1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Staff Emeritus
    Science Advisor

    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 ?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cubic Population Model with steady states
  1. Population Model (Replies: 2)

  2. Population model (Replies: 1)

Loading...