Permanence of an N-species cooperation system with time delays and feedback controls on time scales

In this paper, an N-species cooperation system with time delays and feedback controls is proposed on time scales. In this process, some inequalities, which play a vital roles in the proof of main results, are firstly proved. Then, by applying the theory of differential inequality, sufficient conditions which guarantee the permanence of the system are obtained on time scales. Our results improve and complement some known results to some degree in the literature.     

Permanence of species in nonautonomous discrete Lotka–Volterra competitive system with delays and feedback controls

A nonautonomous N-species discrete Lotka–Volterra competitive system of difference equations with delays and feedback controls is considered. New sufficient conditions are obtained for the permanence of this discrete system. The results indicate that one can choose suitable controls to make the species coexistence in the long run. Moreover, we give some examples to illustrate the feasibility of our result which can be well suited for computational purposes.     

Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays

In this paper, we consider a class of nonautonomous N-species Lotka-Volterra competitive systems with impulses and infinite delays. By developing the methods given in Teng (2002) [25], we give sufficient conditions for permanence and global attractivity of the system.     

PERMANENCE AND GLOBAL ATTRACTIVITY OF N-SPECIES COOPERATION SYSTEM WITH DISCRETE TIME DELAYS AND FEEDBACK CONTROLS

In this paper, we consider the N-species cooperation system with discrete time delays and feedback controls. By using the differential inequality theory and constructing a suitable Lyapunov functional, we obtain sufficient conditions which guarantee the permanence and the global attractivity of the system.     

Persistence and global stability for n-species ratio-dependent predator-prey system with time delays

A n-species ratio-dependent predator-prey food-chain model with time delays is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are obtained for the global stability of the positive equilibrium of the system.     

Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays

This paper is concerned with three 3-species time-delayed Lotka-Volterra reaction-diffusion systems and their corresponding ordinary differential systems without diffusion. The time delays may be discrete or continuous, and the boundary conditions for the reaction-diffusion systems are of Neumann type. The goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems. These conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays. The result of global asymptotic stability implies that each of the three model systems coexists, is permanent, and the trivial and all semitrivial solutions are unstable. Our approach to the problem is based on the method of upper and lower solutions for a more general reaction-diffusion system which gives a common framework for the 3-species model problems. Some global stability results for the 2-species competition and prey-predator reaction-diffusion systems are included in the discussion.     

Global analysis of an impulsive delayed Lotka–Volterra competition system

In this paper, a retarded impulsive n-species Lotka–Volterra competition system with feedback controls is studied. Some sufficient conditions are obtained to guarantee the global exponential stability and global asymptotic stability of a unique equilibrium for such a high-dimensional biological system. The problem considered in this paper is in many aspects more general and incorporates as special cases various problems which have been extensively studied in the literature. Moreover, applying the obtained results to some special cases, I derive some new criteria which generalize and greatly improve some well known results. A method is proposed to investigate biological systems subjected to the effect of both impulses and delays. The method is based on Banach fixed point theory and matrix’s spectral theory as well as Lyapunov function. Moreover, some novel analytic techniques are employed to study GAS and GES. It is believed that the method can be extended to other high-dimensional biological systems and complex neural networks. Finally, two examples show the feasibility of the results.     

Global asymptotical stability of a unique almost periodic solution for enterprise clusters based on ecology theory with time-varying delays and feedback controls

This paper is concerned with some dynamic behavior of enterprises cluster constituted by n satellite enterprises and a dominant enterprise. We present a model involving time-varying delays and feedback controls based on ecology theory, which effectively describe the competition and cooperation of enterprises cluster in real economic environments. Applying the comparison theorem of differential equations and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are obtained. Finally, we present an example to explain the economical significance of mathematical results.     

Permanence extinction and global asymptotic stability in a stage structured system with distributed delays

In this paper we consider a nonautonomous stage-structured competitive system of n-species population growth with distributed delays which takes into account the delayed feedback in both interspecific and intraspecific interactions. We obtain, by using the method of repeated replace, sufficient conditions for permanence and extinction of the species. The global attractivity of the unique positive equilibrium is proved in the autonomous case. Our results extend previous ones obtained by Liu et al. in [Nonlinear Anal. 51 (2002) 1347-1361; J. Math. Anal Appl. 274 (2002) 667-684].     

Permanence for a nonautonomous discrete single-species system with delays and feedback control

A nonautonomous discrete single-species system with delays and feedback control is studied. New sufficient conditions for ensuring the permanence of the system are obtained. A very important fact is found in our results, that is, that the feedback control is harmless to the permanence of species. The corresponding results given in [F.D. Chen, Permanence of a single species discrete model with feedback control and delay, Appl. Math. Lett. 20 (2007) 729–733] are improved and extended.     

Permanence of a delayed discrete mutualism model with feedback controls

This paper discusses a two species discrete model of mutualism with delays and feedback controls. Sufficient conditions are obtained for the permanence of the system. The results show that feedback control variables have no influence on the persistence property of the system.     

PERMANENCE OF A DISCRETE SINGLE SPECIES SYSTEM WITH DELAYS AND FEEDBACK CONTROL

In this paper,a discrete single species system with time delays and feedback control is considered.Sufficient conditions which guarantee the permanence of all positive solutions to this discrete system are obtained.The results show that the feedback control is harmless for the permanence of the species.     

Permanence and global attractivity for discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls

A discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls is proposed and investigated. By using the method of discrete Lyapunov functionals, new sufficient conditions on the permanence of species and global attractivity of the system are established. Particularly, an interesting fact is found in our results, that is, the feedback controls are harmless to the permanence of species for the considered system.     

Permanence for General Nonautonomous Impulsive Population Systems of Functional Differential Equations and Its Applications

In this paper, we investigate general impulsive nonautonomous population dynamical systems of functional differential equations. By utilizing the method of multiple Liapunov-like functionals to construct the permanence region, a general criterion on the permanence for the system is established. Furthermore, as applications of this general criterion, a class of impulsive nonautonomous n-species Lotka–Volterra competitive systems with delays and a class of impulsive nonautonomous 3-species Lotka–Volterra food chain systems with delays are discussed. Some new and useful sufficient conditions on the permanence for these systems are established.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

Almost periodic solutions of n-species competitive system with feedback controls

In this paper, the following almost periodic n-species competitive system with feedback controls