Foxes and Chicken

Yes, foxes and chickens!!   Consider the following problem:  a population of foxes and chickens enjoys a predator/prey relationship which is modelled as a linear system.  The coefficients that model the relationship are determined experimentally, and one parameter, the kill rate  k (of chickens by foxes ) varies with different populations and locales.  If  F_n and C_n are the fox and chicken populations at the beginning of year n, then the populations in the following year are given by 

		F_{n+1} = 0.6*F_n  + 0.5*C_n
		C_{n+1} =  -k*F_n  +  1.2*C_n
		
Can you attach a meaning to each of the coefficients?   In matrix form we can write the column vector [F_i, C_i]  as  x{i}  and express the formula as

		x{i+1}  =  A . x{i}

where the matrix  A  =  {{0.6, 0.5}, {-k, 1.2}}.   Click the following cell:

a = {{0.6, 0.5},{-k, 1.2}}

Now here is the question:  for what values of the kill rate  k  will the fox and chicken populations die out?  For what values will it grow without bound?  Is there a value for which the populations  will stabilize?

Here is where Mathematica really shines!!  We recognize this as an example of a discrete linear dynamical system; that is, the system is described at each discrete time step by a state vector, and one gets the subsequent state vector for the current one by multiplying the current state by a fixed transition matrix.  Later on, we will analyze this phenonemon thoroughly in terms of eigenvectors and eigenvalues.  For the time being, we are going to attack

s[k_] := Max[Abs[Eigenvalues[a]]]

Plot[s[k],{k,0,2}]

Now narrow the graph a little and see if you can find a value of k for which the largest eigenvalue is about 1 and test this value on the population   x   below.

x = {100,1000}

k = .5

a.a.a.a.x

a.a.a.a.a.a.a.a.x

a.a.a.a.a.a.a.a.a.a.a.a.x

a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a.x

Conclusions???