Delay-driven irregular spatiotemporal patterns in a plankton system

Abstract

An inhomogeneous distribution of species density over physical space is a widely observed scenario in plankton systems. Understanding the mechanisms resulting in these spatial patterns is a central topic in plankton ecology. In this paper we explore the impact of time delay on spatiotemporal patterns in a prey-predator plankton system. We find that time delay can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, a phase transition from a regular spiral pattern to an irregular one was observed and the latter gradually replaced the former and persisted indefinitely. The characteristic length of the emergent spatial pattern is consistent with the scale of plankton patterns observed in field studies.

Stochastic models for phytoplankton dynamics in Mediterranean Sea

“…This approach was likely inspired by the prey-predator studies (Liu et al, 2008;Tian and Zhang, 2013;Bengfort et al, 2014), which introduce the prey concentration as a limiting factor of the predator concentration. The crucial role of the limiting resources has been relaunched in recent works (Klausmeier and Litchman, 2001;Huisman et al, 2002Huisman et al, , 2006Ryabov and Blausius, 2008;Ryabov et al, 2010;Ryabov, 2012), in which the authors modified the reaction term to reproduce the effects of two or more limiting factors on the phytoplankton dynamics.…”

Spiral Wave Generation in a Diffusive Predator-Prey Model With Two Time Delays

“…When interaction coefficients of two species are fixed and the conversation rate of predator species is large, the numerical simulations illustrate the existence of both stable and unstable equilibrium near the critical point of the time delay which is in good agreement with our theoretical analysis results. Our numerical results also show that spiral wave patterns are asymptotically stable, whereas in a plankton system spiral wave patterns are not stable [16]. Our study of PDE model simplifies the real ecological processes, which may not always be satisfactory.…”

Delay-induced Turing instability in reaction-diffusion equations

“…Therefore, systems with time delays have received wide attention over the past years. For example, a few of works have recently focused on the investigation of reaction-diffusion equations with small time delays [2,3,9], where the authors reported that small time delays might result in a Turing instability of such systems so that spatial patterns were formed. The Turing instability is a mechanism used to explain how a biological pattern forms and was proposed by the pioneer Turing in Ref.…”

“…In ecology, time delay is commonly used to measure a maturation, gestation period, or reaction time of a predator population [1,4,5], and mathematically it may result in a much richer dynamics for a system, such as inducing the instability of an equilibrium so that Hopf bifurcation occurs [6][7][8][9][10][11][12]. Therefore, systems with time delays have received wide attention over the past years.…”