Differential Equation - Bifurcation

[cse63146]

Homework Statement

The following model describes a fox population:

$$\frac{dS}{dt} = kS(1 - \frac{S}{N})( \frac{S}{M} - 1)$$

a) at what value of N does a bifurcation occur?
b) How does the population behave if the parameter N slowly and continouly decreases towards the bifurcation value?

Homework Equations

The Attempt at a Solution

a) Bifurcation occurs when $$\frac{dS}{dt} = 0$$ and in terms of N, it would be when N = S.

b) as N appraoches the bifurcation point, the population would also deacrese until it reaches S, at which point the population would be 0 (based upon the model)

Is that all?

 

[Mark44]

Homework Statement

The following model describes a fox population:

$$\frac{dS}{dt} = kS(1 - \frac{S}{N})( \frac{S}{M} - 1)$$

a) at what value of N does a bifurcation occur?
b) How does the population behave if the parameter N slowly and continouly decreases towards the bifurcation value?

Homework Equations

The Attempt at a Solution

a) Bifurcation occurs when $$\frac{dS}{dt} = 0$$ and in terms of N, it would be when N = S.

b) as N appraoches the bifurcation point, the population would also deacrese until it reaches S, at which point the population would be 0 (based upon the model)

Is that all?

I don't know if it's relavant, but dS/dt = 0 also when S = M or when S = 0. Otherwise your answer looks fine. You didn't provide any information about what S, N, and M represent, so I don't know if these enter into the bifurcation business.

Your answer for b seems reasonable, based on the limited information provided.

 

[cse63146]

It also says "Suppose that the parameters M and K remain constant over the long term, but as more people move into the aream, the parameter N (carrying capacity) deacreses. Other than that, that's everyting.