Pattern dynamics in a toxin-producing phytoplankton–zooplankton model with additional food

In this paper, we investigate the pattern dynamics in a spatial plankton model including phytoplankton which can release toxins, and zooplankton provided with additional food. The combined effects of toxin liberation and additional food on the stability of the system are explored. It is found that intermediate amounts of additional food and toxin liberation promote the stable coexistence of phytoplankton and zooplankton. A large quantity of additional food can generate Turing patterns more easily, whereas excessive toxin liberation leads to the extinction of zooplankton. Moreover, we obtain the conditions for the occurrence of Turing instability by linear stability analysis. Near the critical value of Turing instability, a multiple-scale method is applied to derive the amplitude equations based on which we can consider the selection and stability of different patterns. The corresponding theoretical results are illustrated by numerical simulations. Furthermore, we show the transitions of pattern formations due to varying the amounts of additional food and toxin liberation, which provides us with particular insight regarding the control of plankton distribution.