**Predator-Prey Model: Foxes and Rabbits**

Adapted by Dr. Sarah from Holly Hirst's paper.

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**with(DEtools): with(plots):**

We begin by looking at a predator-prey model of foxes (the predators) and rabbits (the prey). Let's assume that the initial population of

rabbits is 4, and the initial population of foxes is 2. Let's say that we are given the following models for the growth/death rate of the populations as differential equations:

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**eqn1:=diff(prey(t),t)=0.7*prey(t) - 0.3*prey(t)*predator(t);**

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**eqn2:=diff(predator(t),t)=0.08*prey(t)*predator(t) - 0.44*predator(t);**

Let's plot the solutions to these models with the given initial conditions.

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**a:=DEplot([eqn1,
eqn2],[prey(t),predator(t)],t=0..50,stepsize=0.1,[[prey(0)=4,predator(0)=2]],scene=[t,prey(t)],linecolor=black,labels=[time,population],thickness=1):**

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**b:=DEplot([eqn1,eqn2],[prey(t),predator(t)],t=0..50,stepsize=0.1,[[prey(0)=4,predator(0)=2]],scene=[t,predator(t)],linecolor=green,labels=[time,population],thickness=1):**

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**display(a,b);**

Answer the following questions on a separate piece of paper...

**Question 1**
: What happens to the foxes and the rabbits?

Here is a copy of the Maple code for the remaining questions.

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**eqn1:=diff(prey(t),t)=0.7*prey(t) - 0.3*prey(t)*predator(t);**

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**eqn2:=diff(predator(t),t)=0.08*prey(t)*predator(t) - 0.44*predator(t);**

Let's plot the solutions to these models with the given initial conditions.

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**a:=DEplot([eqn1,
eqn2],[prey(t),predator(t)],t=0..50,stepsize=0.1,[[prey(0)=4,predator(0)=2]],scene=[t,prey(t)],linecolor=black,labels=[time,population],thickness=1):**

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**b:=DEplot([eqn1,eqn2],[prey(t),predator(t)],t=0..50,stepsize=0.1,[[prey(0)=4,predator(0)=2]],scene=[t,predator(t)],linecolor=green,labels=[time,population],thickness=1):**

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**display(a,b);**

**Question 2**
: Describe the effect of increasing the prey birth rate (by changing the .7 rate constant). Is this what you expected to happen? Explain.

**Question 3**
: Describe the effect of increasing the prey death rate (by changing the .3 rate constant). Is this what you expected to happen? Explain.

**Question 4**
: The predator birth rate (of 0.8) is currently much less than the prey birth rate. What effect does having the same birth rate for both species have on the solution?

**Question 5**
: Experiment with the initial popultions for the predators and the prey. Are the behaviors of the popultions over time sensative to small changes to the initial populations? Can you force the extinction of one of the species? both?

**Question 6**
: Suppose that we harvest some proportion of both the predators and the prey. Experiment with incorporating harvesting terms into both of the equations. What happens when you change the harvesting rate? Can you harvest and have the populations increate rather than decrease? Is this what you expected? Explain.