Permanence of the nonautonomous competitive systems with infinite delay and feedback controls

We study the permanence of a class of nonautonomous two-species Lotka–Volterra competitive systems with infinite delay and feedback controls. New results on the permanence of solutions are obtained. The corresponding results given in [F. Chen, Z. Li, Y. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear Anal. RWA 8 (2007) 680–687] are improved and extended.     

Permanence of nonautonomous Lotka–Volterra delay differential systems

In this work, applying the results offered by S. Ahmad and A.C. Lazer [On a property of nonautonomous Lotka–Volterra competition model, Nonlinear Anal. 37 (1999) 603–611] and the recent work of R. Redheffer [Mean values and the nonautonomous May–Leonald equations, Nonlinear Anal. Real World Appl. 4 (2003) 301–306] to an nonautonomous Lotka–Volterra differential system with finite delays, we establish sufficient conditions for the permanence of the system.     

On permanence of Lotka–Volterra systems with delays and variable intrinsic growth rates

For competitive Lotka–Volterra systems, Ahmad and Lazer’s work [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Annali di Matematica 185 (2006) S47–S67] on total permanence of systems without delays has been extended to delayed systems [Z. Hou, On permanence of all subsystems of competitive Lotka–Volterra systems with delays, Nonlinear Analysis: Real World Applications 11 (2010) 4285–4301]. In this paper, existence and boundedness of nonnegative solutions and permanence are considered for general Lotka–Volterra systems with delays including competitive, cooperative, predator–prey and mixed type systems. First, a condition is established for the existence and boundedness of solutions on a half line. Second, a necessary condition on the limits of the average growth rates is provided for permanence of all subsystems. Then the result for competitive systems is also proved for the general systems by using the same techniques. Just as for competitive systems, the eminent finding is that permanence of the system and all of its subsystems is completely irrelevant to the size and distribution of the delays.     

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 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.     

PERMANENCE OF ASYMPTOTICALLY PERIODIC MULTISPECIES LOTKA-VOLTERRA COMPETITION PREDATOR-PREY SYSTEM

In this paper, we consider the permanence of asymptotically periodic mul-tispecies Lotka-Volterra competition predator-prey system. By means of the standard comparison theorem, we improve or extend the corresponding results given by Peng and Chen [1], Teng and Li [2], Zhao and Chen [3]. Also, we obtain the conditions which ensure the permanence and global attractivity of asymptotically periodic multispecies competition predator-prey system.     

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.     

Influence of feedback controls on an autonomous Lotka–Volterra competitive system with infinite delays

In this paper, we consider an autonomous Lotka–Volterra competitive system with infinite delays and feedback controls. The extinction and global stability of equilibriums are discussed using the Lyapunov functional method. If the Lotka–Volterra competitive system is globally stable, then we show that the feedback controls only change the position of the unique positive equilibrium and retain the stable property. If the Lotka–Volterra competitive system is extinct, by choosing the suitable values of feedback control variables, we can make extinct species become globally stable, or still keep the property of extinction. Some examples are presented to verify our main results.     

Global attractivity and almost periodic solution of a discrete mutualism model with delays

In this paper, we consider an almost periodic discrete Lotka–Volterra mutualism model with delays. We first obtain the permanence and global attractivity of the system. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution, which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A new contribution to periodic competition systems with delays

A delayed competition system of Lotka–Volterra type, with periodic coefficients, is considered. The topics of existence and global asymptotic stability of a periodic solution are investigated. The novelty of our results consists in the fact that they require only average conditions. In the study of global attractivity an unusual Lyapunov function is introduced. Our method takes advantage of the fact that there is no deviating argument in the negative feedback terms.     

A 3D competitive Lotka–Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So

For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dyn. Stab. Syst. 8 (1993) 189–217] identified 33 stable nullcline equivalence classes. Among these, only classes 26–31 may have limit cycles. Hofbauer and So [J. Hofbauer, J.W.-H. So, Multiple limit cycles for three dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70] conjectured that the number of limit cycles is at most two for these systems. In this paper, we construct three limit cycles for class 29 without a heteroclinic polycycle in Zeeman’s classification.     

Existence and stability of almost periodic solutions of nonautonomous competitive systems with weak Allee effect and delays

In this paper, a class of nonautonomous Lotka–Volterra type multispecies competitive systems with weak Allee effect and delays are considered. By using Mawhin’s continuation theorem of coincidence degree theory, we obtain some sufficient conditions for the existence of almost periodic solutions for the Lotka–Volterra system. On the case of no delays of Allee effects, by constructing a suitable Lyapunov function, we get a sufficient condition for the globally attractivity of the almost periodic solution for the Lotka–Volterra system. Moreover, we also present an illustrative example to show the effectiveness of our results.     

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.     

Periodic solutions for a Lotka–Volterra mutualism system with several delays

In the present paper, a Lotka–Volterra type mutualism system with several delays is studied. Some new and interesting sufficient conditions are obtained for the global existence of positive periodic solutions of the mutualism system. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions. Our results are different from the existing ones such as those in of Yang et al. [F. Yang, D. Jiang, A. Ying, Existence of positive solution of multidelays facultative mutualism system, J. Eng. Math. 3 (2002) 64–68] and Chen et al. [F. Chen, J. Shi, X. Chen, Periodicity in a Lotka–Volterra facultative mutualism system with several delays, J. Eng. Math. 21 (3) (2004) 403–409].     

Persistence in nonautonomous predator–prey systems with infinite delays

This paper studies the general nonautonomous predator–prey Lotka–Volterra systems with infinite delays. The sufficient and necessary conditions of integrable form on the permanence and persistence of species are established. A very interesting and important property of two-species predator–prey systems is discovered, that is, the permanence of species and the existence of a persistent solution are each other equivalent. Particularly, for the periodic system with delays, applying these results, the sufficient and necessary conditions on the permanence and the existence of positive periodic solutions are obtained. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra systems are strongly improved and extended to the delayed case.     

Delay effect on the permanence for Lotka–Volterra cooperative systems

In this paper, we show that delays can change the permanence for Lotka–Volterra cooperative systems. For certain delays with the same length, the delayed system has a similar property to the corresponding system without delays in the sense of permanence, but for a general delay case, the delays may destroy the permanence for the system.     

Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145–155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1–6] and Banas and Dhage [J. Banas, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.07.038], under weaker conditions with a different method.     

Permanence of a discrete multispecies Lotka–Volterra competition predator–prey system with delays

In this paper, we propose a discrete multispecies Lotka–Volterra competition predator–prey system with delays. For general nonautonomous case, sufficient conditions are established for the permanence of the system.