Coevolution of the reckless prey and the patient predator

TL;DR

This paper models a predator-prey stand-off as an embedded war of attrition, analyzing their coevolution of giving-up rates and revealing three distinct evolutionary outcomes through adaptive dynamics.

Contribution

It introduces a novel predator-prey model incorporating war of attrition dynamics and analyzes coevolutionary outcomes using adaptive dynamics.

Findings

Predator may give up immediately while prey never gives up.

Prey may adopt any giving-up rate above a threshold with a persistent predator.

Predator extinction is a possible long-term outcome.

Abstract

The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator-prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.