Impulsive control of multiple Lotka–Volterra systems

In this work, we consider the control problem of multiple Lotka–Volterra system. Our means to control the population dynamics is via impulses not only in a single species, but also in multiple species, that is, some members of these populations are added to or removed from the environment impulsively at the same time. We establish the strategies for preventing all the species from going extinct by stabilizing some special positive points, which may not be the equilibrium points of the system. We give several Lotka–Volterra systems to illustrate our results by drawing their time-series graphs.     

Global asymptotic stability of a general stochastic Lotka–Volterra system with delays

This paper discusses a general stochastic Lotka–Volterra system with delays. Some conditions for the global asymptotic stability are established.     

Permanence for nonautonomous Lotka–Volterra cooperative systems with delays

In this paper, we consider a class of nonautonomous two species Lotka–Volterra cooperative population systems with time delays, and establish sufficient conditions which ensure the system to be permanent. We improve and extend the known condition of the permanence in [G. Lu and Z. Lu, Permanence for two species Lotka–Volterra cooperative systems with delays, Math. Biosci. Eng. 5 (2008) 477–484] to nonautonomous two-species Lotka–Volterra cooperative systems. Moreover, our conditions need no restriction on the size of time delays.     

Global asymptotic stability of Lotka–Volterra competition reaction–diffusion systems with time delays

This paper is concerned with a time-delayed Lotka–Volterra competition reaction–diffusion system with homogeneous Neumann boundary conditions. Some explicit and easily verifiable conditions are obtained for the global asymptotic stability of all forms of nonnegative semitrivial constant steady-state solutions. These conditions involve only the competing rate constants and are independent of the diffusion–convection and time delays. The result of global asymptotic stability implies the nonexistence of positive steady-state solutions, and gives some extinction results of the competing species in the ecological sense. The instability of the trivial steady-state solution is also shown.     

Almost periodic solutions for Lotka–Volterra systems with delays

This paper studies a general class of delayed almost periodic Lotka–Volterra system with time-varying delays and distributed delays. By using the definition of almost periodic function, the sufficient conditions for the existence and uniqueness of globally exponentially stable almost periodic solution are obtained. The conditions can be easily reduced to special cases of cooperative systems and competitive systems.     

Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays

In the Lotka–Volterra competition system with N-competing species if the effect of dispersion and time-delays are both taken into consideration, then the densities of the competing species are governed by a coupled system of reaction–diffusion equations with time-delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to the competing rate constants to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the competing system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system.     

On permanence of all subsystems of competitive Lotka–Volterra systems with delays

In this paper, permanence for a class of competitive Lotka–Volterra systems is considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. A computable necessary and sufficient condition is found for the permanence of all subsystems of the system and its small perturbations on the interaction matrix. This is a generalization from systems without delays to delayed systems of Ahmad and Lazer’s work on total permanence (S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006) S47–S67). In addition to Ahmad and Lazer’s example showing that permanence does not imply total permanence, another example of permanent system is given having a non-permanent subsystem. As a particular case, a necessary and sufficient condition is given for all subsystems of the corresponding autonomous system to be permanent. As this condition does not rely on the delays, it actually shows the equivalence between such permanence of systems with delays and that of corresponding systems without delays. Moreover, this permanence property is still retained by systems as a small perturbation of the original system.     

Stability of a three-dimensional diffusive Lotka–Volterra system of type-K with delays

A three-dimensional diffusive Lotka–Volterra system of type-K with delays is investigated. We give a stability analysis in detail for all equilibria of the system and obtain some threshold conditions for linear instability and linear asymptotic stability of each equilibrium. We develop the analytical method for stability analysis of reaction–diffusion equations with multi-delays.     

Almost periodic solutions of a discrete Lotka–Volterra competition system with delays

In this paper, we consider a discrete almost periodic Lotka–Volterra competition system with delays. Sufficient conditions are obtained for the permanence and global attractivity of the system. Further, by means of an almost periodic functional hull theory, we show that the almost periodic system has a unique strictly positive almost periodic solution, which is globally attractive. Some examples are presented to verify our main results.     

Global asymptotic stability of positive equilibrium of three-species Lotka–Volterra mutualism models with diffusion and delay effects

In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction–diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction–diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction–diffusion system. Finally, the numerical simulation is given to illustrate our results.     

Permanence and extinction in competitive Lotka–Volterra systems with delays

In this paper, a class of competitive Lotka–Volterra systems are considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. Under the assumption that all proper subsystems are permanent, it is shown that the asymptotic behaviour of the system is determined by the relationship between an equilibrium and a nullcline plane of the corresponding autonomous system: if the equilibrium is below the plane then the system is permanent; if the equilibrium is above the plane then this species will go extinct in an exponential rate while the other species will survive. Similar asymptotic behaviour is also retained under an alternative assumption.     

Geometric approach for global asymptotic stability of three-dimensional Lotka–Volterra systems

In this paper, by applying the geometric criterion and time average property to Lotka–Volterra systems, some results for the global asymptotic stability of the systems are obtained. Furthermore, we consider Li–Wang Conjecture for a three-dimensional system which is transformed from a Lotka–Volterra system.     

Permanence and stability in multi-species non-autonomous Lotka–Volterra competitive systems with delays and feedback controls

In this paper, we consider a multi-species Lotka–Volterra type competitive system with delays and feedback controls. A general criteria on the permanence is established, which is described by integral form and independent of feedback controls. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solution to the model.