时间周期中立型偏泛函微分方程的时空传播特性

Neutral partial functional differential equation is an important kind of mathematical models which describe scientific problems arising from fields of engineering and population biology etc. Ttaveling wave solutions and asymptotic speed of spread can not only reveal the characteristics of equations, but also deeply describe dynamics on spatio-temporal propagation of real objects. However, there are rare results about this theme for this kind of equation resulting from its complicated construction, and periodic factors haven't been considered yet. In this project, we will invesigate characteristics on spatio-temporal propagation for a general neutral partial functional differential equation with time period . It mainly contains the following three parts: (1) establishing the existence of time-periodic traveling fronts by using methods such as degree theory or lower and upper solutions combining with fixed point theorem; (2) studying the uniqeness and stability of time-periodic traveling fronts by using methods of lower and upper solutions and squeezing method; (3) establishing asymptotic spreading speed by using the method of truncation and approximation. Furthermore, basing on these studies, spatio-temporal propagation of species with age structure will be considered by modeling a neutual partial functional differential equation with time period. These obtained results will furtherly reveal properties of solutions of partial functional differential equations with neutral type and patterns of spatio-temporal evolution of species with age structure. And it will be also helpful for protecting species and making them develop sustainablely.

中立型偏泛函微分方程是一类描述工程、种群生物学等众多领域中科学问题的重要数学模型。行波解和渐近播速既揭示了方程本身的性质，又深刻地刻画了实际对象的时空传播动力学特性。但由于中立型偏泛函微分方程结构复杂，目前针对此类方程研究行波解和渐近播速的工作仍属罕见，且未考虑时间周期性因素。本课题着重研究一般的时间周期中立型偏泛函微分方程的时空传播性态。主要包括：(1) 采用度理论、上下解结合不动点定理等建立时间周期行波解的存在性；(2) 利用上下解及夹逼技巧等研究时间周期行波解的唯一性和稳定性；(3) 利用截断、无穷逼近等方法建立渐近播速。在此基础上，通过对具有年龄结构的种群建立时间周期中立型偏泛函微分方程模型，研究种群的时空传播特性。本课题的研究将进一步揭示中立型偏泛函微分方程解的性态及具有年龄结构的种群的时空演变规律，为更好地保护种群的可持续发展提供必要的理论指导。

中立型偏泛函微分方程在工程、种群生物学等众多领域中有着重要的应用。行波解是揭示方程本质特性一类特殊形式的解，能深刻地刻画实际对象的时空传播动力学特性。本项目首先利用线性变换及Ikehara定理等方法研究并获得了一类具有拟单调反应项中立型偏泛函微分方程行波解的指数渐近性态及行波解唯一性；其次，通过构造新型上下解，结合Schauder不动点定理等方法改进了一类具有拟单调反应项中立型偏泛函微分方程行波解存在性结果，并进一步通过构造辅助方程，结合Schauder不动点定理等方法得到了一类具有非单调反应项中立型偏泛函微分方程行波解存在性；最后，获得了一类具有非局部反应项的时滞SIR疾病模型异宿轨及行波解的存在性。另外，进一步研究了几类延迟的分数阶扩散微分方程的数值方法，获得了算法的收敛性和稳定性的理论分析结果；利用代数图论、Lyapunov理论及 Barbalat引理等方法技巧研究并获得了几类无领导者型或领导-跟随者型的多智能体系统的一致性。本项目获得的结果有利于揭示中立型偏泛函微分方程解的性态，为研究此类方程提供一些参考，也为更好地应用于相关领域打下一定基础。