Ground state solutions for a class of fractional Kirchhoff equations with critical growth

In this paper, we study the effect of lower order perturbations in the existence of positive solutions to the fractional Kirchhoff equation with critical growth■ where a, b 0 are constants, μ 0 is a parameter,■ , and V : R~3→ R is a continuous potential function. For suitable assumptions on V, we show the existence of a positive ground state solution, by using the methods of the Pohozaev-Nehari manifold, Jeanjean's monotonicity trick and the concentration-compactness principle due to Lions(1984).     

A CONDITION OF THE EXISTENCE OF STABLE POSITIVE STEADY-STATE SOLUTIONS FOR A ONE PREDATOR TWO PREY SYSTEM

One predator two prey system is a research topic which has both the theoretical and practical values.This paper provides a natural condition of the existence of stable pcsitive steady-state solutions for the one predator two prey system.Under this conditon we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem,discuss the positive stable solution problem bifureated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.     

Positive Solutions for a Competition Model with an Inhibitor Involved

In the paper, we study the positive solutions of a diffusive competition model with an inhibitor involved subject to the homogeneous Dirichlet boundary condition. The existence, uniqueness, stability and multiplicity of positive solutions are discussed. This is mainly done by using the local and global bifurcation theory.     

UNIFORM BOUNDEDNESS AND STABILITY OF SOLUTIONS TO A CUBIC PREDATOR-PREY SYSTEM WITH CROSS-DIFFUSION

Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.     

THREE POSITIVE SOLUTIONS TO BVP FOR p-LAPLACIAN IMPULSIVE FUNCTIONAL DYNAMIC EQUATIONS ON A TIME SCALE

In this paper, we axe interested in the existence of three positive solutions to a BVP for p-Laplacian impulsive functional dynamic equations on a time scale. Using the five-functional fixed theory, we establish a Banach space and an appropriate operator. In this paper, we combine the delta-nabla p-Laplacian BVP with impulsive functional dynamic equations, and obtain some new sufficient conditions for the existence of three positive solutions to the BVP, and our result here generalizes the previous related results.     

The Nehari Manifold and Application to a Quasilinear Elliptic Equation with Multiple Hardy-Type Terms

In this paper, by using the Nehari manifold and variational methods, we study the existence and multiplicity of positive solutions for a multi-singular quasilinear elliptic problem with critical growth terms in bounded domains. We prove that the equation has at least two positive solutions when the parameters A belongs to a certain subset of JR.     

POSITIVE PERIODIC SOLUTION OF A PREDATOR-PREY SYSTEM WITH TIME DELAY

By using continuation theorem in coincidence degree theory,we study the existence of positive periodic solution for a delay and mutual interference predator- prey system with functional response,and obtain sufficient conditions for the existence of positive periodic solution.     

Periodic Solutions of a Model of Mitosis in Prog Eggs

In this paper,we discuss a simplified model of mitosis in frog eggs proposed by M.T. Borisuk and J.J. Tyson in [1]. By using rigorous qualitative analysis, we prove the existence of the periodic solutions on a large scale and present the space region of the periodic solutions and the parameter region coresponding to the periodic solution. We also present the space region and the parameter region where there are no periodic solutions. The results are in accordance with the numerical results in [1] up to the qualitative property.     

Existence and Multiplicity of Positive Solutions for Fractional Differential Equation with Parameter

In this paper, by using the ?xed point theorem for a cone map, we study the existence and multiplicity of positive solutions for a class of fractional di?erential equation with parameter.     

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.     

具有Holling-II型功能反应和交叉扩散的捕食-食饵模型正平衡解的存在性和不存在性

本文主要研究一类在齐次Dirichlet边界条件下带交叉扩散的Holling-II型捕食者-食饵模型正平衡解的存在性, 其中两个交叉扩散系数分别代表食饵远离捕食者的趋势和捕食者追逐食饵的趋势. 应用不动点指标理论得到了正平衡解存在的充分条件, 并进一步研究了正平衡解不存在的条件.     

Coexistence states of a Holling type-II predator-prey system

This paper characterize the existence of coexistence states to a reaction-diffusion predator-prey model with Holling type-II functional response subject to Dirichlet boundary conditions. We find the necessary and sufficient conditions for existence of coexistence states by fixed point index theory and bifurcation theory.     

Positive solutions for a modified Leslie–Gower prey–predator model with Crowley–Martin functional responses

In this paper, we study a modified Leslie–Gower prey–predator model with Crowley–Martin functional response. The stability and instability of the trivial and semi-trivial solutions was studied by analyzing the eigenvalues of the linearized system. The existence, multiplicity and uniqueness of positive steady state solutions were shown by using bifurcation theory, degree theory, energy estimate and asymptotic behavior analysis. Furthermore, all results were characterized in parameter plane.     

A Lotka-Volterra symbiotic model with cross-diffusion

The main goal of this paper is to study the existence and non-existence of coexistence states for a Lotka-Volterra symbiotic model with cross-diffusion. We use mainly bifurcation methods and a priori bounds to give sufficient conditions in terms of the data of the problem for the existence of positive solutions. We also analyze the profiles of the positive solutions when the cross-diffusion parameter goes to infinity.     

Stability and traveling waves of a stage-structured predator–prey model with Holling type-II functional response and harvesting

In this paper, we consider a reaction–diffusion predator–prey model with stage-structure, Holling type-II functional response, nonlocal spatial impact and harvesting. The stability of the equilibria is investigated. Furthermore, by the cross-iteration scheme companied with a pair of admissible upper and lower solutions and Schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant equilibrium.     

A strongly coupled predator–prey system with non-monotonic functional response

The predator–prey system with non-monotonic functional response is an interesting field of theoretical study. In this paper we consider a strongly coupled partial differential equation model with a non-monotonic functional response—a Holling type-IV function in a bounded domain with no flux boundary condition. We prove a number of existence and non-existence results concerning non-constant steady states (patterns) of the underlying system. In particular, we demonstrate that cross-diffusion can create patterns when the corresponding model without cross-diffusion fails.     

一类带有交叉扩散的捕食模型的共存问题

研究带有齐次Dirichlet边界条件的捕食-食饵模型,得到了平凡解存在的条件,并给出半平凡解存在的充分条件以及解的先验估计,最后利用Shauder不动点定理,得到问题至少有一个正解存在的充分条件.该结果说明只要捕获率足够小,物种的交叉扩散相对弱,问题就至少存在一个正解.     

Alternative prey source coupled with prey recovery enhance stability between migratory prey and their predator in the presence of disease

An eco-epidemiological model is considered where the prey population is migratory in nature. To incorporate the temporal pattern of the avian migration into the model, a time dependent recruitment rate was considered with a general functional response. In the numerical simulation we substitute the general functional response with Holling type-I and Holling type-II functional responses. It was observed that the qualitative behaviour of the system does not depend on the choice of the functional responses. The results showed that the system could be made disease free by either decreasing the contact rate or simultaneously increasing the predation and the recovery rate. Moreover, it was observed that the presence of an alternative food source for the predator population helps in the coexistence of all the species.     

Spatial patterns of the Holling–Tanner predator–prey model with nonlinear diffusion effects

In this article, we study the Holling–Tanner predator–prey model with nonlinear diffusion terms under homogeneous Neumann boundary condition. The nonlinear diffusion terms here mean that the prey runs away from the predator, and the predator chases the prey. Nonexistence and existence of nonconstant positive steady states are obtained, which reveal that cross-diffusion can create spatial patterns even when the random diffusion fails to do so. Moreover, asymptotic behaviour of positive solutions as the cross-diffusion tends to ∞ is shown.