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

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
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### References:

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