*(English)*Zbl 0652.92019

The nonlinear behavior of a particular Kolmogorov-type exploitation differential equation system assembled by *R. M. May* [Stability and complexity in model ecosystems. Princeton (1973)] from predator and prey components developed by *P. H. Leslie* [Biometria 35, 213-245 (1948; Zbl 0034.233)] and *C. S. Holling* [Mem. Entomol. Soc. Can. 45, 1-60 (1965)], respectively, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature dependent mite interaction on fruit trees.

The most significant result of this analysis is that, in addition to the temperature ranges over which the single community equilibrium point of the system is either globally stable or gives rise to a globally stable limit cycle, there can also exist a range wherein multiple stable states occur. These stable states consist of a focus (spiral point) and a limit cycle, separated from each other in the phase plane by an unstable limit cycle. The ecological implications of such metastability, hysteresis and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system and the biological control of mite populations are discussed.

It is further suggested that a model of this sort which possesses a single community equilibrium point may be more useful for representing outbreak phenomena, especially in the presence of oscillations, than the non-Kolmogorov predator-prey systems possessing three community equilibrium points, two of which are stable and the other a saddle point, traditionally employed for this purpose.

##### MSC:

92D40 | Ecology |

34C05 | Location of integral curves, singular points, limit cycles (ODE) |

65C20 | Models (numerical methods) |