Mathematical modelling question on predator-prey models

[scooby_r]

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!

 

[mighty2000]

Hello!

To represent the stability of your equilibrium points on a phase diagram, you can use arrows to show the direction of movement for the prey and predator populations. You can also use different colors or symbols to represent the different equilibrium points. For example, you can use a green arrow to show that the populations are increasing and a red arrow to show that they are decreasing. You can also use a green circle to represent a stable equilibrium point and a red circle to represent an unstable equilibrium point.

For the condition a > be/f, where the prey coexists and the predator goes extinct, you would have a stable equilibrium point at E2 (a/b,0) and an unstable equilibrium point at E3 (e/f, 1/c(a-be/f)). The arrows on the phase diagram would show that the prey population is increasing towards E2 and the predator population is decreasing towards extinction. You can also label the equilibrium points on the phase diagram to make it clear which equilibrium point represents which scenario.

For the condition a < be/f, there would be no stable equilibrium point and the phase diagram would show that both the prey and predator populations are decreasing towards extinction. This would be represented by arrows pointing towards the origin (0,0). You can also label this scenario on the phase diagram to make it clear that both populations are going extinct.

I hope this helps! Let me know if you have any further questions or need clarification. Good luck with your research!