# Emergence of spatiotemporal chaos driven by far-field breakup of spiral waves in the plankton ecological systems

## TL;DR

This paper investigates how spiral wave breakup leads to spatiotemporal chaos in a reaction-diffusion model of plankton, revealing complex patterns driven by parameter changes and wave interactions.

## Contribution

It provides a detailed analysis of spiral wave breakup and chaos emergence in a plankton model using simulations and bifurcation theory, expanding understanding of ecological pattern formation.

## Key findings

- Spiral waves exist in the mixed Turing-Hopf bifurcation region.
- Far-field breakup of spiral waves causes spatiotemporal chaos.
- Chaos does not spread uniformly across the entire spatial domain.

## Abstract

Alexander B. Medvinsky \emph{et al} [A. B. Medvinsky, I. A. Tikhonova, R. R. Aliev, B.-L. Li, Z.-S. Lin, and H. Malchow, Phys. Rev. E \textbf{64}, 021915 (2001)] and Marcus R. Garvie \emph{et al} [M. R. Garvie and C. Trenchea, SIAM J. Control. Optim. \textbf{46}, 775-791 (2007)] shown that the minimal spatially extended reaction-diffusion model of phytoplankton-zooplankton can exhibit both regular, chaotic behavior, and spatiotemporal patterns in a patchy environment. Based on that, the spatial plankton model is furtherly investigated by means of computer simulations and theoretical analysis in the present paper when its parameters would be expected in the case of mixed Turing-Hopf bifurcation region. Our results show that the spiral waves exist in that region and the spatiotemporal chaos emerge, which arise from the far-field breakup of the spiral waves over large ranges of diffusion coefficients of phytoplankton and zooplankton. Moreover, the spatiotemporal chaos arising from the far-field breakup of spiral waves does not gradually involve the whole space within that region. Our results are confirmed by means of computation spectra and nonlinear bifurcation of wave trains. Finally, we give some explanations about the spatially structured patterns from the community level.