Equivariant Hopf bifurcation of a time-delayed reaction-taxis-diffusion plankton model with selective grazing and cannibalism on a circular domain

¹ Department of Mathematics, Northeast Forestry University, Harbin, Heilongjiang 150040, PR China
² Engineering Research Center of Agricultural Microbiology Technology, Ministry of Education & Heilongjiang Provincial Key Laboratory of Ecological Restoration and Resource Utilization for Cold Region & School of Mathematical Science, Heilongjiang University, Harbin 150080, PR China

^* Corresponding author: liuming This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 7 June 2025
Accepted: 17 April 2026

Abstract

In this paper, we study a model about non-toxic phytoplankton (NTP), toxin-producing phytoplankton (TPP), and zooplankton defined on a two-dimensional circular domain with predator-taxis and time-delay. The relationship between NTP, TPP and zooplankton is explored by combining selective grazing, cannibalism, Allee effect and additional food. For the proposed model, a rigorous proof of global solution existence with uniform boundedness is provided. By applying the Brouwer fixed-point theorem, we demonstrate the existence of a positive constant steady-state solution. Within the functional analytic framework, we establish conditions for both standard and equivariant Hopf bifurcations occurring in the neighborhood of the spatially homogeneous equilibrium. Next we obtain the existence condition and specific properties of periodic solutions by bifurcation theory and normal form method. Numerically, we select appropriate parameters with practical significance and validate analytical predictions via computational experiments. Specifically, we show different types of rotating waves and standing waves, which are the rather complex dynamic behaviors in the two-dimensional circular domain. And then we reveal the periodic phenomena of NTP, TPP and zooplankton, as well as biological phenomena caused by selective grazing, cannibalism, Allee effect and additional food, which provides valuable results for monitoring and controlling the distribution of plankton populations and protecting aquatic ecosystems.