The $P^*$ rule in the stochastic Holt-Lawton model of apparent competition

TL;DR

This paper generalizes the stochastic Holt-Lawton model to multiple host species with stochastic parasitism and immigration, characterizing persistence, extinction, and the $P^*$ rule for apparent competition.

Contribution

It introduces a comprehensive stochastic model for multiple host species, deriving explicit expressions for parasitoid densities and establishing the $P^*$ rule in this context.

Findings

The system is dissipative and enters a compact set in finite time.

The host species with the largest $P^*$ value stochastically persists.

Other host species tend to extinction when competing with the dominant species.

Abstract

In $1993$, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host density-dependence, we show that this system is dissipative i.e. enters a compact set in finite time for all initial conditions. When there is a single host species, stochastic persistence and extinction of the host is characterized using external Lyapunov exponents corresponding to the average per-capita growth rates of the host when rare. When a single host persists, say species $i$, a explicit expression is derived for the average density, $P_i^*$, of the parasitoid at the stationary distributions supporting both species. When there are multiple host species, we prove that the host species with the largest $P_i^*$ value stochastically persists, while the other host species are asymptotically driven to extinction. A review of the main mathematical methods used to prove the results and future challenges are given.