Introduction: Answer to Example4
Example: Consider the following predator-prey model:
- Does x(t) denote the predator population or the prey population? Justify your answer.
- Find all equilibrium points of the system.
- Suppose the prey population becomes extinct while the predator population is still positive. Describe the long-term behavior of the predator population.
- Suppose the predator population becomes extinct while the prey population is still positive. Describe the long-term behavior of the prey population.
- Describe the long-term behavior of the system when the initial populations are given by
- 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.
- The equilibrium points are solutions of the system
we have the following two cases:
Finally, the system has three equilibrium points
- 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
Hence, one equilibrium point (in this case)
- It will become extinct.
- It will approach the carrying capacity y=2.
- Using the answer to 2, we see that the initial populations correspond to an equilibrium point. Therefore, both populations will remain unchanged
[First Order D.E.]
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