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

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.