Stable patterns with jump-discontinuity for a phytoplankton–zooplankton system with both Allee and fear effect

Conghui Zhang, Jin Lu, Maoxing Liu*, Hanzhi Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This paper is concerned with a phytoplankton–zooplankton system with both Allee and fear effect, in which zooplankton species diffuse but phytoplankton species do not diffuse. We show that this system may lead to a novel pattern formation phenomenon, i.e., far-from-the equilibrium patterns with jump discontinuity. Moreover, the L-stability of these discontinuous stationary solutions are demonstrated under appropriate conditions. In addition, we explore how diffusion, Allee and fear effect affect the system. Our results illustrate that (i) if both species diffuse, then the origin and the positive equilibrium are stable. Furthermore, no discontinuous stationary solutions exist; (ii) in the absence of Allee effect, the phenomenon of bistability disappears and only the positive equilibrium is stable. Besides, any discontinuous stationary solutions may be unstable; (iii) when excluding fear effects from the system, the density of zooplankton will be changed, more precisely, as fear costs increase, zooplankton population density declines.

Original languageEnglish
Article number134481
JournalPhysica D: Nonlinear Phenomena
Volume472
DOIs
Publication statusPublished - Feb 2025

Keywords

  • Allee and fear effect
  • Phytoplankton–zooplankton system
  • Stability
  • Stationary solutions with jump discontinuity

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Zhang, C., Lu, J., Liu, M., & Zhang, H. (2025). Stable patterns with jump-discontinuity for a phytoplankton–zooplankton system with both Allee and fear effect. Physica D: Nonlinear Phenomena, 472, Article 134481. https://doi.org/10.1016/j.physd.2024.134481