Positive solutions of a competition model for two resources in the unstirred chemostat

This paper deals with a competition model between two species for two growth-limiting and perfectly complementary resources in the unstirred chemostat. The main purpose is to determine the exact range of the parameters of two species so that the system possesses positive solutions, and to investigate multiple positive steady states of the system. The main tools used here include the monotone methods and the topological fixed point theory developed by Amann.     

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.     

具有质载和内抑制剂的非均匀恒化器模型共存解的多重性

本文研究一类具有质载和内抑制剂的非均匀恒化器竞争模型. 基于对第二类极限系统分歧的进一步分析, 我们研究了此模型共存解的多重性和稳定性, 从而在一定条件下得到了模型共存解分歧的确切形状, 验证了已有的数值结果.     

Crowding effects on coexistence solutions in the unstirred chemostat

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.     

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.     

一类具有抑制剂的非均匀恒化器模型的共存态

研究了一类具有抑制剂和Beddington-DeAngelis功能反应项的非均匀恒化器模型.根据单调动力系统理论得到了正平衡解的存在性.利用度理论、分歧理论以及摄动理论,分析了抑制剂对系统正平衡解及渐近行为的影响.结果表明当体现抑制作用的参数μ充分大时,此模型或者没有正解,并且一个半平凡晌非负解是全局吸引的;或者模型的所有正解均由一个极限问题决定.     

Asymptotic behavior for a cellular replication and maturation model

We consider a nonlocal first order partial differential equation with time delay that models simultaneous cell replication and maturation processes. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the equation. We then analyze the asymptotic behavior of the solution via upper–lower solution technique.     

Asymptotic behavior and stability of second order neutral delay differential equations

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed point method is introduced as a third approach to study the asymptotic behavior. We conclude the paper with an application to a mechanical model of turning processes.     

Asymptotic behavior of strong solutions of a simplified energy-transport model with general conductivity

In this paper we study a simplified transient energy-transport model in semiconductors with a general conductivity and the Dirichlet boundary conditions on an interval. By using a new iterative scheme, we prove the global existence and uniqueness of strong solutions provided that the variation of the temperature is small. Also, the existence and stability of stationary solutions are proved if the temperature is large.     

Asymptotic stability of a mathematical model of cell population

We consider a simplified system of a growing colony of cells described as a free boundary problem. The system consists of two hyperbolic equations of first order coupled to an ODE to describe the behavior of the boundary. The system for cell populations includes non-local terms of integral type in the coefficients. By introducing a comparison with solutions of an ODE's system, we show that there exists a unique homogeneous steady state which is globally asymptotically stable for a range of parameters under the assumption of radially symmetric initial data.     

一类具有抑制剂和质载的非均匀恒化器模型的共存解

研究了一类具有抑制剂和质载的非均匀恒化器模型.首先,采用锥映射上不动点指标理论得到了物种共存的充分条件.然后,根据度理论及摄动理论,研究了模型正平衡态解的唯一性和稳定性.结果表明当抑制剂的影响充分大时,模型存在唯一的渐近稳定的共存解.最后,利用比较原理和一致持续理论研究了系统的长时行为,并采用数值模拟的方法对所得结论进行了验证和补充.     

Asymptotic behavior of solutions of a periodic diffusion system

This paper deals with a periodic reaction-diffusion system of plankton allelopathy under homogeneous Neumann boundary conditions. Based on the result of Ahmad and Lazer, we show some estimates and nonexistence results for the positive solutions of the system. Furthermore, we investigate the asymptotic behavior of the solutions of the system, that is one species dies out and the other exists as time t tends to infinity.     

Asymptotic behavior of delay differential equations with instantaneously terms

This paper deals with scalar delay differential equation with instantaneously term

     

Asymptotic behavior of certain delay differential equations with forcing term

We study the asymptotic behavior of solutions of the following forced delay differential equation:

Some sufficient conditions that guarantee every solution of the equation to converge to zero are obtained. The results obtained are applied to some well-known delay differential ecological equations with forcing term.     

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as of solutions to a linear, parabolic system of equations with time-dependent coefficients in , where is a bounded domain. On we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function . This includes in particular situations when the coefficients may take different values on different parts of and the boundaries between them can move with t but stabilize as . The main result is an asymptotic representation of solutions for large t. A consequence is that for , the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.     

Asymptotic behavior of large solutions to -Laplacian of Bieberbach–Rademacher type

By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|^p−2u)=b(x)f(u) in a bounded ΩR^N subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular.     

Asymptotic behavior of solutions of a three-species predator-prey model with diffusion and time delays

In this paper, we deal with a reaction-diffusion system with time delays arising from a three-species predator-prey model under the homogeneous Neumann boundary conditions, and study the asymptotic behavior of solutions.     

Global behavior of a three-dimensional linear fractional system of difference equations

We investigate the global asymptotic behavior of solutions of the system of difference equations

where the parameters a, b, c, d, e, and f are in (0,∞) and the initial conditions x₀, y₀, and z₀ are arbitrary non-negative numbers. We obtain some global attractivity results for the positive equilibrium of this system for different values of the parameters.