Population Equilibrium Solutions Differential Equations

[hocuspocus102]

Homework Statement

Consider the following model for the populations of rabbits and wolves (where R is the population of rabbits and W is the population of wolves).

dR/dt = .05R(1-.00025R)-.00109375RW

dW/dt = -.04W+8e-05RW

Find all the equilibrium solutions:
In the absence of wolves, the population of rabbits approaches ____
In the absence of rabbits, the population of wolves approaches ____
If both wolves and rabbits are present, their populations approach r = ____ and w = ____

Homework Equations

The Attempt at a Solution

I don't really have any idea how to do this except that I know in the absence of rabbits the population of wolves approaches 0 because that's common sense. So any suggestions would be helpful! thanks!

 

[mighty2000]

Thank you for posting this interesting problem. I am always excited to see others engaging in mathematical modeling to understand real-world phenomena. In order to find the equilibrium solutions for this model, we can set the derivatives equal to 0 and solve for R and W. This will give us the values of R and W at which the populations of rabbits and wolves will remain constant over time.

Equilibrium solutions:

1) In the absence of wolves (W = 0), the population of rabbits approaches R = 0. This is because there are no predators to limit their growth and they will continue to reproduce until they reach their carrying capacity.

2) In the absence of rabbits (R = 0), the population of wolves approaches W = 0. This is because wolves rely on rabbits as their main source of food and without them, their population cannot be sustained.

3) If both wolves and rabbits are present, their populations will approach the equilibrium values of r = 20000 and w = 5000. This can be found by solving the equations simultaneously. These values represent the point at which the growth rate of rabbits and wolves are equal and their populations will remain stable.

I hope this helps you better understand the dynamics of this population model. Keep up the good work in your studies!