Insurgencies and occupiers model

[allmywuv]

so the model where i(t) = insurgencies and o(t) = occupiers
i' = o(α - Fi),
o' = βi + (C - ρo)

its from this website http://www.idea.wsu.edu/Insurgency/
I tried solving it and
I did find the first excercise which was easy enough

so
for
1) eq. points
this is what i found
[ (α/F), (βα/Fρ) + C/ρ ]
[ (-C/β), 0 ]

2) it ask me to linearize so i did the jacobian and using the eq points
i found the iegenvalues
λ = (βα/ρ + FC/ρ + ρ) /2 +- √((βα/ρ + FC/ρ + ρ) 2 - 4 (αβ + FC)
λ = -ρ/2 +- √(ρ2 + 4 (αβ + FC)

3-4) it ask me how F will affect the model if F < 0 , F = 0 , and F > 0
so i played around with the graph on the website but I couldn't really figure out
what kind of a graph is it
will F be saddle at F =0 and F>0
is there a sink? somewhere

oh I also need to find C
but I don't know how
any help would be appreciated


tnx...

 

[jvicens]

Thank you for your post and for sharing your findings on the model. It seems like you have made some good progress in understanding the equations and finding the equilibrium points and eigenvalues. Let me try to provide some additional insights and help answer your questions.

First, let's define the variables in the model:
- i(t) represents the number of insurgencies at time t
- o(t) represents the number of occupiers at time t
- α is the rate at which occupiers can suppress insurgencies
- β is the rate at which insurgencies can increase in the absence of occupiers
- F is the factor that determines how effective occupiers are in suppressing insurgencies
- C is the amount of support or resources that insurgencies can gather without being suppressed
- ρ is the rate at which occupiers lose support or resources due to their presence

Now, let's take a closer look at the equilibrium points you found:
- [ (α/F), (βα/Fρ) + C/ρ ] represents the point where there are no insurgencies and the occupiers are able to maintain their presence and support. This point is stable as long as ρ > 0.
- [ (-C/β), 0 ] represents the point where there are no occupiers and insurgencies are able to gather support and increase. This point is unstable as any small perturbation will cause the number of insurgencies to increase.

Next, let's look at the eigenvalues you found:
- λ = -ρ/2 +- √(ρ2 + 4 (αβ + FC)) represents the stability of the equilibrium points. If the eigenvalues are both negative, the equilibrium point is stable. If one eigenvalue is positive and the other is negative, the equilibrium point is a saddle point. If both eigenvalues are positive, the equilibrium point is unstable.

Now, to answer your questions about the effect of F on the model:
- If F < 0, the model becomes unstable as the occupiers are not effective in suppressing insurgencies.
- If F = 0, the model becomes a saddle point as the occupiers have no effect in suppressing insurgencies.
- If F > 0, the model can be either stable or unstable depending on the other parameters and initial conditions. If F is large enough, it can lead to a stable equilibrium point where both insurgencies and occupiers coexist.

Lastly, to find