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eigenvalues of the transition matrix

A discrete dynamical system model for the population of cheetahs and gazelles in Namibia is given by the following pair of equations:

.3Ck + .4 Gk = Ck+1
-pCk + 1.3 Gk = Gk+1

where Ck measures the number of cheetahs present in a certain Namibian game reserve at time K, Gk gives the number of gazelles (measured in tens), and k is measured in months.

a) Find a value for p that guarantees a steady- state outcome for this model, then determine the number of cheetahs present for every 1000 gazelles (in the long run).
b) find a value for p that will guarantee 2% growth in both populations. In the long-run, what is the ratio of gazelles to cheetahs?
c) What must be true about the eigenvalues of the transition matrix if both populations die out in the long- run?

Solution Preview

Welcome to BrainMass!

I am going to start by trying to walk you through the problem, and then go about solving it in a hopefully understandable way. I also would appreciate it if you would rate my response on how clearly I explained the problem. It helps me work on things I can improve on, or get an idea of what worked well.

Ok, there's for the introduction.

Let's look at the problem.

So basically what we are looking at is two equations. On the left side of the equations, we have some numbers and the current population of cheetahs and gazelles at time k. On the right side of the equations, we have
k + 1, which in math talk, refers to how many cheetahs and gazelles there will be one year from k. Or the next generation of cheetahs and gazelles.

Now, for part (a). It asks for the value of p, which we see in the second equation, that will guarantee a "steady-state" outcome for this model. I guess we need to know what "steady-state" means. So probably the best way to understand "steady-state", is that the population of gazelles and cheetahs is neither increasing or decreasing over time. Now we need to convert that into math talk.

If the population doesn't change over time, what do you think we could say about G(k) and G(k+1).
And C(k) and C(k+1)? If the population doesn't change over time, then won't the next generation be the same as the current generation? That's right, they're equal, G(k) = G(k+1) and C(k) = C(k + 1).

That was the break we needed!

Let's re-write out equations now--but we can just replace G(k+1) with G(k) and likewise with C(k + 1).

0.3 C(k) + 0.1 G(k) = C(k)
-p C(k) + 1.3 G(k) = ...

Solution Summary

Eigenvalues of the transition matrix are examined closely in this solution.

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