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**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 |

### Keywords:

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

### References:

[1] | Coppel, W, Stability and asymptotic behavior of differential equations, (1965), Heath Boston · Zbl 0154.09301 |

[2] | Grossberg, S, Competition, decision, and consensus, J. math. anal. appl., 66, 470-493, (1978) · Zbl 0425.92017 |

[3] | Hadamard, J, Sur l’iteration et LES solutions asymptotiques des equations differentielles, Bull. soc. math. France, 29, 224-228, (1901) · JFM 32.0314.01 |

[4] | Hale, J.K; Somolinos, A.S, Competition for fluctuating nutrient, J. math. biol., 18, 255-280, (1983) · Zbl 0525.92024 |

[5] | Hallam, T.G; Svoboda, L.J; Gard, T.C, Persistence and extinction in three species Lotka Volterra competitive systems, Math. biosci., 46, 117-124, (1979) · Zbl 0413.92013 |

[6] | Hirsch, M.W, Systems of differential equations which are competitive or cooperative. I. limit sets, SIAM J. math. anal, 13, 2, 167-179, (1982) · Zbl 0494.34017 |

[7] | {\scM. W. Hirsch}, Systems of differential equations which are competitive or cooperative. II. Convergence almost everwhere, SIAM J. Math. Anal., in press. |

[8] | Hirsch, M.W, The dynamical systems approach to differential equations, Bull. amer. math. soc., 11, 1, 1-64, (1984) · Zbl 0541.34026 |

[9] | Hirsch, M.W, Attractors for discrete-time monotone dynamical systems in strongly ordered spaces, (), in press · Zbl 0588.58034 |

[10] | Krasnosel’skii, M.A, Translation along trajectories of differential equations, Amer. math. soc. trans., Vol. 19, (1968), Providence, R.I. |

[11] | Leonard, W.J; May, R, Nonlinear aspects of competition between species, SIAM J. appl. math., 29, 243-275, (1975) · Zbl 0314.92008 |

[12] | de Mottoni, P; Schiaffino, A, Competition systems with periodic coefficients: A geometric approach, J. math. biol., 11, 319-335, (1981) · Zbl 0474.92015 |

[13] | Smale, S, On the differential equations of species in competition, J. math. biol., 3, 5-7, (1976) · Zbl 0344.92009 |

[14] | {\scH. L. Smith}, Invariant curves for mappings, SIAM J. Math. Anal., to appear. · Zbl 0606.47056 |

[15] | {\scH. L. Smith}, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal., to appear. · Zbl 0609.34048 |

[16] | Varga, R.S, Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, N.J · Zbl 0133.08602 |

[17] | Hartman, P, Ordinary differential equations, (1973), Hartman · Zbl 0125.32102 |

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