Detecting transitions from steady states to chaos with gamma distribution

TL;DR

This paper presents a new method using gamma distribution to detect transitions from steady states to chaos in stochastic models, with a focus on the Ricker equation, revealing conditions under which chaos emerges.

Contribution

The paper introduces a gamma distribution-based approach to identify chaos transitions in stochastic population models, specifically analyzing the Ricker model's behavior.

Findings

Ricker model undergoes a transition to chaos when the gamma shape parameter is small.

Stochastic equations converge to deterministic models when variance is low.

Two stable states are identified in the stochastic models based on growth rate branches.

Abstract

In this paper, we introduce a novel method to identify transitions from steady states to chaos in stochastic models, specifically focusing on the logistic and Ricker equations by leveraging the gamma distribution to describe the underlying population. We begin by showing that when the variance is sufficiently small, the stochastic equations converge to their deterministic counterparts. Our analysis reveals that the stochastic equations exhibit two distinct branches of the intrinsic growth rate, corresponding to alternative stable states characterized by higher and lower growth rates. Notably, while the logistic model does not show a transition from a steady state to chaos, the Ricker model undergoes such a transition when the shape parameter of the gamma distribution is small. These findings not only enhance our understanding of the dynamic behavior in biological populations but also provide a robust framework for detecting chaos in complex systems.