Comparison of the global dynamics for two chemostat-like models: random temporal variation versus spatial heterogeneity

TL;DR

This paper compares two chemostat-like models—one with temporal variability and one with spatial heterogeneity—showing they have similar long-term behaviors and identifying invasion rates as key determinants.

Contribution

It introduces and analyzes two fundamentally different models, demonstrating their similar long-term dynamics and the role of invasion rates in both.

Findings

Models exhibit similar long-term behavior despite different nature.

Invasion rates determine species persistence or extinction.

Bistability observed between coexistence and semi-trivial states.

Abstract

This article is dedicated to the study and comparison of two chemostat-like competition models in a heterogeneous environment. The first model is a probabilistic model where we build a PDMP simulating the effect of the temporal heterogeneity of an environment over the species in competition. Its study uses classical tools in this field. The second model is a gradostat-like model simulating the effect of the spatial heterogeneity of an environment over the same species. Despite the fact that the nature of the two models is very different, we will see that their long time behavior is globally very similar. We define for both model quantities called invasion rates which model the growth rate of a species when it is near to extinction. We show that the signs of these invasion rates essentially determine the long time behavior for both systems. In particular, we exhibit a new example of bistability between a coexistence steady state and a semi-trivial steady state.