Rabbits

Rabbits, Foxes and Mathematical Modeling: 

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

Mathematical Modeling of Population Dynamics : 

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

Slide4: 

The growth rate depends on many things, such as Per capita food supply – call it “s” A minimum supply of food, say s0, is needed to sustain life Say growth rate is proportional to s – s0 Call the constant “a” the growth coefficient

Slide6: 

Is infinite growth realistic? Suppose population reaches saturation at x0 Say the growth coefficient is proportional to x0 – x We can interpret the x2 term as a number proportional to the average number of encounters between x individuals. Hence it measures a kind of social friction.

Foxes and Rabbits: 

Foxes and Rabbits

Slide9: 

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”

Slide10: 

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

The Lotka-Volterra (Predator-Prey) Equations: 

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

Slide13: 

Let’s introduce a saturation for the rabbit population:

Functional Response: 

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?

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Michaelis-Menten or Holling type II functional response

Slide16: 

Sigmoidal functional response

Slide17: 

Holling type III functional response

Slide18: 

Holling type IV or Monod-Haldane type functional response

Ratio-Dependent Theory: 

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 :

Spatial Dependence: 

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

Two Predators and One Prey: 

Two Predators and One Prey Defense switching Cross diffusion