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LURE 2009 SUMMER PROGRAM John Alford Sam Houston State University Mathematical Modeling Mathematical modeling is an area of applied mathematics concerned with describing and/or predicting real-world system behavior Examples of real-world systems – – – – an object moving in a gravitational field stock market fluctuations predator-prey interactions nerve impluse propagation Mathematical Modeling Many familiar concepts in mathematics evolved from the desire to answer basic scientific questions using mathematical models – e.g. Newton used calculus to describe and predict how an object of a certain mass will move in response to applied forces Mathematical Modeling Why model? Simplification of a complex system Ease of manipulation (simulation as opposed to experimentation) Gains in understanding of the system – helps to formulate new hypotheses – aids in design of new experiments Mathematical Modeling There is no perfect model! – a model should balance accuracy, flexibility, and cost A general rule of thumb – increasing accuracy decreases flexibility and increases cost Goal – construct a “sufficiently” accurate model with high flexibility and low cost Mathematical Modeling No model is perfect The problem may never be “completely” solved – you need to get used to this if you are going to be doing research Mathematical Modeling Step 1: find a real-world problem – obtain data and general knowledge perform experiments search the literature (journals, books, etc.) – make simplifying assumptions and neglect some complexity Mathematical Modeling Step 2: formulate the model – research the literature for other models (don’t re-invent the wheel) – decide on the model type and form equations- algebraic, differential, integral, difference, etc. deterministic (non-random) or stochastic Mathematical Modeling Step 2: formulate the model (cont.) – model will include variables represent system parts parameters influence variables but are not influenced by them (typically) equations describe behavior and relate variables and parameters Mathematical Modeling Step 3: analyze the model – run computer simulations – apply mathematical theories Step 4: interpret and verify the results to explain or predict – iterate the model for improved accuracy Mathematical Modeling Process real world problem formulate modify report, explain, predict mathematical model analyze interpret and verify mathematical results Population Ecology Unstructured population models – populations are treated as “homogeneous green gunk” (Kot, Elements of Mathematical Ecology) Structured population models – Include effects due to age, spatial location, genetic variation, etc. Population Ecology Unless otherwise stated, we will assume for this discussion that our population models are unstructured. Population Ecology The six axioms of Turchin – Conservation – Individualism – Upper density bound – Mass action – Biomass conversion – Max physiological rates Population Ecology We will now use the first two axioms to construct the exponential growth model for population growth Exponential Growth Model Population density dN Rate of change of population dt N (t ) Per capita rate of change of population 1 dN N dt Exponential Growth Model Conservation – The number of organisms in a population can change only as a result of births, deaths, immigrations, and emigrations Exponential Growth Model Model Using Conservation – The rate of change of a closed population (i.e. no immigration or emigration) is the number of births minus the number of deaths dN BD dt Exponential Growth Model Individualism – Population mechanisms are individual based. That is, all population processes affecting population change (e.g. births, deaths, movement) are a result of what happens to individuals. Exponential Growth Model Model Using Conservation – birth and death rates can be expressed in per capita form B N per capita birth rate D N per capita death rate Exponential Growth Model Model using conservation – Assume the per capita birth rate and the per capita death rate are constant equal to b and d respectively dN B D bN dN dt Exponential Growth Model Define the intrinsic rate of growth (net per capita growth rate) r bd so that dN rN dt Exponential Growth Model Some Mathematics – Solution by separation of variables Exponential Growth Model Separate variables dN dN rN rdt dt N Exponential Growth Model Integrate 1 N dN rdt to get ln N rt C Exponential Growth Model Malthus’ equation (1798) N N 0e where rt N 0 N (t 0) is the initial population density Exponential Growth Model Exponential population growth b d r bd 0 For positive initial populations, there is no limit on population size as time increases lim N0e rt t Exponential Growth Model Exponential population decay b d r bd 0 All initial populations (eventually) become extinct. lim N 0e 0 rt t Exponential Growth Model per capita growth rate vs. N (r>0) Exponential Growth Model growth rate vs. N Exponential Growth Model population growth vs. t Exponential Growth Model Equilibrium solutions are constant solutions N 0 is an equilibrium for * dN rN dt check d d * * [ N ] rN [0] r[0] dt dt Exponential Growth Model Equilibria may be stable or unstable – stability “means” any small perturbation results in a return (over time) to the equilibrium – instability “means” some small perturbation will not result in a return (over time) to the equilibrium Equilibria and stability may depend on parameter(s) in the equation(s) Exponential Growth Model dN rt rN N N 0 e dt – without emigration or immigration, populations that start at 0 stay at 0 – if per capita growth rate is positive, small perturbations from 0 result in large population changes r 0 N 0 is unstable * Exponential Growth Model dN rt rN N N 0 e dt – if per capita growth rate is negative, small perturbations from 0 result in population sizes returning to 0 r 0 N 0 is stable * Population Ecology The six axioms of Turchin – Conservation – Individualism – Upper density bound – Mass action – Biomass conversion – Max physiological rates Population Ecology Limitations exponential growth – Constant per capita growth rate yields unlimited growth – Deterministic nature of the model ignores random (stochastic) effects which are (particularly) important at small population sizes – Model is unstructured and ignores temporal and spatial variability Population Ecology Factors that regulate growth of populations: biotic vs. abiotic – competition within and between species (biotic) – variation in the weather (abiotic) Population Ecology A.J. Nicholson, 1933, The Balance of Animal Populations, Journal of Animal Ecology – population densities are regulated by biotic factors such as competition and disease which affect high-density populations more than low-density populations Population Ecology The six axioms of Turchin – Conservation – Individualism – Upper density bound – Mass action – Biomass conversion – Max physiological rates Logistic Growth Model Per capita growth rate is positive for small population densities Per capita growth rate is negative for large population densities Per capita growth rate decreases as population increases (competition for resources including food, space) Logistic Growth Model per capita growth rate vs. N Logistic Growth Model Per capita growth rate is a linear function of the population density 1 dN N r 1 N dt K Here r>0 and K>0 are parameters – r=intrinsic growth rate – K=carrying capacity Logistic Growth Model Growth rate is quadratic function of N dN N rN 1 dt K Logistic Growth Model growth rate vs. N Logistic Growth Model Exercise 1: solve to get the Verhulst (1838) model of population growth N0 K N rt N 0 ( K N 0 )e where N 0 N (t 0) Logistic Growth Model For positive initial populations, the limiting population is carrying capacity (K) N0 K lim K rt t N ( K N )e 0 0 WHY (mathematically and biologically)? Logistic Growth Model N vs. t N0 K Logistic Growth Model N vs. t N0 K Logistic Growth Model Equilibria There are two equilibria for the logistic model – carrying capacity N K * – zero population density N 0 * Logistic Growth Model Equilibria Multiply right side dN N N rN 1 rN r dt K K 2 Assume 0 N / K 1 – to get dN rN dt Logistic Growth Model Equilibria Because r>0, previous slide shows that a small perturbation of N away from zero will grow exponentially (approximately) Zero equilibrium is unstable Logistic Growth Model Equilibria Let N K N K Substitute and do algebra dN d r 2 N rN 1 r dt dt K K Logistic Growth Model Equilibria Assume very small ( N very close to K ) to get d r dt Logistic Growth Model Equilibria Because r>0, previous slide shows that a small perturbation of away from zero will decay exponentially back to zero decays to zero N decays to K K is a stable equilibrium Population Ecology Interacting Species: aphid infestation Population Ecology A common way ecologists classify species interactions between two species is by denoting positive, negative, or zero (neutral) pairings. (+,+), (+,-), etc. Population Ecology A consumer-resource or trophic interaction is a (+,-) pairing between two species Examples of consumer-resource interactions – predator-prey (e.g. fox and rabbit) – herbivore-plant (e.g. leaf-mining fly and hydrilla) Population Ecology The last three axioms of Turchin – Mass action – Biomass conversion – Max physiological rates These may be used to derive models of consumer-resource interactions Population Ecology Mass action – At low resource densities the number of resource individuals encountered and captured by a single consumer is proportional to resource density capture rate aN as N 0 (N is resource density, a is constant) Population Ecology Biomass conversion – The amount of energy that an individual consumer can derive from captured resources to be used for growth, maintenance, and reproduction, is a function of the amount of captured biomass Population Ecology Maximum physiological rate – No matter how high the resource density is, an individual consumer can ingest resource biomass no faster than some upper limit imposed by its physiology (e.g. the size of its gut) Population Ecology Mass action and max physiological Population Ecology Ecologists call the functional rate at which each predator captures prey as it depends on prey density the functional response. C.S. Holling (ca 1960) described three types of functional response relations Each of Holling’s functional response relations obey Turchin’s mass action and max physiological axioms Population Ecology Holling’s functional response Type II aN f (N ) b N (a, b constants) Population Ecology Population Ecology Holling’s functional response Type II aN f (N ) b N (a, b constants) Population Ecology Holling’s functional response Type III 2 aN f (N ) 2 2 b N (a, b constants) Population Ecology LURE Students – Analyze the following two models for a predator-prey interaction. Treat the number of predators P as a (constant) bifurcation parameter (for now). – Consider r and K to be fixed (for now) – Interpret your results biologically Population Ecology Model 1 dN N rN 1 NP dt K Model 2 dN N N rN 1 P dt K 1 N Population Ecology The number of predators is not (typically) fixed but changes in time. This requires a separate differential equation to describe predator density. Lotka-Volterra Model dN rN cNP dt dP bNP mP dt A classic predator-prey model due to Lotka and Volterra (ca 1925) Lotka-Volterra Model dN rN cNP dt dP bNP mP dt r, c, b, m are all positive constants N is prey (resource) density, P is predator (consumer) density Lotka-Volterra Model Lotka and Volterra model has oscillatory solutions (why biologically?) Lotka-Volterra Model The first equation describes the rate of change of prey (resource) density dN rN cNP dt Let’s consider each term Lotka-Volterra Model The first term shows that in the absence of predation the prey grow exponentially dN rN dt (if P=0) Lotka-Volterra Model The second term describes the loss (minus sign) of prey due to predators dN rN cNP dt The loss is proportional to both the number of prey and the number of predators (linear consumption rate) Lotka-Volterra Model The second equation describes the rate of change of predator (consumer) density dP bNP mP dt Let’s consider each term Lotka-Volterra Model The first term describes the gain of predators due to prey (equals the loss of prey due to predators) The second term shows that the predator population decreases exponentially in the absence of prey dP mP dt (if N=0) Rosenzweig MacArthur Model dN N cN rN 1 P dt K aN dP bN P mP dt a N A classic predator-prey model due to Rosenzweig and MacArthur (ca 1963) Population Ecology Can you describe qualitative differences in these two models?? Lotka-Volterra dN rN cNP dt dP bNP mP dt Rosenzweig-MacArthur dN N cN rN 1 P dt K aN dP bN P mP dt a N Population Ecology A plant-herbivore system can be thought of as a type of predator (=herbivore) and prey (=plant) system The Rosenzweig-MacArthur model has been used to describe plant-herbivore* dynamics where the variables become V=vegetation biomass N=herbivore density * the herbivore here is assumed to be a mammalian grazer Population Ecology The Rosenzweig-MacArthur model for plant-herbivore (mammal) system dV V cV rV 1 N dt K a V dV bV N mN dt a V Population Ecology Stability analysis of the RosenzweigMacArthur model for plant-herbivore system yields the paradox of enrichment (Turchin): As plant standing biomass (= K) is increased, the dynamics of the system become increasingly less stable (i.e. small parameter changes become more likely to result in large qualitative changes in the dynamics) Population Ecology Other models (Turchin) account for plant re-growth as there is a part of the plant that the herbivore typically does not consume (i.e. underground biomass) Population Ecology Still other models (EdelsteinKeshet) use dependent variables to describe the system in terms of plant quality (rather than plant density) and herbivore density Population Ecology LURE students Research (google) plantherbivore models Population Ecology LURE students Propose a plant herbivore model that will account for insect herbivory and plant quality variations (e.g. via fertilizer or sunlight variation)
