Jia Lili, Wang Changyou
Department of Basic Teaching, Dianchi College, Kunming 650228, China.
School of Mathematical Sciences, V.C. & V.R. Key Lab of Sichuan Province, Sichuan Normal University, Chengdu 610066, China.
Biology (Basel). 2025 Apr 24;14(5):462. doi: 10.3390/biology14050462.
In this work, we present a novel four-species periodic diffusive predator-prey model, which incorporates delay and feedback control mechanisms, marking substantial progress in ecological modeling. This model offers a more realistic and detailed portrayal of the intricate dynamics of predator-prey interactions. Our primary objective is to establish the existence of a periodic solution for this new model, which depends only on time variables and is independent of spatial variables (we refer to it as a spatially homogeneous periodic solution). By employing the comparison theorem and the fixed point theorem tailored for delay differential equations, we derive a set of sufficient conditions that guarantee the emergence of such a solution. This analytical framework lays a solid mathematical foundation for understanding the periodic behaviors exhibited by predator-prey systems with delayed and feedback-regulated interactions. Moreover, we explore the global asymptotic stability of the aforementioned periodic solution. We organically combine Lyapunov stability theory, upper and lower solution techniques for partial differential equations with delay, and the squeezing theorem for limits to formulate additional sufficient conditions that ensure the stability of the periodic solution. This stability analysis is vital for forecasting the long-term outcomes of predator-prey interactions and evaluating the model's resilience against disturbances. To validate our theoretical findings, we undertake a series of numerical simulations. These simulations not only corroborate our analytical results but also further elucidate the dynamic behaviors of the four-species predator-prey model. Our research enhances our understanding of the complex interactions within ecological systems and carries significant implications for the conservation and management of biological populations.
在这项工作中，我们提出了一个新颖的四物种周期扩散捕食者 - 猎物模型，该模型纳入了时滞和反馈控制机制，在生态建模方面取得了重大进展。这个模型为捕食者 - 猎物相互作用的复杂动态提供了更现实、更详细的描述。我们的主要目标是确定这个新模型存在一个仅依赖于时间变量且与空间变量无关的周期解（我们将其称为空间均匀周期解）。通过运用针对时滞微分方程的比较定理和不动点定理，我们推导出了一组保证这种解出现的充分条件。这个分析框架为理解具有时滞和反馈调节相互作用的捕食者 - 猎物系统所表现出的周期行为奠定了坚实的数学基础。此外，我们还探讨了上述周期解的全局渐近稳定性。我们有机地结合了李雅普诺夫稳定性理论、带时滞的偏微分方程的上下解技术以及极限的夹逼定理，以制定确保周期解稳定性的额外充分条件。这种稳定性分析对于预测捕食者 - 猎物相互作用的长期结果以及评估模型对干扰的恢复力至关重要。为了验证我们的理论发现，我们进行了一系列数值模拟。这些模拟不仅证实了我们的分析结果，还进一步阐明了四物种捕食者 - 猎物模型的动态行为。我们的研究增进了我们对生态系统内复杂相互作用的理解，对生物种群的保护和管理具有重要意义。
