Zhong Li

In this work, we formulate and analyze a discrete-time commensalism model in which the first species is subject to Michaelis–Menten type harvesting. We first investigate the existence, stability, and global attractivity of all feasible fixed points. Our results demonstrate that when the two species coexist stably, increasing the harvesting intensity leads to the extinction of the harvested species. We then establish the conditions under which various bifurcations occur, including transcritical, pitchfork, and fold bifurcations, with particular emphasis on the period-doubling route to chaos. In addition, a comprehensive analysis of the global dynamics is conducted. Numerical simulations highlight the critical influence of the commensalism strength: Moderate commensalism ensures stable coexistence of species and increases population density, whereas excessive commensalism destabilizes the system, triggering a period-doubling bifurcation cascade to chaos. Furthermore, the intrinsic stability of the host species is shown to profoundly modulate the dynamical complexity of the commensal species. Under certain conditions, instability of the host species can advance the onset of chaos in the commensal species.

In this paper, a predator-prey model with simplified Holling type IV functional response and Allee effect on predator is investigated. We discuss the existence and stability of equilibria, and show that the system has at most two positive equilibria. The results show that the double positive equilibrium is a cusp of codimension 3 and the anti-saddle is a weak focus of order at most 4. Also, we prove that the cusp type degenerate Bogdanov-Takens bifurcation of codimension 3 occurs at the double equilibrium, and the degenerate Hopf bifurcation of codimension 4 occurs at the anti-saddle. Our results show that if the Allee effect is small, predator and prey can achieve a stable coexistence state; conversely, if the Allee effect is large, the predator will be extinct, while prey will survive. Finally, in a changing environment, our research reveals that some complex transient dynamic phenomena such as tracking of unstable equilibria or limit cycles, extinction avoidance, and slow and fast regime shifts.

In this paper, considering constant environment and changing environment, we study the bifurcation and transient dynamics of a Leslie-Gower predator-prey model incorporating Smith growth and Allee effect. In a constant environment, we show that the origin is an attractor, and the system admits at most two positive equilibria: One is a saddle and the other can be a weak focus of order four. The unique equilibrium is a saddle-node or a cusp of codimension three. As system parameters vary, the system undergoes a sequence of bifurcations, including saddle-node bifurcations, degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension four. In a changing environment, by comparing these transient dynamics with the bifurcation structure of the corresponding constant environment system. We show that the system has rich transient dynamics, such as tracking of unstable equilibria or limit cycles, delay or avoid extinction, and slow and fast regime shifts. We show that low Smith growth or Allee effect is conducive to the stable coexistence of species, while the high Smith growth or Allee effect can eventually lead to the extinction of species.

This paper proposes and investigates a two-species amensalism system incorporating a Holling-Ⅱ functional response, which integrates both the saturated fear effect and the Allee effect on the first species. We first establish sufficient conditions for the existence and stability of equilibrium points, analyzing saddle-node, transcritical and pitchfork bifurcations while conducting comprehensive global dynamics analysis. Subsequently, stochastic perturbations are introduced to construct a random system, where we observe noise-induced state transitions and estimate the critical noise threshold for such transitions. Finally, numerical simulations are performed to validate the feasibility of the theoretical results. The study shows that excessively high fear does not necessarily lead to the extinction of the first species. Specifically, when the saturation fear parameter exceeds a certain threshold, the first species persists even under extreme fear conditions; however, when below this threshold, high fear levels still lead to extinction, though weak Allee effects significantly slow this process.

This paper investigates a model of amensalism, in which the first species is influenced by the combined effects of refuge and fear, while the second species exhibits an additive Allee effect. The paper analyzes the existence and stability of the equilibria of the system and derives the conditions for various bifurcations. In the global structure analysis, the stability at infinity is examined, and the phenomena of global stability and bistability in the system are analyzed. Additionally, a sensitivity analysis is employed to evaluate the impact of system parameters on populations. The study reveals that refuge has a significant positive effect on the first population, and refuge’s effect becomes more pronounced as the fear level increases. Under the strong Allee effect, when the initial density of the second species is low, the second species may eventually become extinct; when the initial density is high, if the refuge parameter is below a certain threshold, increasing the refuge parameter slows down the extinction of the first species, whereas, when the refuge parameter exceeds this threshold, the two species can coexist. Under the weak Allee effect, when the refuge parameter surpasses a certain threshold, the two species can achieve long-term, stable coexistence, and the threshold for the weak Allee effect is higher than that for the strong Allee effect.