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Homework Help: Differential Equation - Bifurcation

  1. Mar 25, 2009 #1
    1. The problem statement, all variables and given/known data

    The following model describes a fox population:

    [tex]\frac{dS}{dt} = kS(1 - \frac{S}{N})( \frac{S}{M} - 1)[/tex]

    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?

    2. Relevant equations

    3. The attempt at a solution

    a) Bifurcation occurs when [tex]\frac{dS}{dt} = 0[/tex] 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?
  2. jcsd
  3. Mar 26, 2009 #2


    Staff: Mentor

    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.
  4. Mar 26, 2009 #3
    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.
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