Study of some systems from population dynamics and disease transmission dynamics with heterogeneity

本研究计划由两部分组成。(A)研究两类系统在空间变化异质性下的时空动态，这种异质性可能解释气候变化或工业化造成的环境/资源变化。一类是具有成熟时滞的成熟种群的标量时滞微分方程，另一类是相互作用种群的种群动力学系统，包括竞争种群和捕食-被捕食种群。(B)探索捕食者-食饵系统中的各种恐惧效应以及传染病传播动力学中恐惧效应的相应概念。 对于（A），重点将放在转移速度和轮廓，以及所涉物种的生物学特征，将如何影响管理系统的时空动态。这些特征包括出生函数的类型、死亡率、成熟延迟、相互作用的类型、扩散率和相互作用的强度。可能的主题包括灭绝/持久性，共存模式，传播速度，推动行波及其稳定性，以及初始分布的影响。我们将遇到一些新的动力系统，具有特殊类型的时空异质性与延迟。我希望开发一些新的方法，修改和改进一些现有的方法来实现我们的目标。研究结果有助于空间动力系统的理论和方法的发展，也有助于理解气候变化和工业化的影响。部分（B）是真实的世界问题驱动的。我们将捕食者-被捕食者（P-P）系统中的“恐惧效应”概念扩展到传染病传播动力学模型中的“易感染性”（S-I）相互作用，以反映在传染病流行期间，人类行为（例如，社会活动和旅行）将发生变化，这些变化是由于对疾病的恐惧，并可能随着流行病而演变。这部分的重点将是如何猎物的扩散，觅食和生产策略的演变，以应对感知其捕食者（S）;以及人类的预防水平将如何演变，以及这种演变对长期和短期疾病动态的影响。本研究的结果在农业和公共卫生领域具有潜在的应用价值：P-P系统中恐惧效应的研究结果可能为控制某些农业啮齿动物害虫提供新的和环境友好的方法;以及“恐惧效应”的结果（例如，以媒体报道和公共卫生机构的警告为代表）对人类行为变化及其影响的影响可能有助于向决策者提供信息，在公共卫生方面，控制或减轻传染病流行的有价值的指导方针。

This proposed research program consists of two parts. (A) Study spatial temporal dynamics of two classes of systems under spatially shifting heterogeneity that may account for environment/resource shifting caused by climate change or industrialization. One class are scalar delay differential equations for mature populations with maturation delay, and the other class are systems for population dynamics of interacting species including competing species and predator-prey species. (B) Explore various fear effects in predator-prey systems and the corresponding notion of fear effects in transmission dynamics of infectious diseases. For (A), the focus will be on how the shifting speed and the profile, together with the biological characteristics of the species involved, will affect the spatial temporal dynamics of governing systems. These characteristics include the type of birth functions, death rate, maturation delays, and type of interaction, rate of dispersal  and strengths of interaction. The possible topics include extinction/persistence, pattern of coexistence, spreading speed, pushed traveling waves and their stability, and impact of initial distribution. We will encounter some new dynamical systems with a special type of spatial-temporal heterogeneity with delay. I expect to develop some new approaches/methods, abd to modify and improve some existing ones to achieve our objectives. Thus, the results may contribute to the theory and methods for spatial dynamical systems, and they may also help understand the impact of climate change and industrialization. Part (B) is real world problem driven. We extend the notion of "fear effects" in predator-prey (P-P) systems to "susceptible-infectious" (S-I) interactions in modeling transmission dynamics of infectious diseases to reflect the reality that during the epidemic of an infectious disease, human beings behaviors (e.g., social activities and travel) will change and such changes are due to the fear of disease and may evolve with the epidemic. The focus in this part will be on how a prey's dispersal, foraging and production strategies evolve in response to perceiving its predator(s); and how human being's precaution level will evolve and the impact of such an evolution of precaution level on the long term and short term disease dynamics. I expect the results of this proposed research will have potential applications in agriculture and public health: the results on fear effect in P-P system may suggest new and environmentally friendly approaches for controlling some agriculture rodent pests; and the results on "fear effects" (e.g. represented by media coverage and public health agents' warning) on human's behavior change and its impacts may help inform decision makers,  in public health,  of valuable guidelines for controlling or mitigating the epidemics of an infectious disease.