标题： 具有毒素介导相互作用的浮游植物-浮游动物模型的离散动力学 显示英文标题

标题： Discrete Dynamics of a Phytoplankton-Zooplankton Model with Toxin-Mediated Interactions

摘要： 我们研究了包含Holling类型III捕食和Holling类型II毒素释放的离散浮游植物-浮游动物模型的动力学特性。 正不动点的存在性和稳定性被分析，结果表明当存在两个这样的点，$E_1$和$E_2$时，$E_2$始终是一个鞍点。 在$E_1$处验证了Neimark-Sacker分支，使用规范型方法，表明闭合不变曲线的出现。 这种分支意味着浮游植物和浮游动物种群可能会表现出持续的周期性振荡，这可能对应于自然的浮游植物爆发周期。 边界平衡点$(1,0)$的全局稳定性也得到了确立。 数值模拟被用来说明并确认理论结果。 显示英文摘要

摘要： We investigate the dynamics of a discrete phytoplankton-zooplankton model incorporating Holling type~III predation and Holling type~II toxin release. The existence and stability of positive fixed points are analyzed, and it is shown that when two such points, $E_1$ and $E_2$, exist, $E_2$ is always a saddle. A Neimark-Sacker bifurcation at $E_1$ is verified using the normal form method, indicating the emergence of closed invariant curves. This bifurcation implies that phytoplankton and zooplankton populations may exhibit sustained periodic oscillations, which could correspond to natural plankton bloom cycles. The global stability of the boundary equilibrium $(1,0)$ is also established. Numerical simulations are presented to illustrate and confirm the theoretical findings.