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Predator-Prey Model

  1. Aug 12, 2009 #1
    1. The problem statement, all variables and given/known data

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

    xn = xn-1+0.03xn-1-0.001xn-1yn-1
    yn = yn-1-0.02yn-1+0.005xn-1yn-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.


    3. The attempt at a solution

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

    2. I appreciate it if someone can show me what needs to be done for this question.
     
  2. jcsd
  3. Aug 12, 2009 #2

    Borek

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    Staff: Mentor

    Think what does xn-1yn-1 represent?
     
  4. Aug 12, 2009 #3
    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 xn-1yn-1. :confused: Does this make sense?

    And for part 2, how do we find the fixed points of this model?
     
  5. Aug 12, 2009 #4

    Borek

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    Staff: Mentor

    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).

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

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