一类扩散Holling-Tanner模型行波解的存在性

在生态学中,可以用非线性反应扩散方程来描述种群在时间上的变化和在空间中的分布及扩散情况.对于扩散的生物种群模型,通过研究模型中方程的渐近性态,可以知道该种群是持续生存还是趋向灭绝.在非线性反应扩散方程的研究中,行波解由于其形式简单,研究比较方便,为研究偏微分方程的动力学行为提供了一些途径.文章对一类添加扩散项的扩散Holling-Tanner系统进行了定性分析,得到了系统平衡点局部渐近稳定的充分条件.再通过构造Liapunov函数的方法,得到扩散Holling-Tanner系统平衡点全局渐近稳定的条件,以及该系统行波解存在的充分条件,并进行了数值模拟.     

具有交通流量异质性的网络传染病模型研究

主要研究了异质集合种群网络上的移动和扩散行为对疾病传播的影响.针对现实社会中的网络所具有的异质性,分析了影响城市疾病传播的主要因素为网络拓扑结构以及城市交通流量异质性,建立了依赖于交通流量移出率的传染病动力学模型.通过分析模型的无病平衡点以及正平衡点的存在及其稳定性,发现人口流动会使交通较发达的城市拥有更多的染病者,更容易促使疾病的爆发.     

具有Monod-Haldane功能反应的一类食物链模型的动力学行为

该文研究了一个由食饵种群、捕食者种群和杂食者种群所构成的食物链系统, 其捕食功能反应为Monod-Haldane功能反应. 应用定性分析和Hopf分支理论, 得到了该系统边界平衡点的全局稳定性和周期解存在性的判别准则. 为了概括和归类这个系统的全局动力学行为,该文得到了具有不同动力学行为的参数区域. 应用MATLAB软件,该文提供了一个例子来展示这些结论, 并且表明: 这个系统能够产生非常复杂的动力学行为.     

一类基于集合种群网络的传染病模型研究

研究了一类基于集合种群网络的传染病模型.针对在疾病传播过程中,随着染病者数量的增加,被感染的人数会达到饱和,研究了带有饱和发生率的传染病模型,建立了不同集合种群之间扩散模式,并分析了模型动力学的性态,给出了无病平衡点及其稳定性和正平衡点的存在性.最后用数值模拟验证了理论结果的正确性.     

同类相食对两阶段结构种群模型的动力学影响

研究同类相食对成熟阶段个体具有密度制约的两阶段结构种群的动力学影响,分别分析了不具有同类相食和具有同类相食时两类模型的动力学性态,得到了它们具有相似的动力学性态,即种群灭绝平衡点总存在但不稳定,而种群存在平衡点总存在且是全局渐近稳定的结点.这意味着两个阶段种群密度的最终变化趋势是单调的.同时还讨论了种群存在平衡点的大小对同类相食的依赖性,以及同类相食存在时对种群存在平衡点的大小随自食相关参数的变化.     

恐惧效应对具有避难所和半封闭捕获的捕食模型的影响

研究一类具有恐惧效应及避难所和半封闭捕获项的捕食系统的动力学行为,讨论了系统平衡点的局部稳定性和正平衡点的全局稳定性.证明了系统存在Hopf分支.并考虑恐惧效应,避难所和半封闭捕获项对种群密度的影响,最后举例说明可行性.     

具有阶段结构的Lotka-Volterra合作系统的稳定性和行波解

文建立并研究了一个两物种成年个体相互合作的时滞反应扩散模型.利用线性化稳定性方法和Redlinger上、下解方法证明了该模型具有简单的动力学行为,即零平衡点和边界平衡点是不稳定的,而唯一的正平衡点是全局渐近稳定的.同时, 利用Wang, Li 和Ruan建立的具有非局部时滞的反应扩散系统的波前解的存在性,证明了该模型连接零平衡点与唯一正平衡点的波前解的存在性.     

一类具有相互干扰的食饵-捕食者模型的定性分析

研究了一类具有相互干扰和非线性饱和功能性反应且食饵种群非线性增长的食饵-捕食者模型的动力学行为.利用Pioncare-Bendixson环域定理,张芷芬唯一性定理,旋转向量场理论,获得了保证系统的边界平衡点和正平衡点全局渐近稳定的阀值条件.最后,给出了数值模拟结果.     

一个特殊的三维捕食链非自治扩散系统的持续生存

对一个特殊的三种群非自治捕食链扩散系统进行讨论.其特殊之处在于扩散现象发生在构成捕食链的中间种群间.通过利用比较原理进行微分不等式的比较,得到了该系统持续生存的条件;分析了扩散运动对该系统种群动力学行为的影响.     

具时滞和扩散单种群模型的一致持续生存与全局稳定性

讨论了一类具有时滞的单种群扩散模型,其中扩散依赖于时滞,利用同伦技术得到了模型存在正平衡点和系统一致持续生存的充分条件;同时通过构造适当的liapunov函数证明了系统正平衡点是全局渐渐稳定的.     

Asymptotic behaviors for functionals of random dynamical systems

In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.     

Stability analysis of Riccati differential equations related to affine diffusion processes

We study a class of generalized Riccati differential equations associated with affine diffusion processes. These diffusions arise in financial econometrics and branching processes. The generalized Riccati equations determine the Fourier transform of the diffusion's transition law. We investigate stable regions of the dynamical systems and analyze their blow-up times. We discuss the implication of applying these results to affine diffusions and, in particular, to option pricing theory.     

On nonlinear coupled diffusions in competition systems

A class of reaction-diffusion systems modeling plant growth with spatial competition in saturated media is presented. We show, in this context, that standard diffusion can not lead to pattern formation (Diffusion Driven Instability of Turing). Degenerated nonlinear coupled diffusions inducing free boundaries and exclusive spatial diffusions are proposed. Local and global existence results are proved for smooth approximations of the degenerated nonlinear diffusions systems which give rise to long-time pattern formations. Numerical simulations of a competition model with degenerate/non degenerate nonlinear coupled diffusions are performed and we carry out the effect of the these diffusions on pattern formation and on the change of basins of attraction.     

Diffusion maps tailored to arbitrary non-degenerate Itô processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.     

High-order approximation of Pearson diffusion processes

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.     

Invariant distributions and scaling limits for some diffusions in time-varying random environments

We consider a family of one-dimensional diffusions, in dynamical Wiener mediums, which are random perturbations of the Ornstein–Uhlenbeck diffusion process. We prove quenched and annealed convergences in distribution and under weigh-ted total variation norms. We find two kind of stationary probability measures, which are either the standard normal distribution or a quasi-invariant measure, depending on the environment, and which is naturally connected to a random dynamical system. We apply these results to the study of a model of time-inhomogeneous Brox’s diffusions, which generalizes the diffusion studied by Brox (Ann Probab 14(4):1206–1218, 1986) and those investigated by Gradinaru and Offret (Ann Inst Henri Poincaré Probab Stat, 2011). We point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations.     

Stability analysis in a class of discrete SIRS epidemic models

In this paper, the dynamical behaviors of a class of discrete-time SIRS epidemic models are discussed. The conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. The numerical simulations not only illustrate the validity of our results, but also exhibit more complex dynamical behaviors, such as flip bifurcation, Hopf bifurcation and chaos phenomenon. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models.     

Global directed dynamics of a Lotka–Volterra competition-diffusion system

In this study, we consider a directed–diffusion system describing the interactions between two organisms in heterogeneous environment. We first establish a linearly stability of the co-existence (positive) steady state. Then we further present a classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems.     

Singularly Perturbed Switching Diffusions: Rapid Switchings and Fast Diffusions

Motivated by many problems in optimization and control, this paper is concerned with singularly perturbed systems involving both diffusions and pure jump processes. Two models are treated. In the first model, the jump process changes very rapidly by comparison with the diffusion processes. In the second model, the diffusions change rapidly in comparison with the jump process. Asymptotic expansions are developed for the transition density vectors via a constructive method; justification of the asymptotic expansions and analysis of the remainders are provided.