一类具有Lotka-Volterra竞争模型共存解的存在性与稳定性

该文讨论了在空间分布不均匀的环境下一类具有Lotka-Volterra二维竞争模型的共存解的存在性与稳定性.特别地,两个竞争物种被假设拥有不同的内禀增长率,不同的种内竞争系数和种间竞争系数.结果表明当扰动参数Υ充分小时,该模型的动力学行为被一些函数所刻画.该文使用的数学方法包含Lyapunov-Schmidt分解法,谱理论和单调动力系统理论.     

捕食者带有疾病的入侵反应扩散捕食系统的空间斑图

本文考察了一类在有界区域内在零流边界条件下捕食者带有疾病的入侵反应扩散捕食系统.在没有入侵反应扩散的条件下考虑了这类系统的局部和全局稳定性.找到了具有入侵反应扩散系统的非常数定态解存在性和不存在性的充分条件,其存在预示着空间斑图的形成.文中结论表明当物种的生存空间很大,捕食者的捕食趋向很小时,没有空间斑图出现,两物种不能共存且没有疾病广泛传播.当入侵反应扩散系数很大,自扩散系数固定时,空间斑图出现,两物种能共存,这时疾病也广泛存在.     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.     

一个带毒生化反应竞争模型正平衡解的全局稳定性

研究两个微生物竞争同一营养,而其中一个竞争者会产生毒素抑制另一竞争者且产物系数γi(S)(i=1,2)为一般的非减可导函数时的生化反应模型,指出两篇文献中存在的问题,分析系统平衡点的稳定性,三维系统经历Hopf分支的条件和由此产生的周期解的稳定性.证明系统中某一微生物物种处于竞争劣势而趋于灭绝时另一微生物物种和营养在二维稳定流形上极限环的存在性.并用具体的实例验证文中所得结论.     

一类带有毒素生产的比率型Chemostat模型的定性分析北大核心CSCD

研究了一类带有毒素生产的比率型Chemostat模型.分析了系统平衡点的存在性及局部渐近稳定性.运用Lyapunov-LaSalle不变性原理和极限系统理论,得到了平衡点全局渐近稳定的充分条件.结论表明在满足不同条件时,不仅竞争排斥原理成立,竞争共存也是成立的.     

一类带有毒素生产的比率型Chemostat模型的定性分析

研究了一类带有毒素生产的比率型Chemostat模型.分析了系统平衡点的存在性及局部渐近稳定性.运用Lyapunov-LaSalle不变性原理和极限系统理论,得到了平衡点全局渐近稳定的充分条件.结论表明在满足不同条件时,不仅竞争排斥原理成立,竞争共存也是成立的.     

一类带毒的生化反应竞争模型的定性分析

有些微生物在连续培养中会产生毒素来抑制竞争者,同时竞争中也会产生一些振荡行为.本文研究两个微生物竞争同一营养,而其中一个竞争者会产生毒素抑制另一竞争者且产物系数为一般的形如δ1=A1+B1Sn,δ2=A2+B2Sm的函数时的生化反应模型.分析了系统平衡点的稳定性和当系统的某一微生物物种处于竞争劣势而趋于灭绝时另一微生物物种和营养的二维流形上极限环的存在性.     

具有时滞的周期非均匀竞争恒化器模型的空间动力学

本文提出了一个具有时滞的周期非均匀单种营养基——双种微生物的竞争恒化器模型,利用半群理论, 获得了该模型解的存在唯一性. 进一步, 建立了该模型的竞争排斥原理, 给出了两竞争物种共存的充分条件.     

个体尺度差异下竞争种群系统的稳定性

研究一类基于个体尺度分布的竞争种群系统平衡态的稳定性.利用种群再生数获得了平衡态的存在性条件,借助特征根的分布给出了平衡态的稳定性判据,运用离散化程序展示了两个稳定性实例.     

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.     

Coexistence states of a predator–prey system with non-monotonic functional response

In this paper, we investigate sufficient and necessary conditions for coexistence states of a predator–prey interaction system between two species with non-monotonic functional response under Robin boundary conditions. In view of the results, there is a gap between these two conditions. In this case, we study the multiplicity, stability and some uniqueness of coexistence states depending on some parameters.     

Coexistence and stability of a competition—diffusion system in population dynamics

The coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper, where the effect of dispersion (transportation) is taken into consideration. The mathematical problem involves a coupled system of Lotka-Volterra-type reaction-diffusion equations together with some initial and boundary conditions, including the Dirichlet, Neumann and third type. Necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given. For the stability problem, general criteria for the stability and instability of a steady-state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters. Also given are necessary and sufficient conditions for the existence of multiple steady-state solutions and the stability or instability of each of these solutions. Special attention is given to the Neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized. It is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable. The stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients.     

Existence and stability of coexistence states in a competition unstirred chemostat

In this paper, we consider the two similar competing species in a competition unstirred chemostat model with diffusion. The two competing species are assumed to be identical except for their maximal growth rates. In particular, we study the existence and stability of the coexistence states, and the semi-trivial equilibria or the unique coexistence state is the global attractor can be established under some suitable conditions. Our mathematical approach is based on Lyapunov–Schmidt reduction, the implicit function theory and spectral theory.     

Dynamics and steady-state analysis of an unstirred chemostat model with internal storage and toxin mortality

This study proposes and analyzes a reaction–diffusion system describing the competition of two species for a single limiting nutrient that is stored internally in an unstirred chemostat, in which each species also produces a toxin that increases the mortality of its competitors. The possibility of coexistence and bistability for the model system is studied by the theory of uniform persistence and topological degree theory in cones, respectively. More precisely, the sharp a priori estimates for nonnegative solutions of the system are first established, which assure that all of nonnegative solutions belong to a special cone. Then it turns out that coexistence and bistability can be determined by the sign of the principal eigenvalues associated with specific nonlinear eigenvalue problems in the special positive cones. The local stability of two semi-trivial steady states cannot be studied via the technique of linearization since a singularity arises from the linearization around those steady states. Instead, we introduce a 1-homogeneous operator to rigorously investigate their local stability.     

Long-term coexistence for a competitive system of spatially varying gradient reaction–diffusion equations

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction–diffusion equations. In this work we consider a system of two reaction–diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady states (the time-independent solutions) and examine their stability and bifurcations.     

Coexistence in a two species chemostat model with Markov switchings

This paper formulates a new switched two species chemostat model and discusses the coexistence behavior in the chemostat. A complete classification on the single-species chemostat is carried out firstly, where the stationary distribution with ergodicity is derived to exist and be unique. Then, based on the obtained stationary distribution and the comparison theorem, we put forward some sufficient conditions for the coexistence of microorganisms in the two species chemostat with Markov switchings. Moreover, when the species coexist in the deterministic chemostat for each state and have the same break-even concentrations for all states, they are proved to coexist still in the switched chemostat, which randomized the results of the classical deterministic chemostat. Results in this paper show that Markov switchings can contribute to coexistence of the two species.     

Coexistence states for a diffusive one-prey and two-predators model with B–D functional response

In this paper, we study a diffusive one-prey and two-predators system with Beddington–DeAngelis functional response. The sufficient and necessary conditions for the existence of coexistence states are obtained by means of the fixed point index theory. In addition, the stability and uniqueness of coexistence states are investigated. Finally, we give the sufficient conditions for extinction and permanence of the time-dependent system.     

A competition model for two resources in un-stirred chemostat

This paper studies a un-stirred chemostat with two species competing for two growth-limiting, non-reproducing resources. We determine the conditions for positive steady states of the two species, and then consider the global attractors of the model. In addition, we obtain the conditions under which the two populations uniformly strongly persist or go to extinction. Since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species, it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species. In particular, it is found that both instability and bi-stability subcases of the two semitrivial steady states are included in the coexistence region. The two populations will go to extinction when both possess large diffusion rates. If just one of them spreads faster with the other one diffusing slower, then the related semitrivial steady state will be globally attracting. The techniques used for the above results consist of the degree theory, the semigroup theory, and the maximum principle.     

Analysis of diffusive two-competing-prey and one-predator systems with Beddington-Deangelis functional response

In this paper, a diffusive two-competing-prey and one-predator system with Beddington-DeAngelis functional response is considered. The sufficient and necessary conditions for the existence of coexistence states are provided using the fixed point index theory developed. In addition, the stability and uniqueness of coexistence states are investigated. Finally, this paper discusses the sufficient conditions for extinction and permanence of the time-dependent system.     

On a Lotka-Volterra Competition Diffusion Model with Advection

In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.