Codimension one and codimension two bifurcations analysis of discrete mussel-algae model

In this paper, a continuous mussel-algae model was discretized using the explicit Euler method, yielding a discrete mussel-algae interaction model. Within this work, the stability and bifurcations of a discrete mussel-algae model were studied. First, the existence of the unique positive equilibrium point was given. By choosing the time step $\tau$ as the bifurcation parameter, we investigated several dynamic behaviors of the model. Using the center manifold theorem and bifurcation theory, the conditions for the existence of codimension-one bifurcations (Flip bifurcation and Neimark-Sacker bifurcation) were derived. Then, by substituting several variables and introducing new parameters, the conditions corresponding to the codimension-two bifurcation (1:2, 1:3, and 1:4 strong resonance) were evaluated. One can identify the existence of different bifurcation forms by essentially calculating the critical non-degeneracy coefficients. The numerical simulations validated the proposed results and illustrated the complicated dynamical behaviors of the mussel algae system. In addition, via numerical simulations, we discovered several pathways through which the system could reach chaos. Specifically, both the Flip bifurcation and the strong resonance bifurcations could eventually guide the system into a chaotic state. These results were distinct from those of the corresponding continuous model, providing a novel perspective for studying the relationship between the population densities of mussels and algae.