Effect of delay on the emergent stability patterns in Generalized Lotka-Volterra ecological dynamics

TL;DR

This paper investigates how time delay influences the stability and feasibility of ecological systems modeled by Generalized Lotka-Volterra equations, revealing phase transitions and stability increases with diversity.

Contribution

It provides an analytical and numerical study of delay effects on stability in ecological models, including a critical delay calculation and stability measure for oscillatory regimes.

Findings

Delay induces a transition from equilibrium to oscillatory dynamics.

Critical delay for phase transition is analytically derived and matches simulations.

Stability increases with ecosystem diversity in oscillatory regimes.

Abstract

Understanding the conditions of feasibility and stability in ecological systems is a major challenge in theoretical ecology. The seminal work of May in 1972 and recent developments based on the theory of random matrices have shown the existence of emergent universal patterns of both stability and feasibility in ecological dynamics. However, only a few studies have investigated the role of delay coupled with population dynamics in the emergence of feasible and stable states. In this work, we study the effects of delay on Generalized Loka-Volterra population dynamics of several interacting species in closed ecological environments. First, we investigate the relation between feasibility and stability of the modeled ecological community in the absence of delay and find a simple analytical relation when intra-species interactions are dominant. We then show how, by increasing the time delay, there is a transition in the stability phases of the population dynamics: from an equilibrium state to a stable non-point attractor phase. We calculate analytically the critical delay of that transition and show that it is in excellent agreement with numerical simulations. Finally, we introduce a measure of stability that holds for out of equilibrium dynamics and we show that in the oscillatory regime induced by the delay stability increases for increasing ecosystem diversity.