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Predator-Prey system

  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Scientists studying a population of wolves (y) and a population of rabbits (x) on which the wolves depend for food, have found the sizes x and y of these populations to be modelled well by the equation
    4lny-.02y+3lnx-.001x=37.37

    We are given the initial condition of (x, y) = (5000, 300).

    The origin (0, 0) is a critical point of the system, not a very happy one because it represents the equilibrium solution in which both species are extinct. Find and plot (on a graph given to us on our handout) the other equilibrium (constant) solution.

    2. Relevant equations

    NONE

    3. The attempt at a solution

    See PDF for my solution (sorry if that is an inconvenience).

    So basically I set dy/dt and dx/dt to zero to determine the other critical point, since the question mentioned the origin being a critical point for the system.

    But at the end it's asking for the other equilibrium (constant) solution. You can separate the equations dy/dt and dx/dt and get 4lny-.02y+3lnx-.001x=C, where this C is presumably the constant that the question is referring to, or at least it seems that way to me.

    So is my work helpful for the question at all? Or am I doing something completely different? You can't determine the C in 4lny-.02y+3lnx-.001x=C without having some kind of initial condition. I understand (at least I think) that the equation in the last sentence is the general solution to the Lotka Volterra differential equations in my work. We are given C = 37.37 in our case. So would I just let C = 0 and solve? I can't do anything because we have no initial condition. Let (x,y) = (0,0)? Can't do that either (ln 0 doesn't exist).

    I am certain that my work is incomplete and there is more to it. But I don't know what else to do. I DID find dy/dx in an earlier question, so I might think set dy/dx = 0, but it's a rational function and does not have any zeros (ie, both x and y never equal zero), which is clearly a consequence of the original relation itself not allowing zero.

    So where do I go from here? I'm just not sure about what the question is actually asking.

    Thank you all in advance for your help, much appreciated!
     

    Attached Files:

  2. jcsd
  3. Jan 21, 2012 #2
    I think the lack of sleep is getting to me. It is clear to me now that all I need to do is plug in (x,y) = (3000,200). I wasn't sure what an equilibrium solution was, but now I recall it's when the rate of change wrt time is equal to zero. And I did that, found (x,y), and that is my "initial condition" so I can find my new value of C.

    Seems right to me but if I'm still not on the right track, let me know.
     
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