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

TL;DR

This paper studies a three-species ecological model with two resources and one consumer, revealing how oscillations become unstable and lead to fixed points depending on resource consumption rates and consumer switching dynamics.

Contribution

It introduces a dynamic switching mechanism in the Rosenzweig-MacArthur model, analyzing how resource consumption and consumer preferences affect system stability.

Findings

Oscillations destabilize when resource consumption rates deviate from equal sharing.

The system tends to fixed points with dominant resource preference.

Consumer's inability to switch resources once chosen causes system stabilization.

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.