A predator-prey two-sex branching process

TL;DR

This paper introduces a predator-prey two-sex branching process model that captures the interaction, extinction, and coexistence dynamics of predator and prey populations with sexual reproduction, using control functions based on current population sizes.

Contribution

It develops a novel two-sex controlled branching process model for predator-prey interactions and provides conditions for extinction, fixation, and coexistence of species.

Findings

Conditions for ultimate extinction of both species

Criteria for fixation of one species

Analysis of coexistence scenarios

Abstract

In this paper, we introduce a two-sex controlled branching model to describe the interaction between predator and prey populations with sexual reproduction. This process is a two-type branching process, where the first type corresponds to the predator population and the second one to the prey population. While each population is described via a two-sex branching model, the interaction and survival of both groups is modelled through control functions depending on the current number of individuals of each type in the ecosystem. We provide necessary and sufficient conditions for the ultimate extinction of both species, the fixation of one of the species and the coexistence of both of them. Moreover, the description of the present predator-prey two-sex branching process on the fixation events can be performed in terms of the behaviour of a one-type two-sex branching process with a random control on the number of individuals, which is also introduced and analysed.