Systems: Project 1

Systems of Differential Equations: Models of Species Interaction

Project 1: Predator-Prey Models with Harvesting

Project Requirements

1. Biographical Information

Your project should contain biographical sketchs of the Alfred Lotka and Vito Volterrra. Include any relevant information concerning their work with the predator prey models.

2. Proportional Harvesting

The following is a list of some things that should be included. Note that a satisfactory report will include more than the listed items.

Give a brief summary of the predator-prey model.
Explain how the equations
dx/dt = ax - bxy - qx

dy/dy = -cy + pxy - ry
incorporate proportional harvesting into the predator-prey model. What does "proportional" mean in this context?

Discuss the meaning of the constants q and r in practical terms using specific examples such as fishing with nets, traping or pesticide use.

Find formulas for the coordinates of the equilbrium point for this system in terms of the constants a, b, c, p, q and r.

Let a = 1, b = .03, c = .4, p= .01, r = 0.5. Then, experiment with values of q ranging from 0 to a. Plot representative solution curves for these systems.

Pick different values for a, b, c, p and repeat your experiments with a range of values for q. Investigate enough different cases to answer the following questions:

What happens to the solution curves as q increases?
What happens when q is greater than or equal to a? Does this make sense in terms of the equations?

Let a = 1, b = .03, c = .4, p= .01, q = 0.5. Then, experiment with a range of values of r. Plot representative solution curves for these systems.

Pick different values for a, b, c, p and repeat your experiments with a range of values for r. Investigate enough different cases to answer the following:

What happens to the solution curves as r increases? Does this make sense in terms of the equations?

Describe what happens to the equilibrium values for x(t) and y(t) as q increases? As r increases?

Even though x(t) and y(t) cannot be calculated explicitly, we can show that the average values of x(t) and y(t) are are the equilibrium values for the system.

Below is the skeleton of a proof that the average value of y(t) is the equilibrium value, (a-q)/b, you found earlier. Flesh out the proof to include a justification for each step.

Suppose the period of the solutions x(t) and y(t) is T. (In particular, this means that x(T) = x(0) and y(T) = y(0).

(i) Explain why the average value of y(t) is given by the expression

Your goal is to show that y_avg = (a - q)/b.

(ii) First show that (dx/dt) / x = a - q - by(t).

(iii) Now show that

(iv) Explain why

iv) Conclude that

Therefore,

Give a similar argument to show that the average value of x is the equilibrium value of x.
You have shown that the average values of any solutions are the same as the coordinates of the equilibrium point for the system. Use this fact to describe the effects that proportional harvesting has on the predator and prey populations. In particular, does such a model reflect the increase in the percentage of sharks caught by certain Mediterranean fisheries during World War I?

3. Constant Rate Harvesting

Consider the predator - prey model with harvesting given below.

dx/dt = ax - bxy - q

dy/dy = -cy + pxy - r

Discuss how this model differs from the proportional harvesting model. Your discussion should include, but not be limited to, the following:

Describe the meaning of q and r in practical terms.
How does the insertion of q and rinto the predator-prey system affect the solutions?
Let a = 1, b = .03, c = .4, p = .01, and experiment with values of q and r ranging from 0 to 5, and plot solution curves to this system.
Experiment with harvesting only one of the species at a time, and describe your results.
Change the values of a, b, c, and p and repeat your experiment. Compare the solution curves for this model to the solution curves obtained for the proportional harvesting model.
Find the equations of the nullclines and the equilibrium point(s) for this system using specific values of a, b, c, p, q, and r.

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