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Help in mathematical modelling phase diagrams!

  1. Oct 31, 2009 #1
    Help in mathematical modelling phase diagrams! :(

    Mathematical modelling question on predator-prey models
    logistic prey-predator model with prey logistic growth

    dx/dt= ax - bx^2 -cy
    dy/dt= -ey + fxy

    ax = growth rate of prey in the absence of predation
    -cxy = the death rate per encounter due to predation
    -cy = the natural death rate of predators in the absence of prey
    fxy = is the prey's contribution to the predator's growth rate


    F(x,y) = X (a-bx-cy)=0
    G(x,y) = Y (-e+fx)=0


    Equilibrium points and stability
    E1 (0,0)
    λ1 > 0, λ2 > 0


    E2 (a/b,0)
    λ1 < 0 & λ2 > 0, if fa/b > e (saddle)
    λ1 < 0 & λ2 < 0, if fa/b < e (asymptotically stable node)


    E3 [e/f, 1/c(a-be/f) ]
    For 1/c(a-be/f) to be +ve , a > be/f exists positively
    For a < be/f , then 1/c(a-be/f) doesn't exist

    λ = α+iβ
    α < 0 , β > 0
    E3 can be a stable node or a stable focus.

    Hi guys i need help on representing my stability of my equilibrium points on a phase diagram especially for the condition a > be/f and a < be/f to show the prey coexist and predator extinction as i would be using it for my condition. Hope to hear from you guys..thanks!
     
  2. jcsd
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