• Abstract

The dynamic behavior of alien mussels interacting with algae after arriving in a new environment has long been a focus of invasion ecology research. This paper extends and analyzes a classical mussel-algae model by incorporating a time delay in mussel filter feeding and accounting for environmental variability. We theoretically study the stochastic dynamics, including the global existence and uniqueness of the positive solution, the existence of a unique stationary distribution, and mussel extinction, using tools from stochastic analysis. Furthermore, we derive an explicit expression for the probability density function around the quasi-stable equilibrium by solving the corresponding Fokker-Planck equation. Our theoretical and numerical results indicate that: (a) larger environmental disturbances or artificial removal can effectively prevent the survival of alien mussels in novel habitats, (b) a decreased filter feeding rate leads to an accelerated extinction rate of mussels, and (c) an increased consumption constant c decelerates the transition rate of mussels from the initial state to the extinction state, as analyzed through the mean first passage time of mussels. These findings highlight the complex interaction between intrinsic and extrinsic factors in influencing the invasion dynamics of alien mussels.

• General Summary

• Abstract

Hopf bifurcation in delay differential equations has been a central topic in the study of complex dynamical behaviors in biological and ecological systems. In this review, we revisit Hopf bifurcation phenomena in single-species models that incorporate time delays, emphasizing recent progress in both ordinary and partial differential equation frameworks. We present a comprehensive overview of classic and contemporary models, such as Wright’s equation, Nicholson’s blowflies equation, and diffusive logistic models, highlighting criteria for local and global bifurcations, the geometric and analytical methods used to determine critical values, and the stability of emerging periodic solutions. The review also covers structured models with age, stage, advection, and spatial effects, as well as equations with multiple delays. Through this survey, we aim to consolidate theoretical insights and provide a unified understanding of delay-induced oscillations in population models, laying the groundwork for future developments in delay-driven dynamics.

• General Summary

• Abstract

A mathematical model is proposed that describes the adaptive spatial movement of prey towards higher population density to reduce predation risk. The model admits the increased nonlinearity and the global existence of solutions of the system is established in Sobolev space through analytical estimates. The conditions for the Turing instability from a coexistence steady state are obtained, and sharp conditions for the asymptotical stability of the positive equilibrium in a large region are established with the help of a Lyapunov function. Numerical simulations are presented to support the theoretical results and demonstrate the versatility of spatial models.

• General Summary