一类具有Allee效应的Filippov害虫治理模型的研究

本文构建一类具有Allee效应的Filippov害虫治理模型.首先研究两个子系统的平衡点的存在性和稳定性,并证明了鞍结点的存在.然后运用非光滑动力学系统理论,讨论了真、假、伪平衡点的存在性和稳定性条件.最后应用理论和数值模拟研究与边界结点分支有关的滑动分支,并讨论相应的生物学意义.     

关于Filippov变换的一个注记

本利用Ｆｉｌｉｐｐｏｖ变换导出来一个研究Ｌｉｅｎａｒｄ方程极限环不存在性的判据,它为证明某些Ｌｉｅｎａｒｄ方程不存在极限环提供了一个简捷有效的思想方法。     

有干预措施的Filippov戒烟模型的全局动力学

考虑到媒体和政府对吸烟的不连续干预策略, 在经典传染病模型的基础上, 建立了一类有干预措施的Filippov 戒烟模型. 利用Filippov 定性分析方法, 分析了模型的滑模动力学和全局动力学行为. 在不同的参数区域, 得到了系统无病平衡点、地方病平衡点或伪平衡点的全局渐近稳定性. 结果说明适当的干预能够降低吸烟人数并将吸烟人数控制在阈值之内.     

基于Allee效应诱导的Filippov生态系统的动力学行为研究

该文建立了一类由Allee效应诱导的非光滑Filippov切换系统.运用Filippov系统的定性分析方法,从理论上研究了系统的滑动区域、滑动模态和各类平衡点的存在性.同时用数值方法研究了系统的滑动模态分支、边界焦点分支及全局动力学行为.研究发现:Allee效应的强度可使种群的动态不稳定,不利于濒危生物种群的管理.     

钟摆模型极限环的存在性研究

钟摆系统是一类典型的分段光滑系统,结合Filippov系统刻画语言,解释了当钟摆无能量补充时,钟摆最终会停止在滑动集上的原因.利用数值模拟的方法,给出钟摆系统在有能量补充时,存在极限环的条件.最后,结合环域定理证明了一般的钟摆模型存在唯一稳定的极限环.     

一类具有时滞传染病模型的稳定性

讨论了具有双时滞的SIS传染病模型.研究了一个边界平衡点的全局稳定性和正平衡点的局部稳定性,得到了传染病最终消失和成为地方病的阈值.     

一类非连续病菌与免疫系统竞争模型的动力学分析

建立一类右端不连续的病菌与免疫系统竞争模型,讨论模型滑模的存在性,真假平衡点以及伪平衡点的存在性和全局稳定性及局部滑动边界点分歧等动力学性质.研究结果表明病菌与免疫细胞最终在伪平衡点或真平衡点处共存.最后,运用数学软件进行数值模拟,验证了所得的理论结果.     

一类食饵具有不育控制的捕食模型

讨论了一类食饵具有不育控制的两种群捕食模型,得到了系统平衡点的存在条件,证明了平衡点的局部渐近稳定性和全局稳定性,最后给出了全局稳定的数值模拟,以及对参数进行了分析讨论.     

具有非线性感染力的一类SIR传染病模型的阈值分析

文章针对一类SIR传染病模型进行了改进,考虑了非线性感染力对阈值的影响.主要分四种情形对非线性感染力下的传染病阈值进行了计算与分析.对结果分析可知,传染病的传播阈值与非线性感染力有着密切关系,同时,免疫率μ对传染病阈值λ_c也起着非常关键的决定性作用.     

一类具有时滞的流行病模型分析

讨论了一类带有时滞的SE IS流行病模型,并讨论了阈值、平衡点和稳定性.模型是一个具有确定潜伏期的时滞微分方程模型,在这里我们得到了各类平衡点存在条件的阈值R0;当R0<1时,只有无病平衡点P0,且是全局渐近稳定的;当R0>1时,除无病平衡点外还存在唯一的地方病平衡点Pe,且该平衡点是绝对稳定的.     

Global dynamics of a Filippov plant disease model with an economic threshold of infected-susceptible ratio

This paper presents a Filippov plant disease model incorporating an economic threshold of infected-susceptible ratio, above which control strategies of replanting or removing are needed to be carried out. Based on the Filippov approach, we study the sliding mode dynamics and further the global dynamics. It is shown that there is a unique equilibrium, which is a disease-free equilibrium, an endemic equilibrium or a pseudo-equilibrium. Moreover, the equilibrium is proved to be globally asymptotically stable. Our results indicate that the control goal can be achieved by taking appropriate replanting and removing rate.     

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.     

Global dynamics of Filippov-type plant disease models with an interaction ratio threshold

A Filippov-type plant disease model is developed by introducing a interaction ratio threshold, the number of susceptible plants infected by per diseased plant, which determines whether control measures including replanting or roguing are carried out. The main purpose of this paper is to give a completely qualitative analysis of the model. By employing Poincaré maps, our analysis reveals rich dynamics including a global attractor bounded by a touching closed orbit, which is convergent in finite time from its outside, a global attractor bounded by two touching closed orbits and a pseudo-saddle, and a globally asymptotically stable pseudo-node. Moreover, we give biological implications of our results in implementing control strategies for plant diseases.     

Equilibrium,pseudoequilibrium and sliding-mode heteroclinic orbit in a Filippov-type plant disease model

Plant diseases have caused tremendous crop losses and have massive impacts on food security and environment. Modeling the spread of plant diseases and understanding the dynamics of the resulting plant disease models may provide practical insights on designing effective control measures. In this paper, by incorporating cultural strategies and economic threshold policy, we present a Filippov-type plant disease model. The resulting model has state dependent discontinuous right-hand side and thus non-smooth analysis and generalized Lyapunov approach are employed for model analysis. We show that the model exhibits the phenomena of stable equilibrium, unstable pseudoequilibrium as well as sliding-mode heteroclinic orbit. Biological implications of our results in implementing control strategies for plant diseases are also discussed.     

Global analysis for an epidemical model of vector-borne plant viruses with disease resistance and nonlinear incidence

Vector-borne disease models play an important role in understanding the mechanism of plant disease transmission. In this paper, we study a vector-borne model with plant disease resistance, disease exposed period and nonlinear incidence. We compute the basic reproduction number, determine the implicit locations of equilibria and then investigate their global stability by generalizing a classic geometric approach to higher dimensional systems. Higher dimensions cause greater difficulties such as the construction of the transformation matrix and the estimate of the $Lozinski\tilde{\iota}$ measure in this geometric approach. For a complete control of vector-borne diseases, a quantitative way is provided by the given expression of the basic reproduction number, from which we need not only increasing plant disease resistance but also decreasing the contact rate between infected plants and susceptible vectors instead of a single one of them.     

Modeling plant disease with biological control of insect pests

Abstract

Introduction: This article discusses the problem of plant diseases that pose major threat to agriculture in several parts of the World. Herein, our focus is on viruses that are transmitted from one plant to another by insect vectors. We consider predators that prey on insect population leading to reduction in infection transmission of plant diseases. Methods: We formulate and analyze a deterministic model for plant disease by incorporating predators as biological control agents. Existence of equilibria and the stability of the model are discussed in-detail. Basic reproduction number R₀ of the proposed model is also computed and this helps in determining the impact of different key parameters on the transmission dynamics of disease. Additionally, the proposed model is extended to stochastic model and simulation results of both deterministic and stochastic models are compared and analyzed. Results: Our results of stochastic model show the less number of infected plants and insects compared to corresponding results for deterministic model. Also, our results analyze the impact of different key parameters on the equilibrium levels of infected plants and identify the key parameters. Discussion: Presented results are used to conclude and demonstrate that the biological control is effective in reducing the infection transmission of plant disease and there is a need to use plant-insect-specific predators to get desirable results.     

Simple Filippov superalgebras of type B(m, n)

This account is a first step toward a classification of finite-dimensional simple Filippov superalgebras over an algebraically closed field of characteristic 0. Here, n-ary Filippov superalgebras with nonzero even and odd parts are treated for the case n ⩾ 3. Supported by RFBR (grant No. 05-01-00230) and by SB RAS (Integration project No. 1.9 and Young Researchers Support grant No. 29). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 240–261, March–April, 2008.     

Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems

This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems.