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Chapter 1 GENERAL INTRODUCTION Community ecology is often perceived as a “mess", given the seemingly vast number of process that can underlie the many patterns of interest, and the apparent uniqueness of each study system. However, at the most general level, patterns in the composition and diversity of species - the subject matter of community ecology – are influenced by only four classes of process: selection, drift, speciation, and dispersal. Vellend (2010) 1.1 INTRODUCTION AND BASIC CONCEPTS Ecology describes the relationships among organisms, and between them and the environment, thus determining aspects such as species abundance, compositions of biological community that represents a group of organisms living in a specified place and time. The word ‘Ecology’ has attracted attention of scientists and philosophers from the ages of human civilization. It was first defined by the German biologist Ernst Haeckel in 1869; According to him ecology is the science of interrelation between living organisms and their environment. The word ‘ecology’ owes its origin to the Greek word ‘Oikos’ meaning ‘house’ or ‘place to live’ (Tansley, A.G.,1946). In ecology, biological interactions or the relationships between two species in an ecosystem can be categorized into many different classes based either on the effects or on the mechanism of the interaction and can be classified as: i. Neutralism describes as the relationship between two species which do interact but do not affect each other. 1 ii. Competition is an interaction between two species that is mutually detrimental. Competition between organisms occurs in case of limited supply of essential resource, such as food, in case of animals and water, nutrients and light in the case of plants. iii. Predation is the consumption of one organism by another where the consumed organism (prey) was alive when the predator first attacked it. Predators may or may not kill their prey prior to feeding on them, but the act of predation always results in the death of the prey. iv. Parasitism is a relationship between two species in which one benefits at the expense of the other. A parasitoid is an insect that parasitizes and kills other insects. Predation and parasitism are an example of antagonistic ecological interactions in which one species take advantage of another species. Predators use their prey as a source of food only, whereas parasites use their hosts both as a food and as a habitat. v. Mutualism is a relationship between two species that have developed a positive, reciprocal dependency and both populations benefit from this association. Through the relationship both the species strengthen their chances for survival, fitness or growth. A facultative Mutualist is a species that benefit from interaction with other species, but does not absolutely require the interaction, whereas an obligate Mutualist is a species that cannot survive without the Mutualist species. 2 vi. Amensalism between two species involves one impeding or restricting the success of the other without being affected positively or negatively by the presence of the other. It is a type of symbiosis. The application of mathematical biology has an immense impact towards the study of ecology and in development of commonly used biological resources like fisheries, wildlife and forestry. Recently Scientists and researchers give emphasis on the interaction between mathematics and biology which initiate a new research area. Most of our knowledge of biological systems is comparatively recent, and most biological theories evolve rapidly; therefore it is necessary to develop some useful mathematical models to describe the consequences of these biological systems. It is observed that these newly developed mathematical models are significantly influenced through the biological theories in the past and the consequent expansion of those theories in recent time. The traditional mathematical model of predator-prey interactions consists of the following system of two differential equations: dx P( x) eg ( x, y ) y ; dt dy eg ( x, y ) y D( y ) ; dt x(0) x0 0, (i) y (0) y 0 0. Where x and y represent the prey and the predator population sizes respectively, and functions P( x), D( y ) describe the intrinsic growth rate of the prey and the mortality rate of the predator, respectively. The function D( y ) is assumed to be linear while the function P (x ) may have a linear P( x) ax D( y) dy , x or logistic P( x) ax1 K expression. The function g ( x, y ) is called “functional response” or “feeding rate” and 3 represents the prey consumption per unit time. The model assumes a linear correspondence between the prey consumption and the predator production through the positive constant e. In the literature, there are various forms for functional responses which are classified into the following categories: Prey Dependent: Prey dependent functional response is unaffected by predator density. This includes: Holling type I: in which the attack rate of the individual predator (consumers) increases linearly with prey density, however when the consumers are satisfied, it consumes prey at a constant rate. x g ( x, y) g ( x) c for 0 x K for xK Here K is level of prey at which predator is satisfied at c . Holling type II: in which the attack rate increases with decreasing prey density until it becomes constant at satiation. This response is a representation of the typical behaviors of predators that specialize on one or a few prey. It is also called hyperbolic functional response. g ( x, y) g ( x) ax Ax 1 1 Ahx b1 x Here A is search rate, h is handling time, a1 is a maximum consumption rate and b1 is half saturation level. 4 Holling type III: in which attacks rate accelerate at first and then decelerates towards satiation. It is also called sigmoid functional response. Sigmoid functional responses are representation of typical behavior of generalist’s natural enemies which readily switch from one food species to another and or which concentrates their feeding in areas where certain resources are not abundant. a1 x 2 Ax 2 g ( x, y ) g ( x ) . 1 Ahx 2 b1 x 2 Holling type IV: When the nutrient concentration reaches a high level, an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, Andrews (2006) proposed the response function , called the Monod-Haldane function, and also called a Holling type-IV function. The Holling type IV functional response is of the form. g ( x, y) g ( x) a1 x . b1 x 2 Ivelev-Type: Ivelev (1955) derived the functional response of a consumer by arguing that the rate of consumption should depend on how hungry it is. This functional response is of the form. g ( x, y) g ( x) 1 e a1 x The prey dependent functional responses fail to model the interference amongst predators so we study the predator dependent response in which effect of both predator and prey populations are considered. Predator dependent functional responses may be of following kinds : 5 Beddington-De Angelis Type: This functional response was introduced by Beddington (1975) and DeAngelis et al. (1975), independently. It is similar to Holling type II functional response but contains an extra term describing mutual interference by predators. g ( x, y) a1 x . m x b1 y Here a1 and m are positive parameters, b1 is a predator interference parameter. b1 0 is the case where predator benefit from co-feeding, m is saturation constant and a1 is the maximum consumption rate. Hassell-Varley Type: This is an elaboration of Holling type II, using a variable power (sigma) of the amount of predators. It is of the form. g ( x, y ) a1 x b1 x y Where a1 0, b1 0 and 0 . Crowley–Martin type: This type of functional response was introduced by P.H. Crowley and E. K. Martin (1989). This is defined as g ( x, y) a1 x . 1 b1 x b2 y b1b2 xy Here constants ai , bi (i 1,2) are positive, stands for the effect of capturing rate, handling time and the magnitude of interference among predators respectively. 6 Ratio-dependent Functional Response: This function response is predator – dependence response , in which it only depends on the ratio of prey population size to predator population size, not on the absolute number x of species. In particular g ( x, y) g . There is growing evidences that in some y situations when predators have to search for food (and therefore have to search or compete for food), the per capita growth rate should be function of the ratio of prey to predator abundance. It is observed that ratio – dependent models provide richer dynamics in comparison to the predator–dependent models. This is strongly supported by numerous fields and laboratory experiments and observations (Arditi and Ginzburg, 1989; Arditi et al., 1991). It is given by x A y a1 x . g ( x, y ) x b1 y x 1 Ah y In the natural world, many species have a life history that takes their individual members through two stages: immature and mature with a time lag. In ecology, more realistic models should include some of the past states of the system: that is, ideally, a real system should be modeled by differential equations with time-delay (Freedman and Gopalasmay, 1986). The time delay is the inherent property of the dynamical systems and plays an important role in almost all branches of science and particularly in the biological sciences (e.g., population dynamics, epidemiology, etc.). Delay differential equations differ from ordinary differential equations in that the derivative at any time depends on the solution at prior time. The simplest constant time delay equations have the form 7 y (t ) f (t , y(t ), y(t 1 ), y(t 2 ),.......... y(t k )), where the time delays (lags) , j , are positive constants. More generally, state dependent delays may depend on the solution, that is i i (t , y(t )) . The objective of studying delay differential equations is to assess the qualitative or quantitative results for models with delays representing different biological mechanisms (Maynard Smith, 1974). The delay in the prey-predator model was first included by Volterra, 1926, who took into account time taken for pollutants produced by a population to build up, eventually increasing the death rate of the population. Delay may also be due to development time of the population itself or of its resources. Kuang, 1993 mentioned that animals must take time to digest their food before further activities and responses take place and hence any model of species dynamics without delays is an approximation at best. Detailed arguments on the importance of time delays in realistic models may be found in classical books of Gopalsamy, 1992; Macdonald, 1989 and Kuang, 1993. 1.2 MATHEMATICAL TECHNIQUES The following qualitative methods, lemmas, definitions and theorems have been used to analyze various nonlinear dynamical models proposed in the thesis 1.2.1 REGION OF ATTRACTION A part of the phase space including the attractor (sets of population states that are invariant, they can be equilibria) itself, as well as all initial conditions from which a trajectory will converge to the attractor. 8 1.2.2 EQUILIBRIUM POINT AND STABILITY Equilibrium is a point at which a variable (or variables) remains unchanged over time. A particular value of a variable is called an equilibrium value if, value of variable never changes when it start at this value. It is the graph of a constant solution. In mathematics, the point x 0 n is an equilibrium point for the differential equation dx f ( x), if f ( x 0 ) 0 for some value x 0 . Equilibria are sometimes called fixed points dt or steady states. Stability is a very important issue in the study of nonlinear systems. Stability means, that a small change (a small disturbance) of a physical system at some instant time, changes the behavior of the system only slightly at all future time t. Various stability concepts frequently associated with a dynamical systems are: local stability, global stability, absolute stability and Lyapunov stability of equilibrium points. An equilibrium point is said to be locally stable if small perturbations remain close to the equilibrium, and locally asymptotically stable if small perturbations eventually returns to the equilibrium. As such it does not predict the overall behavior of the system. On the other hand, when small perturbations continue to move away from the equilibrium, the equilibrium point is said to unstable. Equilibrium is globally stable if a system approaches the equilibrium regardless of its initial position, For example consider the following system of differential equations for n- interacting species: dx f ( x, ). dt (1.2.1) 9 Here x ( x1 ,..., x n ) t is a state variable and is control parameter. The function f captures the interaction amongst species. The equilibrium point x e (say) of system (1.2.1) is a point, satisfying f ( xe , ) 0 . Linear stability is used to study the stability of the steady state xe in a small neighborhood. The system (1.2.1) can be linearized at xe , as dx J x, dt (1.2.2) Where, J is the Jacobian matrix (Variational matrix) of the system (1.2.2) whose elements are given by f a ij i x j . The characteristic equation corresponding to the xe system (1.2.2) is obtained as n A1 n 1 A2 n 2 ... An 0 (1.2.3) The stability of equilibrium point x e can be determined by looking at the signs of eigenvalues i of the Variational matrix J at each of the equilibrium points of the system. The equilibrium point is asymptotically stable if the real parts of all eigenvalues of characteristic equation corresponding to this equilibrium point are negative and strongly stable if every eigenvalue of characteristic equation is pure imaginary and of simple type. An equilibrium point is hyperbolic if none of the eigenvalues have zero real part. If one or more of the real parts of the eigenvalues are positive then the state is unstable. 10 The Routh-Hurwitz Stability Criterion provides the necessary and sufficient conditions under which all roots of the characteristic equation (1.2.3) lie in the left half of complex plane: A1 0, A1 A3 1 A2 A1 A3 A5 ... 0 0 A1 A3 A5 1 A2 A4 ... 0 0 0, 1 A2 A1 A3 ... 0 0 0. 0 A1 A4 0,..., 0 A3 0 0 ... 0 ... 0 An In particular, the Routh – Hurwitz criterion for n 2, n 3 and n 4 are n2 A1 traceJ 0 ; A2 det J 0 . n3 A1 0 ; A3 0 and A1 A2 A3 0. n4 A1 0 ; A2 0 ; A3 0 ; A4 0 ; A1 A2 A3 0 and A3 A1 A2 A3 A12 A4 . 1.2.3 BIFURCATION ANALYSIS The term bifurcation generally refers to something “splitting apart”. With regard to differential equations or systems involving a parameter, it refers to abrupt changes in the character of the solutions as the parameter is changed continuously. Bifurcation analysis of nonlinear models addresses the effects of parameter changes on the qualitative nature of the attractor (stable vs. unstable, fixed point vs. cycle, etc.). Bifurcations are said to occur when small quantitative parameter changes produce large qualitative changes in the dynamics. In one sense, bifurcation analysis is the next step beyond local stability analysis, in that it describes the fact of large perturbations from equilibrium. In the mathematical theory of bifurcations, a Hopf bifurcation is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate 11 eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane resulting in a stable limit cycle (oscillations). The Hopf bifurcation break the stability of system and give rise to oscillation that are uniform in space and periodic in time. 1.2.4 PERSISTENCE Biologically, persistence means the survival of all populations in future time. Mathematically, persistence of a system means that strictly positive solutions do not have omega limit points on the boundary of the non – negative cone. Persistence may be defined mathematically as Freedman and Waltman (1984, 1985). A population P (t ) is said to persist (sometimes called strongly persist) if P (0) 0 implies P(t ) 0 and lim inf P(t ) 0. More related definitions in both ecological and abstract situations are t given in Butler, et al., 1986; Butler and Waltman, 1986; Freedman and Moson (1990). A population P (t ) is said to persist uniformly if P (t ) persists and there exists P(t ) . Finally, we say that a system 0 independent of P(0) 0, such that limt inf persists (uniformly) whenever each component persists (uniformly). To prove the persistence of populations, we have also used Butler – McGehee lemma. Before the statement of Butler – McGehee lemma, we would like here to give some definitions in regard of this lemma. Isolated Hyperbolic Equilibrium: Equilibrium point ( x10 , x 20 , x 30 ,...x n0 ) is isolated if there exists a neighborhood N of ( x10 , x 20 , x 30 ,...x n0 ) such that the only equilibrium of 12 x i f i in N is ( x10 , x 20 , x 30 ,...x n0 ), and a point is hyperbolic if none of the eigenvalues of the variational matrix at equilibrium point have zero real part. Omega Limit Set ( Limit Set): If x(t ) is the solution of the system x (t ) f ( x) and if there exists a number t 0 such that x(t ) is defined for all t t 0 , then an omega limit point of solution x(t ) is a point x 0 such that there exists a sequence of real numbers {tn } with lim t n and lim x(t n ) x 0 . If S denote the solution x(t ), then the set of omega limit n n points of S will be denoted by (S ) or ( x(t )). Orbit: the orbit C ( x 0 ) of x (t ) f ( x) through x 0 is defined by C ( x 0 ) {x R n : x x(t , x 0 ), t R}. Statement of Butler – McGhee: Let E be an isolated hyperbolic equilibrium in the limit set ( X ) of an orbit ( X ). Then either ( X ) E or there exists points E and E in ( X ) with E M (E ) and E M (E ). 1.2.5 PONTRYAGIN’S MAXIMUM PRINCIPLE The maximum principle is an optimization technique that was first proposed in 1956 by Pontryagin and its associates for various types of time-optimizing continuous processes. This is powerful method for the computation of optimal control, which has the crucial advantage that it does not require prior evaluation of the infimal cost function and its proof is based on maximizing the Hamiltonian. It has as a special case the EulerLagrange equation of the calculus of variations. 13 We assume from now on that b, c and C are differentiable in t and x with continous derivaties, and that the stopping set D is a hyperplane, thus D {y} , for some y R d and some vector subspace of R d . Define for R d the Hamiltonian H (t , x, u, ) T b(t , x, u) c(t , x, u ) Pontryagin's maximum principle states that, if ( xt , u t ) is optimal, then there exist adjoint paths ( t ) t in R d and ( t ) t in R with the following properties: for all t , i. H (t , xt , u, t ) t , has maximum value 0 , achieved at u u t , ii. Tt Tt b(t , x t , u t ) c(t , x t , u t ) , iii. t Tt b(t, xt , ut ) c(t , xt , u t ) , iv. x t b(t , xt , u t ) . Moreover the following transversality conditions hold: v. (T C (t , x )) 0 for all , and, in the time-unconstrained case, vi. C (t , x ) 0 . Note that, in the time-unconstrained case, if b, c and C are time-independent, then t 0 for all t . 14 The Hamiltonian serves as a way of remembering the first four statements, which could be expressed alternatively as (i) 0 H , u (ii) H , t (iii) H , t (iv) x H . 1.2.6 CHAOS Apparently simple deterministic systems with only a few elements can generate random behavior and this randomness is fundamental to the systems. This randomness cannot be removed by gathering more and more information about the elements of the system. This fundamental randomness is called chaos. Thus dynamical systems whose behavior is not stationary, nor periodic nor quasi periodic are called chaotic. Definition: Let J be a set. Then a flow defined by the vector function f : J J defined in a certain region / volume of space V is said to be chaotic if i. f has sensitive dependence on initial condition. ii. f is topologically transitive. iii. f has periodic points which are dense in V . A function f : J J is said to have sensitive dependence on initial conditions, if there exists a 0 such that for any x J and any neighborhood N of x , there exists y N and n 0 such that f n ( x) f n ( y ) . A function f : J J is said to be topologically transitive, if for any pair of open sets, U , V J there exists K 0 such that f K U V . 15 A chaotic system is unpredictable because of sensitive dependence on initial conditions. It cannot be broken down or decomposed into two subsystems (two invariant subsets) which do not interact under f because of topological transitivity. And in the midst of this random behavior we nevertheless have an element of regularity, namely the periodic points which are dense. If f f k be a map on R then we call ' p ' to be a periodic point of period k , if p p, k being the smallest such positive integer. In other words, we can say that, ‘Chaos is qualitatively more or less random behavior which is intrinsic to a system and not due to externally imposed noise”. 1.3 LITERATURE SURVEY Mathematical population models have been used to study the dynamics of preypredator systems. Lotka (1925) and Volterra (1927) proposed a simple model known as Lotka-Volterra model of prey-predator interactions. Since then, many mathematical models have been constructed based on more realistic explicit and implicit biological assumptions. Modeling is a frequently evolving process, to gain a deep understanding of the mathematical aspects of the problem and to yield non trivial biological insights, one must carefully construct biologically meaningful and mathematically tractable population models (Kuang, 2002). Some of the aspects that need to be critically considered in a realistic and plausible mathematical model include; carrying capacity which is the maximum number of prey that the ecosystem can sustain in the absence of predator, competition among prey and predators which can be intra specific or inter specific, harvesting of prey or predators and functional responses of predators . 16 Continuous predator–prey models have been studied mathematically. The principles of Lotka (1925) and Volterra (1927) model, conservation of mass and decomposition of the rates of change into birth and death processes, remain valid until today and many theoretical ecologists adhere to these principles. Modifications were limited to replacing the Malthusian growth function, the predator per capita consumption of prey or the predator mortality by more complex functions such as the logistic growth, Holling type I, II, III and IV functional responses or density-dependent mortality rates. The mentioned functional responses all depend on prey-abundance X only, but soon it became clear that predator abundance Y can influence this function (Curds and Cockburn, 1968; Hassell and Varley, 1969; Salt, 1974) by direct interference while searching or by pseudo-interference [in the sense of Freedman et al. (1977)] and models were developed incorporating this effect (Hassell and Varley, 1969; DeAngelis et al., 1975; Beddington, 1975). However, these models usually require more parameters and their analysis is complex. Therefore, they are, on one side, rarely used in applied ecology and, on the other side, have received little attention in the mathematical literature. A simple way of incorporating predator dependence into the functional response was proposed by Arditi and Ginzburg (1989) who considered this response as a function of the ratio X / Y . Interesting properties of this approach have emerged that are in contrast with predictions of models where the functional response only depends on prey abundance (e.g., Arditi et al., 1991; Ginzburg and Akcakaya, 1992; Arditi and Michalski, 1995). The interaction of multiple predators, predation rate and prey selection are also modified by the habitat complexity (Hughes and Grabowski, 2006). For aquatic predatory insects, rice fields, ponds, and temporary pools are habitats with ample heterogeneity 17 with regard to spatial structures and prey species abundance (Bambaradeniya et al., 2004). When ample prey species are available structural complexity is more important in determining predator success. Zhang et al. (2006) studied the stability of three species population model consisting of an endemic prey (bird), an alien prey (rabbit) and an alien predator (cat). It may be pointed out here that all the above studies are based on the traditional prey dependent models. Recently, it has been observed that in some situations, especially when a predator have to search for food and have two different choice of food, a more suitable predator-prey theory should be based on the so-called ratio-dependent theory, in which the per-capita growth rate should be function of the ratio of prey to predator abundance, and should be the so-called predator functional response (Abrams and Ginzburg, 2000; Akcakaya et al., 1995; Arditi et al., 1991; Arditi and Saiah, 1992). Kesh et al. (2000) proposed and analyzed a mathematical model of two competing prey and one predator species where the prey species follow Lotka-Volterra dynamics and predator uptake functions are ratio dependent. They derived conditions for the existence of different boundary equilibria and discussed their global stability. They also obtain sufficient conditions for the permanence of the system. Hsu et al. (2001) studied the qualitative properties of a ratio dependent predator-prey model. They showed that the dynamic outcome of interactions depends upon parameter values and initial data. Three general forms of functional response are commonly used in ecological models: linear, hyperbolic, and sigmoidal. How predators respond to changes in prey availability (functional response) is an issue of particular importance. There is evidence from several models that the type of functional response specified can greatly affect model predictions (Gao et al., 2000; Kar and Chaudhury, 2004).Many authors (Rijn and Sabelis 1993; Rijn 18 and Tanigoshi 1999; Eubanks and Denno 2000; Srinivasu and Prasad 2011) studied that the alternative food has no dynamics of its own, that is, alternative food is always available in constant amounts, unaffected by consumption. This simplification is justified for many arthropod predators because they can rely on plant-provided alternative food sources such as pollen or nectar, the availability of which is unlikely to be influenced by the predator's consumption. The theory of harvesting is important in natural resource management and bioeconomics. Most species have a growth rate which more or less maintains a constant population equal to the carrying capacity of the environment K (this of course depend on the population). In this case the growth and death rates are nearly equal. The harvesting of species affects their mortality rates and if the harvesting is not too much the population will adjust to a new equilibrium X * K . It has been evident that there is need to develop ecologically acceptable strategies for harvesting any renewable resources such as fish, plants, animals etc. It is interesting to note that even if the excess harvest does not threaten extinction, it can cause damage to the resource in the long run. Massive fruit collection from the forest has an adverse effect on regeneration. The problem then is to determine a strategy which ensures steady harvest year after year without a progressive decline in the abundance of the resource. The problem here is how to maximize the sustainable yield (SY) by determining the population growth dynamics so as to obtain a harvesting rate which keeps the population at its maximum growth. We consider a logistic population growth model in which the mortality rate is enhanced by harvesting; by a term that is proportional to the existing population N . We consider a logistic growth 19 X model F X rX 1 , where r is the intrinsic growth rate and K the carrying K capacity of the environment. Considering the effort E of harvesting, we assume that the harvest is proportional to the stock level as well as the effort i.e. h qEX , where q is the constant of proportionality called the catchability constant. The effort is measured in man days. If grass is cut with strokes of a sickle, the harvest depends on the number of strokes E on the grass density. If the effort is constant then the harvest as a function of the stock is a straight line passing through the origin. The intersection of the line or the growth curve gives the sustainable yield. The net growth rate after harvest is given by dX X rX 1 qEX . There are numerous studies on the effects of harvesting on dt K population growth. In the context of predator-prey interaction, some studies that treat the populations being harvested as a homogeneous resource include those of Dai and Tang, 1998; Myerscough et al., 1992; Chaudhuri (1996) and Leung (1995). Theoretical ecology remained silent about the astonishing array of dynamical behaviors of three species models for a long time. Of course, the increasing number of differential equations and the increasing dimensionality raise considerable additional problems both for the experimentalist and theoretician. Freedman and Waltman (1984) considered three level food webs – two competing predators feeding on a single prey and a single predator feeding on two competing prey species. They obtain criteria for the system to be persistent. Kar and Chaudhuri (2004) considered a two-prey one-predator harvesting model with interference. The model is based on Lotka-Volterra dynamics with two competing species which are affected not only by harvesting but also by the presence of a 20 predator, the third species. Optimal harvesting policy and the possibility of existence of a bioeconomic equilibrium is discussed. Dubey and Upadhyay (2004) proposed a two predator one prey system with ratio dependent predator growth rate. Criteria for local stability, instability and global stability of the non-negative equlibria are obtained. They also discussed about the permanent co-existence of the three species. Braza (2008) considered a two predator; one prey model in which one predator interferes significantly with the other predator is analyzed. The analysis centers on bifurcation diagrams for various levels of interference in which the harvesting is the primary bifurcation parameter. Bioeconomic modelling of the exploitation and management of biological resources, like fisheries and the techniques associated with the exploitation of these resources have been discussed by Clark (1990) in models of single species fisheries submitted to harvest, but it is difficult to construct and study a realistic model of a multispecies interaction because the model may not be analytically tractable. It is also difficult to determine the optimal harvest policy of a n-dimensional model having more than one control variable. The studied problem deals with the sustainable exploitation of two interacting species, one is the prey and the other is the predator involving combined or nonselective harvesting of the two populations considering that the fishing effort is the same for both species by using the Schaefer’s hypothesis (CPUE) (Schaefer, 1957). The dynamical behavior of the exploited model has already been studied. The optimal policy of exploitation is derived using Pontryagin’s maximal principle (Pontryagin et al.,1964). On the other hand, the functional response of predators or consumption rate function refers to the change in the density prey attacked per unit time per predator as prey density 21 changes. In most predator–prey models considered in the ecological literature, the predator response to prey density is assumed to be monotonic increasing; this inherent assumption means that the more prey population in the environment the better for the predator population. The problem of harvesting of competitive species has drawn the attention of bio – economists. Bhattacharya and Begum (1996) obtained the feasible dynamic equilibrium to obtain a limit of cost per unit effort and maximum value of effort for the joint harvesting of a logistic growth model of two competitive systems. Kar et al. (2006) has studied the bio-economic model of a ratio – dependent prey – predator system with optimal harvesting. They proved that optimal equilibrium populations lead to a situation where total user’s cost of harvest per unit effort equals the discounted value of the future profit. Kar et al. (2010) proposed a biological economic model based on a prey-predator dynamics where prey species are continually harvested and predation is considered type II functional response. Palma and Olivares (2012) have studied a predator-prey model with Allee effect and sigmoid functional response. In general a dynamic model consists of a biological (or biophysical) model that describe the behavior of a living system and an economic model that relates the biological system to market prices, resource and institutional constraints. Bio-economic models often contain a single equation to represent biological process. Nowadays, the biological resources in the prey – predator ecosystem are commercially harvested and sold with the aim of achieving economic interest. However, these models are less realistic. To make model realistic one should include some of the past states of these systems; ideally a real system may modeled by differential equations with time-delays. Time-delays occur so often in almost every 22 situation, that to ignore them is to ignore reality. The time delay is the inherent property of the dynamical systems and plays an important role in almost all branches of science and particularly in the biological sciences (e.g., population dynamics, epidemiology, etc.) see (Yan and Zhang, 2008; Yang, 2009; Meng et al. 2011). The importance derives from the fact that many of the phenomena around us do not act instantaneously from the moment of their occurrence. For example, a change in the resources or environment does not affect the survival of existing populations immediately. There is always a time lag between the moment an action takes place and its effect is observed. In ecology, more realistic models should include some of the past states, i.e., a real system should be modeled by differential equations with time delays. Kuang (1993) mentioned that animals take some time to digest their food before further activities and responses take place and hence any model of species dynamics without delays is an approximation at best. Now it is beyond doubt that in an improved analysis, the effect of time-delay due to the time required in going from egg stage to the adult stage, gestation period, et cetera, has to be taken into account. Detailed arguments on the importance and usefulness of time-delays in realistic models may be found in the classical books of Macdonald, 1989; Gopalsamy, 1992; and Kuang, 1993. Recently, it is of interest to investigate the possible existence of chaos in biological population. The subjects of chaos and chaos control are growing rapidly in many different fields such biological systems, structural engineering, ecological models, aerospace science, and economics .Food chain modeling provides challenges in the fields of both theoretical ecology and applied mathematics. Determining the equilibrium states and bifurcations of equilibria in a nonlinear system is also an important problem in 23 mathematical models. Two-species continuous time models have only two basic patterns: approach to equilibrium or to a limit cycle. In contrast, simple discrete time models of even a single species can exhibit chaotic behavior. But research of the last 25 years (Gilpin, 1979; Gucken-Heimer and Holmes, 1983; Schaffer, 1985) demonstrates the very complex dynamics that can arise in model systems (in continuous time) with three or more species. Several of the early mathematical investigations of chaos were of ecological models (May, 1974). An investigation by Gilpin (1979) showed that a system of one predator and two competing prey can exhibit chaotic behavior. Schaffer and Collaborators (1986 a) have been especially persuasive in their view that chaos may be a much more important phenomenon than ecologists had earlier believed. May (1974) argued that the reason why field ecologists have not been able to get any reliable proof of existence of chaos is that they have preoccupied notions and that they are observing what they want. It was argued by many authors that poor data quality (short and noisy character of the time-series) makes the techniques of nonlinear dynamics an unsuitable tool for the analysis of ecological data, and thus, leads to failure of such attempts. One common hindrance (in the study of nature populations) to understand the underlying dynamical process has been the non-availability of data of suitable length and precision. Another difficulty is that there is no method which can fix these parameters a priori .On the other hand, the existence of chaos in almost all the physical systems motivates one to critically study the same in natural populations. Hastings and Powell (1991) have studied three species food chain with double Holling type II functional responses; they found that there is ‘tea-cup’ attractor in the system. Gakkhar and Naji (2003) investigated a three species ratio dependent food chain (Holling- Tanner Type) model, they also found that 24 there is ‘tea-cup’ attractor in the system .Wang et al., (2005) studied the three food chain model with impulsive effects on top predator and found that the impulsive perturbations bring to these simple systems chaos solutions. Wang and Pang (2008) studied a model of a hybrid ratio dependent three species food chain model and they also found chaotic attractor in the system. Ranjit Kumar Upadhyay (1997) studied why chaotic dynamics are rarely observed in natural populations. Many papers studied a predator-prey model with the Michaelis–Menten functional response. Naji and Balasim (2007) studied dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response and investigated bifurcation and chaotic behavior at conversion rate of prey from predator The control of chaotic dynamics with various ecological factors, namely, migration, predation, conversion, refuge, omnivory, habitat-heterogeneity is the most challenging task in such studies and thus it receives a good deal of attention from various scientists .Vandermeer (1993) investigated a three species ratio-dependent food chain model and show that there is chaotic attractor. Hastings and Powell (1991) have studied three species food chain with double Holling type II functional responses,they found that there is ‘tea-cup’ attractor in the system. Gakkhar and Naji (2003) investigated a three species ratio dependent food chain (Holling- Tanner Type) model, they also found that there is ‘tea-cup’ attractor in the system. In the last few years, researchers have shown keen interest to investigate the direction and stability of Hopf bifurcation arising from a delay-induced neural network. Study of the bifurcation and its stability, however, is relatively new in the ecological system. Recently, Song and Wei, (2004); Yang and Tian, (2005); Qu and Wei, 2007; Celik (2008, 2009) have studied the direction and stability of Hopf bifurcation of delay-induced ecological systems. In most of the studies, delay 25 occurs in a first degree term. Sun et al. (2006) investigated the direction and stability of a delay-induced eco-epidemiological system with Type I response function, where delay occurred in the term of degree two. 1.4 SUMMARY AND SCOPE OF WORK DONE Mathematical modeling as a tool in the study of population dynamics has a long and diverse history, spanning at least three centuries. Although multitude of models has been put forward and many challenging problems have been solved, the endeavor to explain and formulate general principles underlying the dynamics of populations in space and time is far from over. The pros and cons of particular models and concepts are still subject of lively debates, which makes population dynamics a wide-open field for researchers interested in mathematical modeling. This thesis attempts to provide new insight into some population-dynamical problems and also proposes a new perspective on certain models of ecological communities. The specific research questions have been classified according to the level of modeling complexity on which they are situated and the organization of this thesis as outlined below intends to reflect this classification. Chapter 2. In this chapter, a nonlinear mathematical model is proposed and analyzed to see the effect of harvesting on food chain model. We have also studied the effect of the habitat complexity and gestation delay on the stability of a food chain model. It is observed that there are stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. It is observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the gestation period exceeds some critical value. We have studied a delay-induced food-chain model in presence of habitat complexity of space. Using stability theory of differential equations, 26 we have obtained conditions for the existence of different equilibria and discussed their stabilities. Using differential inequality, conditions have been obtained under which system persists. Chapter 3. In this chapter we have discussed how additional food to predator changes the structure of prey and predator system. Criteria for local stability, instability and global stability of the equilibria are obtained. Using differential inequality, we have obtained sufficient conditions that ensure the persistence of the system. The presence of additional food incorporated in the shape of the functional response, it may also increase equilibrium predator densities, which, in turn, will lead to a reduction in equilibrium prey density. The effects of additional food may help to devise strategies for biological control involving a supply of additional food. To complement our analytical findings numerical simulations are carried out. Chapter 4. In this chapter two time delays ( 1 and 2 ) are considered in the model to describe the time that juveniles of prey and predator take to mature. The stability analysis of the proposed model is carried out. The Hopf bifurcation conditions of the interior equilibrium point are established. We have obtained the sufficient conditions for stability of the interior equilibrium point. Moreover, the conditions for the existence of Hopf bifurcation of the cases 1 0, 2 0 and 1 2 0 , respectively are determined. We have found critical value of delay and observed that under certain conditions, system bifurcates from the steady states of the system at some critical values of 1 , 2 . It is concluded from analysis that the stability properties of the system could switch with the time delay that is incorporated on different densities in the model. 27 Chapter 5. In this chapter, we have studied the effect of harvesting on dynamics of preypredator model with Holling Type III functional response. A combined harvesting policy for prey and predator species is discussed by using the Pontryagin’s Maximum principle. Using differential inequality, we have obtained sufficient conditions that ensure the persistence of the system. It has been shown that the total user’s cost of harvest per unit effort is equal to the present value of marginal revenue of effort at the optimal equilibrium level. It has also been noted that the increase in discount rate decreases the economic rent and even it tend to zero if the discount rate tends to infinity. Finally, numerical simulations is done to support the analytical findings. Chapter 6. This chapter aims to study the effect of Harvesting on predator species with time-delay on a prey-predator model with Holling type-IV functional response. Two delays are considered in the model to describe the time that juveniles of prey and predator take to mature. The dynamics of the system are studied in terms of local and Hopf bifurcation analysis. We have concluded from our analysis that the stability properties of the system could switch with the Harvesting with time delay that is incorporated on different densities in the model. Chapter 7. In this chapter, we have considered Beddington-De Angelis response in a food chain model and investigated the Hopf bifurcation and Chaos of the model. The mortality rate of predator has ability to control the chaotic oscillations. We have also studied the effect of Harvesting on prey species. Our analysis shows that, using the parameter mortality rate as control, it is possible to break the stable behavior of the 28 system and drive it in an unstable state. Also it is possible to keep the levels of the populations in a stable state using the above control. Chapter 8. In this chapter we have formulated and analyzed prey-predator model with one prey dispersal in a two patch environment. Each patch is supposed to be homogeneous. Patch two constitutes a reserved area of prey and no predation is permitted in this zone whereas patch one is an open access predation zone. Criteria for local stability, instability and global stability of the non-negative equilibria are obtained. Using differential inequality, we have obtained sufficient conditions that ensure the persistence of the system. We have found that prey reserve and team approach are most important factor of ecology. A deep insight into mathematical biology has opened up different areas of research in the last few decades. Mathematical modeling of the dynamics of ecological system has played a pivotal role in attempting an inter-relationship between a predatorprey system in an environment and the consequences of interaction of interspecies as well as intraspecies. A great deal of mathematical modeling has been accompanied by a rich theory of differential equations. For example in chapter 2, if we assume that predators are impulsively released with some impulsive period, then model will be more realistic. We leave it for future studies. Similarly a model in chapter 3 and 6 can be modified. In chapter 5, we have used ‘tax’ as only control instrument, but there are other instruments also like a catch- quotes and subsidies etc to apply to the problem of populations from over exploitations. All these factors may be used in future research. 29