  This paper is the first to propose an allelopathic phytoplankton competition
ODE model influenced by a fear effect based on natural biological phenomena. It
is shown that the interplay of this fear effect and the allelopathic term cause
rich dynamics in the proposed competition model, such as global stability,
transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation.
We also consider the spatially explicit version of the model and prove
analogous results. Numerical simulations verify the feasibility of the
theoretical analysis. The results demonstrate that the primary cause of the
extinction of non-toxic species is the fear of toxic species compared to
toxins. Allelopathy only affects the density of non-toxic species. The
discussion provides guidance for the conservation of species and the
maintenance of biodiversity.
