Global analysis of a system of predator-prey equations. (English) Zbl 0608.92016

Conway, E. D.; Smoller, J. A.

SIAM J. Appl. Math. 46, 630-642 (1986).

A special case of the general Kolmogorov form of simple interaction of one predator and one prey, the so-called Rosenzweig-MacArthur system: \[ (*)\quad u'=u[f(u)-v],\quad v'=v[u-\gamma], \] is studied. Under the assumption that the prey population is asocial, the authors obtain an analogue of the Kolmogorov theorem, as well as several other aspects of the richer dynamics.
They discovered most of the phenomena discussed in §§2 and 3 by numerical studies of (*), and this paper is a direct consequence of their effort to understand their numerical discoveries. In §4, the authors show that ”chaos” is present for a ”forced” version of (*).
This study gives illustrative natural examples of recent results on the continuation of periodic orbits obtained by J. C. Alexander and J. A. Yorke [J. Differ. Equations 49, 171-184 (1983; Zbl 0516.34040)].

Reviewer: B.Li

MSC:

92D25	Population dynamics (general)
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
92D40	Ecology
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

predator-prey equations; equilibrium points; limit cycles; Hopf bifurcation; Rosenzweig-MacArthur system; Kolmogorov theorem; chaos; examples of recent results on the continuation of periodic orbits

Citations:

Zbl 0516.34040