preditor prey Model... Need Help with stability analysis.
Hi im Matt, im new to this website.
I have a project i am working on at the momement invloving building a model to represent the population rates of change of the following animals. Rabbits, Foxes, Stoats.
i have got my system of differentials for each animal and i am assuming there is no cxompetition between the two preditors (foxes and stoats).
my ODE's are such:
rabbits: d(x)/d(t)=ax-hxy-mxz where a=population growth constant, h=killing efficency constant of foxes, m=killing efficency constant of stoats. x,y,z=population size at time t
foxes: d(y)/d(t)=-by+kxy where b=death rate constant(through natural corse's), k=population growth constant, x,y=population size at time t
stoats: d(z)/d(t)=-cz+nxz where c=death rate constant(through natural corse's), n=population growth constant, x,z=population size at time t
i believe i get four points of equilibrium these being
(0 , 0 , 0) , (c/n , 0 , a/m) , (b/k , a/h , 0) , (0 , (a-mz)/h , (a-hy)/m)
from using a Jocobian Matrix i recieve the following eigen-values: and what i believe to be the correct interpretation of the stability analysis of the eigenvalues.
at point (0 , 0 , 0) => lambda = a , or lambda = -b , or lambda = -c
therefore indicating a unstable saddle point with the instability being in rabbits.
at point (c/n , 0 , a/m) => lambda = -b+(kc)/n , or lambda = i*sqrt(ac) , or lambda = -i*sqrt(ac)
therefore indicating a stable centre point between rabbits and stoats (shown from the two purly imaginary eigenvalues)
and that foxes will die out unless the population growth constant(n) becomes significantly small enough to turn b(death rate constant) positive. shown by lambda = -b+(kc)/n
at point (b/k , a/h , 0) => lambda = -c+(nb)k , or lambda = i*sqrt(ab) , or lambda = -i*sqrt(ab)
therefore indicating a stable centre point between rabbits and foxes (shown from the two purly imaginary eigenvalues).
and that stoats will die out unless the population growth constant(k) becomes significantly small enough to turn c(death rate constant) psotive. shown by lambda = -c+(nb)k
at point (0 , (a-mz)/h , (a-hy)/m) => lambda = -a+mz+hy , or lambda = -b , or lambda = -c
i am unsure of this particular one but i belive it to be indicating an unstable saddle point .
any guidence on my stability analysis would be greatly appriciated.
on a further note.. i now have to go on to produce phase path diagrams ,, which would you consider to be most appropriate for this scenario. a 3 dimensional phase plot showing all through variables or a series of 2 dimensional phase plots comparing rabbits v foxes , rabits v stoats and foxes v stoats?
again any comments will be greatly appricated
Re: preditor prey Model... Need Help with stability analysis.
from recapping ive realised ive made some errors in my math. from re-calculating my new eigenvalues correspond to the situation much better and make the stability analysis much more straight forward.