Invasion dynamics of mussels in deterministic and stochastic environments: A mussel-algae model with manual removal

To address whether alien mussels can successfully invade a new habitat, we investigate the dynamic behaviors of the mussel-algae interaction using both deterministic and stochastic models of differential equations by simultaneously incorporating the manual removal of mussels. In the deterministic model, we derive the ecological invasion index of mussels, denoted as R₀. We find that when R₀<1, the invasion of the mussel population may fail, whereas when R₀>1, the invasion is successful. Additionally, we observe a backward bifurcation phenomenon, where the success of the invasion also depends on the initial population size of mussels; sufficiently large initial populations can lead to successful invasions even when R₀<1. Similarly, in parallel to the deterministic model, the stochastic invasion index (R₀^s) of mussels serves to distinguish model outcomes. When R₀^s<1, it is highly likely that the mussel invasion fails, while when R₀^s>1, the invasion is successful, facilitated by the presence of an ergodic stationary distribution. Theoretical analysis demonstrates that the rate at which all solutions converge to the stationary distribution is polynomial. Moreover, we conclude that human intervention represents the most effective approach to mitigate invasion, and it is possible to sustain both mussel and algae populations through proper mussel removal strategies.