Univ. Oldenburg,Germany Horst Malchow Univ. Osnabrück, Germany Mark Lewis Nick Britton Sergei Petrovskii Ulrike Feudel Vitaly Volpert Wilson C. Ferreira Jr. Univ. Alberta, Canada Univ. Bath, UK Univ. Leicester, UK Unvi. Oldenburg, Germany Univ. Lyon, France UNICAMP, Brasil [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] Mini-Symposia Speakers Brenda Tapia Santos Fabio Chalub Gandhi Viswanathan Jorge X. Velasco-Hernandez José Fernando Fontanari Luca Giuggioli Marcos Capistrán Ocampo Marcos da Luz Max O. Souza Univ. Veracruz., México UNLisboa, Portugal UFRN, Brazil IMP, México USP, Brazil Univ. of Bristol, UK CIMAT, México UFPR, Brazil UFF, Brazil 99 [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] Contributed Talk Speakers Armando G. M. Neves Artur César Fassoni Bernardo A. Mello Bourget Romain Carlos M. Viriato Neto Chakib Jerry Claire Dooley Claudia Pio Ferreira Claudio Arancibia-Ibarra Crysttian A. Paixão Daniel Bearup Daniel Cowley Eduardo Gonzalez Emanuelle A. Paixão Fernando A. Oliveira Fernando Córdova-Lepe Frédéric Hamelin Gilberto Corso Graciela A. Canziani Jacques A. L. Silva Jaime Huincahue-Arcos João F. C. A. Meyer João P. Juchem Neto José D. Flores Kévin Cazelles Lourdes Esteva Luiz F. S. Freitas Marat Rafikov Marı́a L. Maestri Mario I. Simoy Mark Kelbert Matheus J. Lazo Matthew Adamson Miguel A. Dumett Paulo R. A. Campos Roberta R. Delboni Robin N. Thompson Rodrigo Del Valle Ruth Becerra-Klix Salvador Lou Vega Sebastián Valenzuela Sudipta Tung Ulrike Schlaegel UFMG, Brazil UNICAMP, Brazil UnB, Brazil Univ. d’Angers, France UFSJ, Brazil Moulay Ismail Univ., Morocco Univ. Oxford, UK UNESP, Brazil PUC Valparaı́so, Chile FGV, Brazil Univ. Leicester,UK Univ. Bristol, UK PUC Valparaı́so, Chile UFLA, Brazil UnB, Brazil PUC Maule, Chile Agrocampus Rennes, France UFRN, Brazil UNCPBA, Argentina UFRGS, Brazil PUC Valparaı́so, Chile UNICAMP, Brazil UFRGS, Brazil Univ. South Dakota, USA Univ. Montpellier, France UNAM, Máxico UNICAMP, Brazil UFABC, Brazil UNCPBA, Argentina UNCPBA, Argentina Swansea University, UK FURG, Brazil Univ. Leicester, UK UFP, Brazil UFPE, Brazil UNICAMP, Brazil Univ. Cambridge, UK PUC Maule, Chile PUC Valparaı́so, Chile 100 UNICAMP, Brazil PUC Valparaı́so, Chile IISER, India Univ. Alberta, Canada [email protected] [email protected] [email protected] [email protected] viriato [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] jacques [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] Poster Presenters Adriano Gomes Garcia Anderson A. Ferreira Deccy Y. T. Angel Divane Marcon Evandro E. Marquesone Francisco Q. Camargo Irene Duarte Gandica Jaqueline M. Silva João B. Soares João N. N. Quaresma John A. León Marı́n Josemeri A. Jamieniak Luciano M. Peres Lucy T. Takahashi Luverci N. Ferreira Maria L. G. Araújo Marina Z. Gregorini Oscar A. R. Cespedes Otonio Dutra da Silva Romuel F. Machado Tatiana Straatmann Vinicius A. G. Bastazini Wesley A. C. Godoy UNESP, Brazil UFPel, Brazil UFG, Brazil UNICAMP, Brazil UTFPR, Brazil USP, Brazil Univ. Quindı́o, Colombia UFVJM, Brazil UFPA, Brazil UFPA, Brazil Univ. Quindı́o, Colombia UNESP, Brazil FURG, Brazil UFJF, Brazil FURG, Brazil UFRPE/UFS, Brazil UFG, Brazil UFG, Brazil UFSM, Brazil UFOP, Brazil INPA, Brazil UFRGS, Brazil ESALQ/USP, Brazil [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] tati [email protected] [email protected] [email protected] Participants Adriane Frank Aline Parigi Andres Quiroga Danilo F. F. Leonel Elisa Regina Cara Felipo Bacani Marcelo Awade Maria Cristina Varriale Mario Rocha Retamoso Fernanda Somavilla UFSM, Brazil UFSM, Brazil UNC, Argentina UFPR, Brazil UFRGS, Brazil UNICAMP, Brazil USP, Brazil UFRGS, Brazil FURG, Brazil UFSM, Brazil 101 [email protected] aline [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

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