Lotka–Volterra

May 2011
42
0
Hello,

\(\displaystyle \dot{x} = ax\left(1-\dfrac{x}{K}\right)-\dfrac{bxy}{1+Ax}\)
\(\displaystyle \dot{y} =cy \left(1-J\dfrac{y}{x}\right)\)

This system desciribes population growth of two species. This is quite similar to Lotka–Volterra equation modelling prey and predator population growth, but I am having hard time finding meaning for \(\displaystyle \dfrac{bxy}{1+Ax}\) as i have found only \(\displaystyle bxy\) in literature instead. What could be these species be and what does \(\displaystyle \dfrac{bxy}{1+Ax}\) stands for intuitevily?

My guess:
Well \(\displaystyle y\) stays there as it is.. proportional. So.. for big \(\displaystyle x\) it becomes close to \(\displaystyle \dfrac{by}{1+A}\).. meaning that predators can eat only limited amount?
 
May 2011
42
0
I have another problem concerning the same equation system.

I would like to prove that for some parametr values there exsist limit cycle for this system. My idea was to consturct (x,y) domain such that flow only goes in it, use Poincare-Bendixson theorem, then show that there is an equilibrium point inside this domain and that it is repelling. That would be enough. However it is not so easy to consturct this domain, because you can't take x or y axis for domain boundary.

Is there an easier way to prove that there is limit cycle for some parametr values?

Equlibrium point is close to boundaries though {y = 1.363636364, x = .6000000000} - as per paramters i choosed.

Do you have a better idea... or maybe you can at least confirm that i am doing something right?
 

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