**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*'=2*y*(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)

.

**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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