[Paper Review] Persistence in the Moran model with random switching

Arnaud Guillin, Arnaud Personne+1 more|arXiv (Cornell University)|Nov 4, 2019

Mathematical and Theoretical Epidemiology and Ecology Models34 references4 citations

TL;DR

This paper studies the long-term behavior of the Moran model with random environmental switching, showing that stochastic persistence—coexistence of all species—can emerge even when no species is consistently favored, provided environmental switching offers sufficient diversity. Using a large-population limit leading to a Piecewise Deterministic Markov Process (PDMP), the authors apply stochastic persistence theory to prove that biodiversity is preserved when selection parameters switch between multiple environments, even if one species is never the fittest in any single state.

ABSTRACT

The paper is devoted to the study of the asymptotic behaviour of Moran process in random environment, say random selection. In finite population, the Moran process may be degenerate in finite time, thus we will study its limiting process in large population which is a Piecewise Deterministic Markov Process, when the random selection is a Markov jump process. We will then study its long time behaviour via the stochastic persistence theory of Benaïm \cite{benaimpersistence}. It will enable us to show that persistence can occur, i.e. asymptotic coexistence of all species, when there are enough switching possibilities. This is true even if one species has never a predominant selection.

Motivation & Objective

• To analyze the asymptotic behavior of the Moran process under random environmental switching.
• To establish the large population limit of the discrete Moran process as a Piecewise Deterministic Markov Process (PDMP).
• To investigate under what conditions stochastic persistence—long-term coexistence of all species—can occur despite no species being dominant in any single environment.
• To extend persistence criteria beyond two-species systems using stochastic persistence theory and accessibility arguments.

Proposed method

• Derive the large population limit of the discrete Moran process with random selection, showing convergence to a PDMP driven by a Markov jump process for selection parameters.
• Use the stochastic persistence framework of Benaïm [9] to analyze long-time behavior, focusing on whether the process spends infinite time away from absorbing boundaries.
• Characterize the vector fields governing species dynamics under constant selection, and analyze their flows and compositions under switching.
• Apply a key lemma on flow equivalence under positive scaling of vector fields to preserve accessibility properties across different selection regimes.
• Compute invasion rates and persistence conditions for two- and three-species systems under various environmental switching patterns.
• Use ergodic measures and composite flows to assess accessibility and invariant measures in multi-environment settings.

Experimental results

Research questions

• RQ1 Under what conditions does stochastic persistence occur in a two-species Moran model with random environmental switching?
• RQ2 Can a species persist even if it is never the fittest in any single environment, provided switching occurs between multiple states?
• RQ3 What are the necessary and sufficient conditions for a species to successfully invade or be excluded in a multi-environment setting?
• RQ4 Is coexistence of three species possible under two environmental states, and if not, why?
• RQ5 How does the number of environmental states affect the possibility of long-term persistence in multi-species systems?

Key findings

• Stochastic persistence can occur in the Moran model with random switching even when no species is ever the fittest in any single environment, provided there is sufficient environmental diversity.
• For two species and two environments, persistence is guaranteed if the selection parameters satisfy the inequality $-s_1 < s_2 < -s_1/(1+s_2)$, as shown in Example 1.
• In the three-species case with only two environmental states, persistence is impossible if two species alternate between being favored and disadvantaged while the third remains neutral.
• With three environmental states, coexistence of three species becomes possible, demonstrating that increased environmental switching complexity supports biodiversity.
• The accessibility of the system’s state space under different selection regimes is preserved under positive scaling of vector fields, enabling the use of flow composition techniques to analyze long-term behavior.
• The large-population limit of the Moran process converges to a PDMP, enabling the use of continuous-time tools to study discrete stochastic processes in population dynamics.