AI 中文总结
研究均匀和非均匀资源环境下觅食者 - 剥削者系统的周期动力学，通过构建模型描述三者相互作用，分析不同资源更新机制及条件下的周期解情况，揭示了资源更新、可用性与空间非均匀性对动力学行为的影响。
AI 中文摘要
我们研究了在均匀和非均匀资源环境下具有趋化级联的觅食者 - 剥削者系统中的时间周期动力学。该模型描述了觅食者、剥削者和环境资源之间的相互作用，其中觅食者向更高资源密度移动，而剥削者向觅食者密度更高的区域聚集。结果表明，不同的资源更新机制以根本不同的方式塑造周期动力学。对于周期性资源更新率，我们证明了对于任何正更新率都存在正时间周期解，并在参数的适当条件下进一步证明了它们的全局稳定性。对于均匀环境，只有大的资源更新率才能通过霍普夫分岔使常数稳态不稳定，从而产生非常数时间周期解。对于空间非均匀和时间均匀的环境，数值模拟表明空间非均匀性根据总资源可用性对周期动力学表现出相反的影响。当资源足够丰富时，空间非均匀性倾向于抑制时间振荡的出现，而当资源相对稀缺时，它可能促进振荡行为，并且足够集中的局部资源供应可以触发局部甚至全局时间振荡。这些发现揭示了资源更新机制、资源可用性和空间非均匀性在塑造动力学行为中的微妙相互作用。
英文摘要
We investigate time-periodic dynamics in a forager-exploiter system with a taxis cascade under both homogeneous and heterogeneous resource environments. The model describes the interactions among foragers, exploiters, and environmental resources, where foragers move toward higher resource densities while exploiters aggregate toward regions with higher forager densities. Our results show that different resource renewal mechanisms shape periodic dynamics in fundamentally different ways. Precisely, for time-periodic resource renewal rates, we establish the existence of positive time-periodic solutions for any positive renewal rate and further prove their global stability under suitable conditions on the parameters. In contrast, for homogeneous environments, only large resource renewal rates can destabilize the constant steady state through the Hopf bifurcation, thereby generating non-constant time-periodic solutions. Interestingly, for spatially heterogeneous and temporally homogeneous environments, our numerical simulations indicate that spatial heterogeneity exhibits opposing effects on periodic dynamics depending on total resource availability. When resources are sufficiently abundant, spatial heterogeneity tends to suppress the emergence of temporal oscillations, whereas when resources are relatively scarce, it may instead promote oscillatory behaviors, and sufficiently concentrated local resource supplies can trigger local or even global temporal oscillations. These findings reveal a delicate interplay among resource renewal mechanisms, resource availability, and spatial heterogeneity in shaping dynamics behavior.
