Introduction: Answer to Example4

Example: Consider the following predator-prey model:

.

1.

Does x(t) denote the predator population or the prey population? Justify your answer.

2.

Find all equilibrium points of the system.

3.

Suppose the prey population becomes extinct while the predator population is still positive. Describe the long-term behavior of the predator population.

4.

Suppose the predator population becomes extinct while the prey population is still positive. Describe the long-term behavior of the prey population.

5.

Describe the long-term behavior of the system when the initial populations are given by

.

Answer:

1.

Recall that in the absence of prey, the population of predators decrease. It is clear that if y=0, then we have x'(t) = -x, meaning that x(t) will decrease. While, if we set x=0, we have y'=2y(1-y/2). Here we recognize the logistic equation which implies that y should get closer and closer to the carrying capacity y=2. Conclusion x represents the predator population.

2

The equilibrium points are solutions of the system

.

Since,

,

we have the following two cases:

• Case 1: x=0, then the second equation gives

  .

  Hence, we have two equilibrium points

  .

• Case 2: y=10/9, then the second equation gives

  ,

  which gives

  .

  Hence, one equilibrium point (in this case)

  .

Finally, the system has three equilibrium points

3.

It will become extinct.

4.

It will approach the carrying capacity y=2.

5.

Using the answer to 2, we see that the initial populations correspond to an equilibrium point. Therefore, both populations will remain unchanged

.

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