Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach

Abstract

In this paper, we propose and analyse a mathematical model to study the mathematical aspect of reaction diffusion pattern formation mechanism in a predatorprey system. An attempt is made to provide an analytical explanation for understanding plankton patchiness in a minimal model of aquatic ecosystem consisting of phytoplankton, zooplankton, fish and nutrient. The reaction diffusion model system exhibits spatiotemporal chaos causing plankton patchiness in marine system. Our analytical findings, supported by the results of numerical experiments, suggest that an unstable diffusive system can be made stable by increasing diffusivity constant to a sufficiently large value. It is also observed that the solution of the system converges to its equilibrium faster in the case of two-dimensional diffusion in comparison to the one-dimensional diffusion. The ideas contained in the present paper may provide a better understanding of the pattern formation in marine ecosystem.

Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion

“…Hillary and Bees [3] compared the dynamics of the plankton communities with and without the delay associated with a zooplankton maturation period and found that time delay can lead to progressively replacing regular plankton dynamics by chaos. There are also other causes of delays in population dynamics at various time scales [5,7,[11][12][13][14]18]. Consequently, delay needs to be considered.…”

“…Furthermore, we have noted that the ecological system is often deeply perturbed by human exploiting activities over recent years [5,19,20]. For the view of human needs, the exploitation of ecological resources and harvest of population are commonly carried out in the fields of wildlife, fishery and forestry management [14]. As a consequence, people are facing the problems about shortage of resource and worsening environment.…”

“…It means that diffusion can cause chaos to appear or disappear. [14] and [32] In this subsection, we consider the parameter values as the same with [14,32]: H P = 0.3, b = 2, c = 0.8, F = 0.0001, H Z = 2.5, D = 1. According to [32] system is locally asymptotically stable at its interior equilibrium.…”

“…For this reason, the spatio-temporal dynamics of plankton model is necessary to be studied. So far, there have been a lot of excellent papers about plankton dynamics with diffusion in a plankton system [12][13][14][15]. Most of them mainly focus on determining the diffusion-driven spatiotemporal patterns.…”

“…Spatiotemporal chaos is thought to have important implications for ecosystem functioning not only in terms of their predictability but also by affecting straightforwardly species persistence and ecosystem's stability with respect to unfavorable environmental changes [17]. In [14] the authors incorporated diffusion terms into the minimal model and considered the following reaction-diffusion model…”