Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.     

Positive periodic solutions for impulsive predator-prey model with dispersion and time delays

In this paper, we study the existence and global attractivity of positive periodic solutions for impulsive predator-prey systems with dispersion and time delays. By using coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solutions, and by means of a suitable Lyapunov functional, the uniqueness and global attractivity of positive periodic solution is presented. Some known results subject to the underlying systems without impulses are improved and generalized.     

Global attractivity of positive periodic solutions for nonlinear impulsive systems

In this paper, the existence, uniqueness and global attractivity of positive periodic solutions for nonlinear impulsive systems are studied. Firstly, existence conditions are established by the method of lower and upper solutions. Then uniqueness and global attractivity are obtained by developing the theories of monotone and concave operators. And lastly, the method and the results are applied to the impulsive n$n$-species cooperative Lotka–Volterra system and a model of a single-species dispersal among n$n$-patches.     

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.     

Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays

The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator-prey reaction-diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays.     

Periodic solutions of a class of degenerate parabolic system with delays

This paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition. Under locally Lipschitz condition on reaction functions, we apply Schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem. With quasi-monotonicity in addition, we also show that the periodic problem has a maximal and a minimal periodic solutions. Applications of the obtained results are also given to some nonlinear diffusion models arising from ecology.     

On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations. We obtain some new results relative to the lower solution α and upper solution β with α?β.     

An extension of kamake theorem and periodic lotka-volterra systems with diffusion

Based on an extended Kamake theorem, the global attractivity of a positive periodic solution for a Lotka-Volterra prey-predator periodic system with diffusion is given. This gives a partial answer to a problem of Hess.     

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.     

Existence and global attractivity of positive periodic solutions for delay Lotka-Volterra competition patch systems with stocking

By means of Mawhin's continuation theorem of coincidence degree and Lyapunov functional, a set of easily verifiable criteria are established for the existence and global attractivity of positive periodic solutions for delay Lotka-Volterra competition patch system with stocking

     

Existence and global attractivity of periodic solution for an impulsive delay differential equation with Allee effect

Sufficient conditions are obtained for the existence and global attractivity of positive periodic solution of an impulsive delay differential equation with Allee effect. The results of this paper improve and generalize noticeably the known theorems in the literature.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

Positive periodic solution for a multispecies competition-predator system with Holling III functional response and time delays

In this paper, a delayed multispecies ecological competition-predator system with Holling-III functional response is studied. By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functionals, some sufficient conditions are derived for the existence, uniqueness and global asymptotic stability of positive periodic solutions to the system.     

Existence and asymptotic stability of periodic solution for evolution equations with delays

In this paper, we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a Hilbert space H

     

Existence and global attractivity of almost periodic solutions for neural field with time delay

In this paper, the problem of existence and attractivity of almost periodic solutions for delayed neural field (DNF) with variable coefficients is discussed. First, the model of DNF is established as a modified neural field model. Secondly, by combining the theory of the exponential dichotomy, employing Banach fixed point theory and using differential inequality technique, some conditions for the the existence and attractivity of almost periodic solutions for the DNF are proposed.     

Time delayed parabolic systems with coupled nonlinear boundary conditions

The aim of this paper is to show the existence and uniqueness of a solution for a system of time-delayed parabolic equations with coupled nonlinear boundary conditions. The time delays are of discrete type which may appear in the reaction function as well as in the boundary function. The approach to the problem is by the method of upper and lower solutions for nonquasimonotone functions.

     

Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model

Sufficient conditions are obtained for the existence and global attractivity of periodic positive solution of an impulsive Lasota-Wazewska model for the survival of red blood cells.     

Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis

In this paper, we consider the generalized model of hematopoiesis