Existence and Instability of Bifurcation Solutions for a Partial Diffusion Phytoplankton–Zooplankton Model

In this paper, a partial diffusion phytoplankton–zooplankton interaction model is proposed by incorporating the Allee effect as well as fear effect, and we consider the situation that zooplankton species can move while phytoplankton species cannot. First, we analyze the stability of constant stationary solutions via the linearization theory. Then by regarding the diffusion coefficient as the bifurcation parameter, we construct nonconstant steady-state bifurcation solutions and prove that they are always unstable. This is opposite to the result for general predator–prey models when both species diffuse. Moreover, by choosing the Allee threshold as the bifurcation parameter, we prove that there exist Hopf bifurcation solutions around a positive constant solution, and their directions are also investigated. Finally, we validate the theoretical results through numerical simulations.