Solving the Predator-Prey Model: Foxes & Rabbits

[roam]

Homework Statement

The difference equations below model the yearly populations of foxes and rabbits, measured in tousands. The foxes kill the rabbits for food.

x_n = x_n-1+0.03x_n-1-0.001x_n-1y_n-1
y_n = y_n-1-0.02y_n-1+0.005x_n-1y_n-1

1. Which population does x represent and which population does y represent? Give a reason for your answer.

2. Find the fixed points of this model.

The Attempt at a Solution

1. I believe "x" represents the population of rabbits (the prey) and "y" represents fox (predator). I know that x_n decreases by 0.001x_n-1y_n-1 whereas y_n increases by 0.005x_n-1y_n-1. But I don't know how to explain this in a proper way...

2. I appreciate it if someone can show me what needs to be done for this question.

 

[Borek]

Think what does x_n-1y_n-1 represent?

 

[roam]

Think what does x_n-1y_n-1 represent?

I think maybe it's the rate at which the predators and the prey meet, since "x" is the population of prey (rabbits) then it decreases by the x_n-1y_n-1. Does this make sense?

And for part 2, how do we find the fixed points of this model?

 

[Borek]

I think maybe it's the rate at which the predators and the prey meet

Good - now, which population does grow and which goes down in effect of such meetings? Sign of the coefficient will help you decide what is what (could be it will just confirm your earlier idea, I have not bothered with checking if you are right or wrong, just tried to show how to analyze the equation to get to the answer).

And for part 2, how do we find the fixed points of this model?

Not sure what they mean by fixed points - probably that the populations are stable (not oscillating). That means x_n=x_n-1...

Funny thing, I wrote a paper about this model for Polish monthly computer magazine Bajtek, back in late eighties or early nineties Copy must be less than 10 feet from me, but I have no idea where.