Global dynamics of some evolution equations with temporal and sptatial structure

一些具有时空结构的演化方程的全局动力学

• 批准号: RGPIN-2016-04665
• 负责人: Zou, Xingfu
• 金额: $ 2.91万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615223
• 关键词: Global; dynamics; some; evolution; equations

This proposed research program consists of two parts: (A) investigate the dynamics of some dynamical systems arising from biology in the forms of delay differential equations and reaction-diffusion equations; (B) re-examine the predator-prey interactions by incorporating the anti-predator defence responses of the prey into the model systems. For (A), the focus will be on the impact of the temporal delay (maturation delay or latency delay) and spatial diffusion, and among the topics are the equilibria or steady states and their stability, multiplicity and spatial patterns of steady states, basins of attraction, various bifurcations, travelling wave solutions and spreading speeds. The new methodology developed and new results obtained will enhance our knowledge on the effects of temporal delays and spatial diffusion on solution behaviours and enrich the theory and methodology of dynamical systems; the results will also help us better understand the evolution processes of populations, their interactions and their spatial spreads, as well as infection and spread mechanisms of infectious diseases. All these will help us predict and control future populations of some species and some infectious diseases. For (B), some exiting well-know predator-prey models will be modified to include various anti-predator defence responses, such as reducing productions, decreasing parental care and increasing vigilance and mobility etc. The resulting new models will be analysed and compared to the existing ones. Since such responses widely exist but have been largely overlooked in existing models, such re-examinations will help us better understand predator-prey interactions and take effective measures to prevent some species from going to extinction due to direct (predation) and indirect effects (fear) from their predators.

本研究计划包括两部分：（A）研究生物学中的时滞微分方程和反应扩散方程形式的动力系统的动力学;（B）通过将捕食者的反捕食防御反应引入模型系统，重新研究捕食者-食饵相互作用。对于（A），重点将放在时间延迟（成熟延迟或潜伏延迟）和空间扩散的影响上，主题包括平衡或稳态及其稳定性，稳态的多重性和空间模式，吸引盆地，各种分叉，行波解和传播速度。所开发的新方法和所获得的新结果将增强我们对时滞和空间扩散对解行为影响的认识，丰富动力系统的理论和方法;其结果也将有助于我们更好地理解种群的演化过程、相互作用和空间传播，以及传染病的感染和传播机制。所有这些都将有助于我们预测和控制某些物种的未来种群和某些传染病。对于（B），我们将修改一些现有的著名的捕食者-食饵模型，使之包括各种各样的反捕食者防御反应，如减少生产、减少亲代照料、增加警惕性和机动性等。由于这种反应广泛存在，但在现有的模型中被忽视了，这种重新检查将有助于我们更好地了解捕食者-猎物相互作用，并采取有效措施，防止一些物种因直接（捕食）和间接影响（恐惧）而灭绝。