Survival and complete convergence for a spatial branching system with local regulation

TL;DR

This paper analyzes a spatial branching process with local regulation, demonstrating conditions for survival, equilibrium uniqueness, and convergence, with implications for long-term coexistence in complex systems.

Contribution

It introduces new results on survival probabilities, equilibrium classification, and convergence for a spatial branching system with local regulation.

Findings

System survives with positive probability under small competition.

Uniqueness of nontrivial equilibrium established for certain parameters.

Complete convergence and long-term coexistence demonstrated.

Abstract

We study a discrete time spatial branching system on $\mathbb{Z}^d$ with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the nontrivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model. Along the way we classify the equilibria and their domain of attraction for the corresponding deterministic coupled map lattice on $\mathbb{Z}^d$.