# Cubic Population Model with steady states

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 ?

## Answers and Replies

HallsofIvy
Science Advisor
Homework Helper

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".)

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 towards 0.

You mean towards 1 right ? positive = unstable ?