Periodic competitive differential equations and the discrete dynamics of competitive maps. (English) Zbl 0596.34013

Smith, Hal L.

J. Differ. Equations 64, 165-194 (1986).

The author considers systems of differential equations \(x_ i'=x_ if_ i(t,x),\) \(1\leq i\leq n\), \(x\in R^ n_+\), for which \(f_ i(t+2\pi,x)=f_ i(t,x)\) and \(\partial f_ i/\partial x_ j(t,x)\leq 0,\) \(i\neq j\), for all values of (t,x). Such systems model competition between biological species in a periodic environment. The focus of the paper is on the asymptotic behavior of these systems. Our approach is to consider the Poincaré (period) map, T, for such a system and to study the associated discrete dynamical system: \(x_{p+1}=Tx_ p\), on \(R^ n_+\). The competitive hypothesis on the system of differential equations implies that if Tx\(\leq Ty\) then \(x\leq y\), i.e., that \(T^{-1}\) is a monotone map. In the case \(n=2\), P. De Mottoni and A. Schiaffino showed that all bounded orbits (of T) converge eventually monotonically to a fixed point of T (all bounded solutions are asymptotically \(2\pi\)- periodic) for the special case of Lotka-Volterra system and J. Hale and A. Somolinos extended that result to general planar periodic competitive systems. For \(n>2\), this result fails. We study the attractor for the discrete dynamical system generated by T in the case \(n>2\).

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
37-XX	Dynamical systems and ergodic theory
92B05	General biology and biomathematics

Keywords:

mathematical models of competition; first order differential; equation; Lotka-Volterra system; planar periodic competitive systems; attractor; discrete dynamical system

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