Help in mathematical modelling phase diagrams

[scooby_r]

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!

 

[jvicens]

Hello,

Thank you for reaching out for help with your mathematical modelling question. I would be happy to assist you with representing the stability of your equilibrium points on a phase diagram.

First, let's review what equilibrium points and stability mean in this context. An equilibrium point is a point at which the values of both prey (x) and predators (y) do not change over time. This means that the population sizes of both species remain constant at this point. Stability refers to the behavior of the system around the equilibrium point. A stable equilibrium point means that if the system is disturbed from this point, it will return to it over time. An unstable equilibrium point means that if the system is disturbed, it will move further away from this point.

Now, to represent these equilibrium points and their stability on a phase diagram, we can plot the values of x and y on the x-axis and y-axis, respectively. The equilibrium points can be marked as points on this graph. For example, E1 (0,0) would be represented as the origin point (0,0) on the graph.

To show the stability of these points, we can use arrows or vectors. These arrows will indicate the direction in which the system will move if it is disturbed from the equilibrium point. For a stable equilibrium point, the arrows will point towards the point, showing that the system will return to it. For an unstable equilibrium point, the arrows will point away from the point, indicating that the system will move further away from it.

In your case, for the equilibrium point E2, you mentioned that it can either be a saddle or an asymptotically stable node. For a saddle, the arrows will point in different directions, showing that the system can move away from this point in multiple directions. For an asymptotically stable node, the arrows will all point towards the point, indicating that the system will return to it over time.

For the equilibrium point E3, you mentioned that it can be a stable node or a stable focus. For a stable node, the arrows will point towards the point, as the system will return to it. For a stable focus, the arrows will form a spiral pattern, indicating that the system will oscillate around the point before returning to it.

I hope this helps in representing the stability of your equilibrium points on a phase diagram. If you have any further questions, please don't hesitate to ask. Good luck with your modelling!