Consider the following discrete model for the interaction between the populations xn and yn:
xn+1 = f(xn, yn) = (α1 + 1)xn/1 + xn + β1yn
yn+1 = g(xn, yn) = (α2 + 1)yn/1 + yn + β2xn
where α1, α2, β1 and β2 are positive constants.
(a) Interpret this model and explain the type of interaction that is described by these equations.
Explain the assumptions that are implicitly made and meaning of the each of the model constants
in terms of the physical problem.
(b) Give an example of two species whose interaction can be represented by such model.
(c) Determine all the steady states of the model and obtain appropriate condition(s) that ensure(s)
physically meaningful steady state(s).
(d) Determine whether total extinction could occur in this mode
(e) Determine the condition in which yn goes extinct.
(f) Determine the condition in which xn goes extinct.
(g) Find the condition for coexistence.
(h) For the specific case of α1 = 1, α2 = 2, β1 = 1/4 and β2 = 1/2 sketch the phase portrait and discuss if coexistence is possible between the two species in this case.
Sample Solution
