Attractivity Of Positive Periodic Solution Research Articles

OPTIMAL IMPULSIVE HARVESTING OF A SINGLE-SPECIES WITH GOMPERTZ LAW OF GROWTH

In this paper we investigate the optimal harvesting problems of a single species with Gompertz law of growth. Based on continuous harvesting models, we propose impulsive harvesting models with constant harvest or proportional harvest. By using the discrete dynamical systems determined by the stroboscopic map, we discuss existence, stability and global attractivity of positive periodic solutions, and obtain the maximum sustainable yield and the corresponding optimal population level. At last, we compare the maximum sustainable yield of impulsive harvest with that of continuous harvest, and point out that proportional harvest is superior to constant harvest.

Existence and global attractivity of positive periodic solutions for a generalized predator–prey system with time delay

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for the generalized two-predator and one-prey system with time delay is established. Further, by constructing a Lyapunov functional and using the result in the existence part of positive periodic solutions, the attractivity of positive periodic solutions for the above system is obtained.

Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model

In this paper we shall consider the following nonlinear impulsive delay population model: (0.1) x ′ ( t ) = - δ ( t ) x ( t ) + p ( t ) x ( t - m ω ) e - α ( t ) x ( t - m ω ) a.e. t > 0 , t ≠ t k , x ( t k + ) = ( 1 + b k ) x ( t k ) , k = 1 , 2 , … , where m is a positive integer, δ ( t ) , α ( t ) and p ( t ) are positive periodic continuous functions with period ω > 0 . In the nondelay case ( m = 0 ), we show that (0.1) has a unique positive periodic solution x * ( t ) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of x * ( t ) . Our results imply that under the appropriate linear periodic impulsive perturbations, the impulsive delay equation (0.1) preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.

Global attractivity of positive periodic solutions for an impulsive delay periodic “food limited” population model

We will consider the following nonlinear impulsive delay differential equationN′(t)=r(t)N(t)((K(t)−N(t−mw))/(K(t)+λ(t)N(t−mw))), a.e.t>0,t≠tk,N(tk+)=(1+bk)N(tk),K=1,2,…, wheremis a positive integer,r(t),K(t),λ(t)are positive periodic functions of periodicω. In the nondelay case(m=0), we show that the above equation has a unique positive periodic solutionN*(t)which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity ofN*(t). Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.

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 -species cooperative Lotka–Volterra system and a model of a single-species dispersal among n -patches.

Periodic solutions and almost periodic solutions of a neutral multispecies Logarithmic population model

A neutral multispecies Logarithmic population model is proposed in this paper. By using the contraction mapping principle and constructing a suitable Lyapunov functional, a set of easily applicable criteria are established for the existence, uniqueness and global attractivity of positive periodic solution (positive almost periodic solution) of the model.

Existence and global attractivity of positive periodic solution for competition-predator system with variable delays

Abstract In this paper, Lotka–Volterra competition-predator system with variable delays is considered. Some sufficient conditions ensuring the existence and global attractivity of periodic solution for this system are obtained by using coincidence degree theory and Lyapunov functional method. An example is also worked out to demonstrate the advantages of our results.

Existence and global attractivity of positive periodic solutions of a survival model of red blood cells

In this paper, we deal with a model for the survival of red blood cells with periodic coefficients, x ( t ) = − μ ( t ) x ( t ) + P ( t ) e − r ( t ) x ( t − r ( t ) ) , t ≥ 0 . Some new sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the equation.

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.

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka–Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.

Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control

Sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with feedback control d x d t = - b ( t ) x ( t ) + F ( t , x ( t - τ 1 ( t ) ) , … , x ( t - τ n ( t ) ) , u ( t - δ ( t ) ) ) , d u d t = - η ( t ) u ( t ) + a ( t ) x ( t - σ ( t ) ) . The method involves the application of Krasnoselskii's fixed point theorem and estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.

Existence and global attractivity of positive periodic solutions of functional differential equations with impulses

Existence and global attractivity of positive periodic solutions of functional differential equations with impulses

Existence and global attractivity of positive periodic solution of an impulsive delay differential equation

Sufficient conditions are obtained for the existence and global attractivity of periodic positive solutions of an impulsive delay differential equation of the form Our theorems have generalized some known results on special delay population models. It is shown that under appropriate linear periodic impulsive perturbations, the impulsive delay differential equation preserves the original periodicity and global attractivity properties of the nonimpulsive delay differential equation.

Existence and global attractivity of positive periodic solutions of functional differential equations with impulses

Sufficient and realistic conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with impulses d y ( t ) d t = y ( t ) F ( t , y ( t - τ 1 ( t ) ) , … , y ( t - τ n ( t ) ) ) , a.e. t > 0 , t ≠ t k , y ( t k + ) - y ( t k ) = b k y ( t k ) , k = 1 , 2 , … . The method involves the application of the Gaines and Mawhin's coincidence degree theory, the constructing suitable Lyapunov Functionals and estimations of uniform upper bounds on solutions. When these results are applied to some special delay population models, some new results are obtained, and some known results are generalized. In particular, our results indicate that under the appropriate linear periodic impulsive perturbations, the above impulsive delay differential equation preserves the original periodicity and global attractivity of the nonimpulsive delay differential equation d y ( t ) d t = y ( t ) F ( t , y ( t - τ 1 ( t ) ) , … , y ( t - τ n ( t ) ) ) .

Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics

In this paper we shall consider the following nonlinear impulsive delay differential equation x′(t)+ αV(t)x(t)x n(t−mω) θ n+x n(t−mω) =λ(t), a.e. t>0, t≠t k,x(t k +)= 1 (1+b k) x(t k), k=1,2,…, where m and n are positive integers, V( t) and λ( t) are positive periodic continuous functions with period ω>0. In the nondelay case ( m=0), we show that the above equation has a unique positive periodic solution x ∗(t) which is globally asymptotically stable. In the delay case, we present sufficient conditions for the global attractivity of x ∗(t) . Our results imply that under the appropriate periodic impulsive perturbations, the impulsive delay equation shown above preserves the original periodic property of the nonimpulsive delay equation. In particular, our work extends and improves some known results.

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 x′ 1(t)=h 1(t,x 1(t))[a 1(t)−b 1(t)x 1(t)−c 1(t)y(t)]+D 1(t)[x 2(t−τ 1)−x 1(t)]+S 1(t), x′ 2(t)=h 2(t,x 2(t))[a 2(t)−b 2(t)x 2(t)]+D 2(t)[x 1(t−τ 2)−x 2(t)]+S 2(t), y′(t)=h 3(t,y(t))[a 3(t)−b 3(t)y(t)−β(t) ∫ −τ 0 k(s)y(t+s) ds−c 3(t)x 1(t)]+S 3(t). Some known results are generalized and improved.

Global behaviors of a generalized periodic impulsive Logistic system with nonlinear density dependence

The global behaviors of a generalized periodic impulsive Logistic system with nonlinear density dependence are studied. Conditions for the existence and global attractivity of positive periodic solution are obtained via the method of comparison and Liapunov function. The corresponding results for the periodic impulsive Logistic system, which are dependent on solving the system, are extended.

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 positive periodic solutions of periodic n-species Lotka–Volterra competition systems with several deviating arguments

In this paper, we study the existence and global attractivity of positive periodic solutions of periodic n-species Lotka–Volterra competition systems. By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka–Volterra competition systems with several deviating arguments and the existence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka–Volterra competition system with several delays. Some new results are obtained. As an application, we also examine some special cases of the system we considered, which have been studied extensively in the literature. Some known results are improved and generalized.