1. Rabbits, Foxes and Mathematical Modeling Peter Pang University Scholars Programme and Department of Mathematics, NUS SMS Workshop July 24, 2004

2. Mathematical Modeling of Population Dynamics How does population grow? Denote the population by “x”. Suppose the population x(t) at time t changes to x + Δx in the time interval [t, t + Δt]. Then the growth rate is

4. The growth rate depends on many things, such as Per capita food supply – call it “s” A minimum supply of food, say s 0, is needed to sustain life Say growth rate is proportional to s – s 0 Call the constant “a” the growth coefficient

6. Is infinite growth realistic? Suppose population reaches saturation at x 0 Say the growth coefficient is proportional to x 0 – x We can interpret the x 2 term as a number proportional to the average number of encounters between x individuals. Hence it measures a kind of social friction.

8. Foxes and Rabbits

9. Let’s look at the fox population Let’s assume that the fox population doesn’t get really huge so that the issue of “population saturation” can be ignored Recall that the model for unlimited growth is Now, suppose that the only food for foxes is rabbits; then s is proportional to the population of rabbits Denote rabbit population by “y”

10. As for the rabbit population, let’s again assume we have unlimited growth while the rabbits are being eaten by the foxes -- we further assume that the number of rabbits eaten is proportional to the fox population

11. The Lotka-Volterra (Predator-Prey) Equations Population of foxes – x Population of rabbits – y where c, d, f, g are constant parameters

13. Let’s introduce a saturation for the rabbit population:

14. Functional Response Lotka-Volterra System (with unlimited growth for prey) The term p(y) measures the number of prey susceptible to each predator as the prey population changes -- this is known as the functional response p(y) = gy says that the number of prey for each predator is a constant proportion of the prey population What if the prey population really shoots up?

15. Michaelis-Menten or Holling type II functional response

16. Sigmoidal functional response

17. Holling type III functional response

18. Holling type IV or Monod-Haldane type functional response

20. Ratio-Dependent Theory Recall that functional response measures the number of prey susceptible to each predator as the prey population changes. We have seen various types of functional response functions of the type p(y). The ratio-dependent theory asserts that functional response should be dependent of the ratio of prey to predator (especially if the predator needs to search for the prey), i.e., instead of being just a function of y, p should be a function of y/x :

22. Spatial Dependence What can happen to a spatially inhomogeneous population? Diffusion Cross Diffusion Partial differential equations

23. Two Predators and One Prey Defense switching Cross diffusion