M. Farkas

Zip bifurcation in a competition model

On considere le systeme S˙=γS(1−S/K)−m 1 / x 1 / S/(a 1 +S)−m 2 x 2 S/(a 2 +S) x˙ 1 =m 1 x 1 S/(a 1 +S)−d 1 w 1 , x˙ 2 =m 2 x 2 S/(a 2 +S)−d 2 x 2 . Soit λ i =a i d i /(m i −d i ). On etudie le cas λ=λ 1 =λ 2

Stable oscillations in a predator-prey model with time lag

Abstract The Lotka-Volterra model with carrying capacity at the prey and time delay in the equation concerning the predator is considered. The time delay is taken into consideration by an integral with the weight function a exp(−at). It is shown that under certain conditions imposed upon the parameters of the system a supercritical Hopf bifurcation takes place at a certain value a0, of a and the bifurcating closed paths are orbitally asymptotically stable for values of a below a0.

On the stability of stationary age distributions

Age structured nonlinear single species and predator-prey population models are treated and a straightforward relatively simple method is shown to arrive at a condition of stability of stationary solutions.

The stable coexistence of competing species on a renewable resource

Abstract We consider a system of three autonomous ordinary differential equations modeling two competing species for a prey resource. We obtain a simple criterion in terms of the specific growth of the prey for there to be an asymptotically stable equilibrium. In a special case, we show that a Hopf bifurcation about this equilibrium could occur. In another special case we show that the interior equilibrium could be globally asymptotically stable. The results are illustrated by a numerical example. Finally it is shown that in the general case, not all competition models posess the properties described in this paper.

Competitive Exclusion by Zip Bifurcation

A model that describes the competition of two predator species for a single regenerating prey species was introduced by Hsu et al. (1978 a,b; see also Koch, 1974 a,b) and has been studied since then by several workers, e.g., Butler (1983), Keener (1983), Smith (1982), and Wilken (1982). In this model of a three-dimensional system of ordinary differential equations the prey population is assumed to have a logistic growth rate in the absence of predators, and the predator populations are assumed to obey a Holling-type functional response (Michaelis-Menten kinetics). Butler (1983) has shown that most of the results concerning the model of Hsu et al. can be achieved for a whole class of two-predator-one-prey models whose common feature is that the prey’s growth rate and the predators’ functional response are arbitrary functions satisfying certain natural conditions.

CONTROLLABLY PERIODIC PERTURBATIONS OF AUTONOMOUS SYSTEMS

1. This paper is dealing with the behaviour of a periodic solution of a system of differential equations under perturbation. The case when the unperturbed system is autonomous and the perturbation periodic and non-autonomous will be considered, However, it will be assumed that the period of the perturbation is controllable. The results are stated and proved for D-periodic (derivo-periodic) solutions of cylindrical systems first (cf. [1]). Since the latter are, in a certain sense, generalizations of periodic solutions of arbitrary systems, the corresponding results for the ordinary case follow readily. The generalization involved is justified by the fact that D-periodic solutions of cylindrical systems occur frequently in applications (cf. e.g. [1], [2], [3]). It is to be noted here that already in [4] Vol. I., p. 80, H. POINCAt~ has mentioned a case which is basically the one called here a D-periodic solution of a cylindrical system.

Political and economic rationality leads to velcro bifurcation

In this paper political and economic rationality is modelled regarding an economic problem with a four-dimensional dynamical system taking into consideration the information about the problem spread among the people who support the political alternatives. Under special parameter conditions velcro bifurcation occurs, which destabilizes the equilibrium points when information is going to spread. The last stable equilibrium point is related to the economically rational equilibrium point.

Estimates on the Existence Regions of Perturbed Periodic Solutions

The n dimensional perturbed system of differential equations

STABILITY OF BIFURCATING ORBITS IN A PREDATOR PREY MODEL

\dot x = f(x) + \mu g({t / {\tau ,x}})

On isolated periodic solutions of differential systems

is considered. It is assumed that g is periodic in the variable t with period