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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!

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# Help in mathematical modelling phase diagrams!

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