Mathematical model of livestock and wildlife: Predation and competition under environmental disturbances

TL;DR

This paper develops a mathematical model of a trophic web involving livestock, wild herbivores, and predators, incorporating spatial heterogeneity and competition, to analyze population dynamics under environmental disturbances in Patagonia.

Contribution

It introduces a hierarchical competition model with patches and metapopulations, including new terms to account for habitat heterogeneity, extending previous models.

Findings

Model accurately predicts population control scenarios.

Habitat heterogeneity significantly impacts species coexistence.

Numerical simulations align with mean field calculations.

Abstract

Inspired by real scenarios in Northern Patagonia, we analyze a mathematical model of a simple trophic web with two herbivores and one predator. The studied situations represent a common practice in the steppes of Argentine Patagonia, where livestock are raised in a semi-wild state, either on the open range or enclosed, coexisting with competitors and predators. In the present work, the competing herbivores represent sheep and guanacos, while the predator is associated with the puma. The proposed model combines the concepts of metapopulations and patches dynamics, and includes an explicit hierarchical competition between species, which affects their prospect to colonize an empty patch when having to compete with other species. We perform numerical simulations of spatially extended metapopulations assemblages of the system, which allow us to incorporate the effects of habitat heterogeneity and destruction. The numerical results are compared with those obtained from mean field calculations. We find that the model provides a good theoretical framework in several situations, including the control of the wild populations that the ranchers exert to different extent. Furthermore, the present formulation incorporates new terms in previously analyzed models, that help to reveal the important effects due to the heterogeneous nature of the system.