On the dynamics of two classes of periodic ecological models (Q1342931)

A Lotka-Volterra system of the type \(\dot x = x(a(t) - b(t)x - c(t)y)\), \(\dot y = y(d(t) - e(t)x - f(t)y)\) is considered. The coefficients of the system are \(p\)-periodic. The dynamics of this system is described in detail when the coefficients \(b,c,e,f\) are constant. For the cooperating- species case a complete description of the dynamics is obtained. For the predator-pray case the author derives a necessary and sufficient condition for existence of a positive \(p\)-periodic solution. For the equation \(\dot x = x(x - a(t)) (b(t) - x)\) sufficient conditions for existence of two positive \(p\)-periodic solutions are found. Moreover \(0 \leq x_ 1 (t) \leq x_ 2 (t)\) for all \(t\) and \(\min_ t a(t) \leq x_ 1 (t) \leq \max_ t a(t)\), \(\min_ t b(t) \leq x_ 2 (t) \leq \max_ t b(t)\).