Nonautonomous Competitive Lotka Volterra System Research Articles

Chaotic and stable behavior around the equilibrium points of a non-autonomous competitive Lotka-Volterra system

Chaotic and stable behavior around the equilibrium points of a non-autonomous competitive Lotka-Volterra system

Extinction and survival in competitive Lotka-Volterra systems with constant coefficients and infinite delays

The qualitative properties of a nonautonomous competitive Lotka-Volterra system with infinite delays are studied.By using a result of matrix theory and the fluctuation lemma, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernel, which are sufficient to ensure the survival and the extinction of a determined number of species. The surviving part is stabilized around a globally stable critical point of a subsystem of the system under study. These conditions also guarantee the asymptotic behavior of the system.

Permanence and extinction of stochastic competitive Lotka–Volterra system with Lévy noise

This paper derives sufficient conditions for stochastic permanence and extinction of a stochastic non-autonomous competitive Lotka–Volterra system with Levy noise. For the autonomous case, the results show that stochastic permanence and extinction are characterized by two parameters $$\mathcal {B}_{1}$$ and $$\mathcal {B}_{2}$$ : if $$\mathcal {B}_{1}\mathcal {B}_{2} \ne 0$$ , then the system is either stochastically permanent or extinctive. That is, it is extinctive if and only if $$\mathcal {B}_{1}<0$$ and $$\mathcal {B}_{2}<0$$ ; otherwise, it is stochastically permanent. Some existing results are included as special cases. An example and its simulations are given to support our theoretical results.

Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays

The qualitative properties of certain type of nonautonomous competitive Lotka-Volterra systems with infinite delay are considered. By constructing suitable Lyapunov-type functional, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernels, which are sufficient to ensure the extinction and survival of a determined number of species. The surviving part stabilizes around any solution of a subsystem of the systems in study. These conditions also guarantee the persistence, extreme stability and asymptotic behavior of the systems.

Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive Lotka–Volterra system

A three dimensional nonautonomous competitive Lotka–Volterra system is considered in this paper. It is shown that if the growth rates are positive, bounded and continuous functions, and the averages of the growth rates satisfy certain inequalities, then any positive solution has the property that one of its components vanishes. Moreover, if one of the above inequalities is changed, then all components of any positive solution have positive infimum.

Extinction in nonautonomous competitive Lotka–Volterra systems with infinite delay

The principle of competitive exclusion is extended to n -species nonautonomous Lotka–Volterra competition systems of differential equations with infinite delay. It is shown that if the coefficients are bounded, continuous and satisfy certain inequalities, then any solution with initial function in an appropriate space will have n − 1 of its components tend to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation.

Extinction in a nonautonomous competitive Lotka–Volterra system

A nonautonomous competitive Lotka–Volterra system is considered in this work. Sufficient conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction. It is shown that these conditions are weaker than those of Montes de Oca and Zeeman [F. Montes de Oca, M.L. Zeeman, Extinction in nonautonomous competitive Lotka–Volterra systems, Proc. Amer. Math. Soc. 124 (1996) 3677–3687].

Extinction in two dimensional nonautonomous Lotka–Volterra systems with the effect of toxic substances

In this paper, we consider a nonautonomous competitive Lotka–Volterra system of two species with the effect of toxic substances. It is shown that toxic substances play an important role in the extinction of species. We prove that one of components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation under some conditions.

Average growth and extinction in a competitive Lotka–Volterra system

In this paper, we answer an earlier conjecture. Consider a class of nonautonomous competitive Lotka–Volterra systems. We show that certain inequalities involving the interaction coefficients and averages of the growth rates imply that the system is persistent. However, reversing one of these inequalities implies that the Nth species goes to extinction.

Average growth and total permanence in a competitive Lotka-Volterra System

In this paper, we consider a class of non-autonomous competitive Lotka-Volterra Systems. Such a system is called strongly permanent if small perturbations of the system are permanent. We define such a system to be totally permanent if the system as well as its subsystems are strongly permanent. When the growth rates have averages and the interaction coefficients are non-negative constants, we give a computable necessary and sufficient condition for the system to be totally permanent.

Average conditions for permanence and extinction in nonautonomous Lotka–Volterra system

An n species nonautonomous competitive Lotka–Volterra system is considered in this paper. The average conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction. The generalization for the result is presented, that is, for each r ⩽ n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n − r are driven to extinction. It is shown that these average conditions are improvement of those of Ahmad and Montes de Oca [Appl. Math. Comput. 90 (1998) 155–166] and Montes de Oca and Zeeman [Proc. Amer. Math. Soc. 124 (1996) 3677–3687] and [J. Math. Anal. Appl. 192 (1995) 360–370].

Almost necessary and sufficient conditions for survival of species

We consider a nonautonomous competitive Lotka–Volterra system of two species, satisfying two inequalities involving averages of the growth rates and the interaction coefficients, which imply persistence. We introduce a third species and give a third inequality, involving the average of the growth rate of the third species and solutions of a linear algebraic system, which guarantees persistence of the system. It is also shown that reversing this inequality implies non-persistence; more specifically, extinction of the third species with small positive initial values, in the autonomous case. Our conditions are simple and computable.

Extinction in nonautonomous T-periodic competitive Lotka-Volterra system

A nonautonomous T-periodic competitive Lotka-Volterra system of n species is considered. It is shown that if the coefficients are T-periodic, continuous and satisfy certain inequalities, then any solution with strictly positive initial conditions has the property that all but one of its components vanish while the remaining component approaches the canonical solution of a certain logistic differential equation.

Extinction in nonautonomous competitive Lotka-Volterra systems

It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to nonautonomous systems of arbitrary finite dimension. That is, for thennspecies nonautonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction. The restriction of the system to the remaining axis is a nonautonomous logistic equation, which has a unique solutionu(t)u(t)that is strictly positive and bounded for all time; see Coleman (Math. Biosci.45(1979), 159–173) and Ahmad (Proc. Amer. Math. Soc.117(1993), 199–205). We prove in addition that all solutions of thenn-dimensional system with strictly positive initial conditions are asymptotic tou(t)u(t).

Balancing Survival and Extinction in Nonautonomous Competitive Lotka-Volterra Systems

We generalise and unify some recent results about extinction in nth-order nonautonomous competitive Lotka-Volterra systems. For each r ≤ n, we show that if the coefficients are continuous, bounded by strictly positive constants, and satisfy certain inequalities, then any solution with strictly positive initial values has the property that n − r of its components vanish, whilst the remaining r components asymptotically approach a canonical solution of an r-dimensional restricted system. In other words, r of the species being modeled survive whilst the remaining n − r are driven to extinction.

On the nonautonomous Volterra-Lotka competition equations

A nonautonomous competitive Lotka-Volterra system of two equations is considered. It is shown that if the coefficients are continuous and satisfy certain inequalities, then any solution that is positive at some point has the property that one of its components vanishes while the other approaches a certain solution of the logistic equation.