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Homework Help: Cubic Population Model with steady states

  1. Apr 15, 2012 #1
    Cubic Population Model with steady states !!

    I am unsure as what this question means:

    Consider the cubic population model: dN/dt = cN(N-k)(1-N) where c>0 and 0<k<1

    If the the initial populations is N_0 describe without proof the future of the population, distinguish the various cases on the size of N_0 relative to the steady states N_1, N_2 and N_3.


    Now I have found and classified the 3 steady states. But am not sure how to proceed. Solving the equation does not seem to be feasible so what do I do ?
     
  2. jcsd
  3. Apr 15, 2012 #2

    HallsofIvy

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    Re: Cubic Population Model with steady states !!

    I presume you have determined that the steady states are N=0, N= k, and N= 1.

    Now, look at what happens between those values.

    We can write the equation as dN/dt= (-1)(N- 0)(N- k)(N- 1).

    If N< 0, all three of those factors are negative. dN/dt is the product of 4 negative numbers so dN/dt is positive. N moves toward 0.

    If 0< N< k, N-0 is positive while the other two factors are still negative. dN/dt is the product of one positive and three negative numbers so dN/dt is negative. N moves down toward 0 (N= 0 is a "stable" equilibrium).

    If k< N< 1, both N- 0 and N- k are positive while N-1 is still positive. dN/dt is the product of two positive and two negative numbers so dN/dt is positive. N moves away from k toward 1. (k is an "unstable equilibrium".)

    Finally, if N> 1, all terms, except that original (-1), are positive so dN/dt is the product of three positive and one negative term. N moves down toward 1. (1 is a "stable equilibrium".)
     
  4. Apr 15, 2012 #3
    Re: Cubic Population Model with steady states !!

    You mean towards 1 right ? positive = unstable ?
     
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