Some further results for a class of weighted nonlinear elliptic equations

Following our previous work [M. Lin, Multiple positive solutions for a class of weighted nonlinear elliptic equations, preprint, 2006], we give an exact growth order near zero for positive solutions of a class of weighted elliptic equations and use it to give the existence of multiple positive solutions and sign changing solutions of the considered problem.     

Positive solutions to generalized Dickman equation

The paper investigates large-time behaviour of positive solutions to a generalized Dickman equation. The asymptotic behaviour of dominant and subdominant positive solutions is analysed and a structure formula describing behaviour of all solutions is proved. A criterion is also given for sufficient conditions on initial functions to generate positive solutions with prescribed asymptotic behaviour with values of their weighted limits computed.     

一类奇异次线性边值问题正解存在的充分必要条件

本文研究一类奇异次线性边值问题正解的存在性，得到Ｃ［０，１］正解和Ｃ１［０，１］正解存在的充分必要条件，也得到正解的唯一性．     

一类糖酵解模型正平衡解的存在性分析

研究生化反应中具有代表性的一类糖酵解模型.运用先验估计讨论非常数正平衡解的不存在性,得到非常数正平衡解存在的必要条件.在常数平衡解Turing不稳定的基础上,利用度理论方法和解的先验估计,进一步给出非常数正平衡解存在的充分条件.     

一类四阶次线性奇异边值问题的正解

本文利用极大值原理和通过构造上下解给出了一类四阶次线性微分方程的奇异边值问题有C2[0,1]和C3[0,1]正解存在的充分必要条件.     

Positive Solutions for a Lotka–Volterra Prey–Predator Model with Cross-Diffusion of Fractional Type

In this paper we consider a Lotka–Volterra prey–predator model with cross-diffusion of fractional type. The main purpose is to discuss the existence and nonexistence of positive steady state solutions of such a model. Here a positive solution corresponds to a coexistence state of the model. Firstly we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system. Secondly we derive some necessary conditions to ensure the existence of positive solutions, which demonstrate that if the intrinsic growth rate of the prey is too small or the death rate (or the birth rate) of the predator is too large, the model does not possess positive solutions. Thirdly we study the sufficient conditions to ensure the existence of positive solutions by using degree theory. Finally we characterize the stable/unstable regions of semi-trivial solutions and coexistence regions in parameter plane.     

A qualitative study on general Gause-type predator–prey models with non-monotonic functional response

In this paper, we study a diffusive predator–prey model with general growth rates and non-monotonic functional response under homogeneous Neumann boundary condition. A local existence of periodic solutions and the asymptotic behavior of spatially inhomogeneous solutions are investigated. Moreover, we show the existence and non-existence of non-constant positive steady-state solutions. Especially, to show the existence of non-constant positive steady-states, the fixed point index theory is used without estimating the lower bounds of positive solutions. More precisely, calculating the indexes at the trivial, semi-trivial and positive constant solutions, some sufficient conditions for the existence of non-constant positive steady-state solutions are studied. This is in contrast to the works in previous papers. Furthermore, on obtaining these results, we can observe that the monotonicity of a prey isocline at the positive constant solution plays an important role.     

Positive solutions for ratio-dependent predator-prey interaction systems

In this paper, we study the dynamics of predator-prey interaction systems between two species with ratio-dependent functional responses. First we provide sufficient and necessary conditions for positive steady-state solutions, and then we investigate the relationships between positive equilibria and positive solutions of the system over a large domain. Furthermore, we deal with the uniqueness and the stability of positive steady-states solutions with some assumptions. In addition, we discuss the extinction and the persistence results of time-dependent positive solutions to the system.     

On the semilinear reaction diffusion system arising from nuclear reactors

In this work, the initial-boundary value problem for a class of semilinear reaction-diffusion systems is considered. By an abstract fixed point theorem on positive cone together with an accurate a priori estimate, we establish the coexistence of the positive stationary solutions and the uniqueness of ordered positive stationary solutions. Next, we study the global existence and blowup of positive solutions and obtain a threshold result. Finally, we give the blowup rate estimate of positive blowup solutions.     

A note on positive non-constant steady-state solutions to a reaction-diffusion model

In the present paper, we investigate a reaction-diffusion system with feedback effect subject to the homogeneous Neumann boundary condition and study the positive steady-state solutions. We establish a priori estimates for positive steady-state solutions and derive some results for non-existence of positive non-constant steady-state solutions. Our analysis complements the existing results on this model.     

具有阶段结构且食饵有避难所的模型分析

研究了捕食者具有阶段结构且食饵有避难所的非自治捕食系统.利用Lyapunov函数方法得到了系统持续生存的条件,以及在一定条件下存在唯一全局渐进稳定的周期正解.对于更广泛的概周期现象,也得到了存在唯一全局渐进稳定的概周期正解的充分条件.     

Solutions to the system of operator equations AXB=C=BXA

In this paper, we present some necessary and sufficient conditions for the existence of solutions, hermitian solutions and positive solutions to the system of operator equations AXB=C=BXA in the setting of bounded linear operators on a Hilbert space. Moreover, we obtain the general forms of solutions, hermitian solutions and positive solutions to the system above.     

Stability of stationary solutions in models of the Calvin cycle

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in the model of Zhu et al. (2009) for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models derived from the work of Grimbs et al. (2011), for which it was known that more than one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.     

Steady-state solutions of a reaction–diffusion system arising from intraguild predation and internal storage

Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.     

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.     

一类具第三类功能反应且食饵具有避难所的捕食系统的分析

研究了一类具第三类功能反应且食饵具有避难所的非自治捕食系统.利用Lyapunov函数方法得到了系统持续生存的条件,以及在一定条件下,系统存在全局渐进稳定的周期正解.对于更广泛的概周期现象,也得到了存在唯一全局渐进稳定的概周期正解的充分条件.     

Positive solutions of a diffusive prey–predator model in a heterogeneous environment

In this paper, we investigate a diffusive prey–predator model in a spatially degenerate heterogeneous environment. We are concerned with the positive solutions of the model, and obtain some results for the existence and non-existence of positive solutions. Moreover, the multiplicity, stability and asymptotical behaviors of positive solutions with respect to the parameters are also studied.     

二阶非线性中立型时滞差分方程的正解

本文研究了一类二阶非线性中立型时滞差分方程的正解,得到了最终正解的存在性判据及存在正解的必要条件,建立了一些正解不存在性定理,所得结论推广并改进了已知的一些结果。     

Positive solutions of singular boundary value problem of negative exponent Emden—Fowler equation

This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden-Fowler equation. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C¹[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.     

关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^＊X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.