Predator-Prey Model
A predator-prey model couples a prey population and a predator population , in the form and for positive constants ; the fixed points and their linearised stability classify the dynamics.
Consider a population of prey and predators with the equations
where are positive constants.
Consider a specific case with
The fixed points satisfy
There are three fixed points: , and .
We have
-
At , we have
The eigenvalues are and , with eigenvectors and .
Thus, it is a saddle point.
-
At , we have
The eigenvalues are and , with eigenvectors and .
Thus, it is a saddle point.
-
At , we have
The eigenvalues are found by solving
Since , the fixed point is a stable spiral.
At , we have , so the motion is counter-clockwise.
Now, we can sketch the overall phase portrait.
Interpretation of the terms
In the prey equation , the term models excess births over natural deaths, competition over scarce resources, and deaths due to predation. In the predator equation , the birth rate increases with predation while is the natural death rate. All of are positive constants.
Related
Stated in
- Example 9.14 (Predator-Prey Model)§9.5 Non-Linear Dynamical Systems