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.     

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.     

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.     

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.     

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.     

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.     

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.     

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.     

Hopf bifurcation and multiple periodic solutions in Lotka–Volterra systems with symmetries

The purpose of this paper is to study Hopf bifurcations in a delayed Lotka–Volterra system with dihedral symmetry. By treating the response delay as bifurcation parameter and employing equivariant degree method, we obtain the existence of multiple branches of nonconstant periodic solutions through a local Hopf bifurcation around an equilibrium. We find that competing coefficients and the response delay in the system can affect the spatio-temporal patterns of bifurcating periodic solutions. According to their symmetric properties, a topological classification is given for these periodic solutions. Furthermore, an estimation is presented on minimal number of bifurcating branches. These theoretical results are helpful to better understand the complex dynamics induced by response delays and symmetries in Lotka–Volterra systems.     

Integrable deformations of the Bogoyavlenskij–Itoh Lotka–Volterra systems

We construct a family of integrable deformations of the Bogoyavlenskij–Itoh systems and construct a Lax operator with spectral parameter for it. Our approach is based on the construction of a family of compatible Poisson structures for the undeformed systems, whose Casimirs are shown to yield a generating function for the integrals in involution of the deformed systems.We show how these deformations are related to the Veselov–Shabat systems.     

Hopf bifurcation in time-delayed Lotka–Volterra competition systems with advection

In this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s).