Instability of oscillations in the Rosenzweig-MacArthur model of one consumer and two resources

Abstract

The system of two resources $R_1$, $R_2$ and one consumer $C$ is investigated within the Rosenzweig-MacArthur model with Holling type II functional response. The rates ${\beta }_i$ of consumption of resources $i=1,2$ are coupled by the condition ${\beta }_1+{\beta }_2=1$. The dynamic switching is introduced by a maximization of $C$: $d{\beta }_1/dt=(1/\tau )dC/d{\beta }_1$, where the characteristic time $\tau$ is large but finite. The space of parameters where both resources coexist is explored numerically. The results indicate that oscillations of $C$ and mutually synchronized $R_i$ which appear at ${\beta }_i=0.5$ are destabilized for ${\beta }_i$ larger or smaller. Then, the system is driven to one of fixed points where either ${\beta }_1>0.5$ and $R_1<R_2$ or the opposite. This behaviour is explained as an inability of the consumer to change the preferred resource, once it is chosen.

14 pages, 8 figures