Periodic solutions of a Lotka–Volterra type multi‐species population model with time delays

A delayed periodic Lotka–Volterra type population model with m predators and n preys is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and by constructing suitable Lyapunov functionals, sufficient conditions are derived for the existence, uniqueness and global stability of positive periodic solutions of the model. Numerical simulation is presented to illustrate the feasibility of our main results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability and Hopf bifurcations in a competitive Lotka–Volterra system with two delays

We consider a Lotka–Volterra competition system with two delays. We first investigate the stability of the positive equilibrium and the existence of Hopf bifurcations, and then using the normal form theory and center manifold argument, derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions.     

On a neutral Lotka–Volterra system

Sufficient conditions are obtained for the linear stability of the positive equilibrium of the neutral system in terms of the parameters of the system. The case n=2 is considered in detail and the general case is discussed briefly.     

Global dynamics behaviors for a nonautonomous Lotka–Volterra almost periodic dispersal system with delays

A nonautonomous Lotka–Volterra dispersal system with continuous delays and discrete delays is considered. By using a comparison theorem and delay differential equation basic theory, we obtain sufficient conditions for the permanence of the population in every patch. By constructing a suitable Lyapunov functional, we prove that the system is globally asymptotically stable under some appropriate conditions. Using almost periodic functional hull theory, we get sufficient conditions for the existence, uniqueness and globally asymptotical stability for an almost periodic solution. This implies that the population in every patch exhibits stable almost periodic fluctuation. Furthermore, the results show that the permanence and global stability of system, and the existence and uniqueness of a positive almost periodic solution, depend on the delay; then we call it “profitless”.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

Conservation laws for Lotka–Volterra models

We derive necessary and sufficient conditions on a Lotka–Volterra model to admit a conservation law of Volterra's type. The result and the proof for the corresponding linear algebra problem are given in graph‐theoretical terms; they refer to the directed graph which is defined by the coefficients of the differential equation system. Copyright © 2003 John Wiley & Sons, Ltd.     

Global asymptotic stability in n-species nonautonomous Lotka–Volterra competitive systems with infinite delays

By means of Lyapunov functional, we have succeeded in establishing the global asymptotic stability of the positive solutions of a delayed n-species nonautonomous Lotka–Volterra type competitive system without dominating instantaneous negative feedbacks. As a corollary, we show that the global asymptotic stability of the positive solution is maintained provided that the delayed negative feedbacks dominate other interspecific interaction effects with delays and the mean delays are sufficiently small.     

Analysis of a Monod–Haldene type food chain chemostat with periodically varying substrate

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with periodically varying substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Furthermore, we numerically simulate a model with sinusoidal input, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halfing.     

Global attractor in competitive Lotka–Volterra systems

For autonomous Lotka–Volterra systems of differential equations modelling the dynamics of n competing species, new criteria are established for the existence of a single point global attractor. Under the conditions of these criteria, some of the species will survive and stabilise at a steady state whereas the others, if any, will die out (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extinction in a two dimensional Lotka–Volterra system with infinite delay

A nonautonomous two dimensional Lotka–Volterra system with infinite delay is considered. An extension of the principle of competitive exclusion is obtained.     

Survival of three species in a nonautonomous Lotka–Volterra system

In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra system of three species with constant interaction coefficients. In this paper, we study a nonautonomous Lotka–Volterra model with one predator and two preys. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution.     

Global asymptotic stability in n-species non-autonomous Lotka–Volterra competitive systems with infinite delays and feedback control

A non-autonomous Lotka–Volterra competition system with infinite delays and feedback control and without dominating instantaneous negative feedback is investigated. By means of a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the system. Some new results are obtained.     

Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance

In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra‐type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size‐selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long‐term simulation of evolutionary dynamics to demonstrate the emergence of high‐level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching. Copyright © 2015 John Wiley & Sons, Ltd.     

Multi‐dimensional Lotka–Volterra systems for carcinogenesis mutations

In the paper we consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of (n+1) equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second—as a system of reaction–diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka–Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one—the last equation expresses competition between the pre‐malignant and malignant cells and the environment is also unbounded, while for the third one—it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared. It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability. Copyright © 2009 John Wiley & Sons, Ltd.     

Stability of Volterra diffusion equations with time delays

This paper is concerned with asymptotic stability of a system of reaction-diffusion equations which is expressed in terms of Volterra integrals, under homogeneous Dirichlet boundary conditions. The effect of diffusion and delays on the stability of the system are analyzed, and sufficient conditions are given for the existence of positive solutions of the corresponding steady-state problem. Their global attraction with respect to nonnegative solutions of the time-dependent system is discussed. The main tools are Lyapunov functionals, positive definite kernels, and Laplace transforms.     

Singularity of Lotka‐Volterra models under unfoldings

In this paper, we classify the singularity of a Lotka‐Volterra competitive model with a Gaussian competition function and non‐Gaussian carrying capacity functions. These functions need not be completely different to affect adaptive dynamics of the model. For instance, it will be seen how ostensibly similar models can actually give rise to quite different behaviors due to their properties under unfolding. The use of Gaussian‐like carrying capacity functions can also show the sensitivity of the model to assumptions on the carrying capacity function's shapes. The classification is achieved using singularity theory of fitness functions under dimorphism equivalence. We also investigate the effect of the presence of unfolding and bifurcation parameters on the evolution of the system near its singular points. Particularly, we study the adaptive dynamics of the system near the singularity by focusing on ESS and CvSS types, and dimorphisms. Mutual invasibility plots are used to show regions of coexistence.     

Traveling wave solutions of Lotka–Volterra type two predators‐one prey model

In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non‐trivial boundary equilibria. One is the saturated state E_K of the prey without any predator. Other two equilibria, E₁ and E₂, are the coexistence states of the prey with only one species of predators. Using a high‐dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from E_K to E₂ and from E₁ to E₂. These results show that the advantageous species v₂ always win in the competition and exclude species v₁ eventually. Finally, some numerical simulations are presented, and biological interpretations are given. Copyright © 2016 John Wiley & Sons, Ltd.     

Coexistence states for a Lotka‐Volterra symbiotic system with cross‐diffusion

In this work, we study coexistence states for a Lotka‐Volterra symbiotic system with cross‐diffusion under homogeneous Dirichlet boundary conditions. By using topological degree theory and bifurcation theory, we prove the existence and multiplicity of positive solutions under certain conditions on the parameters. Asymptotic behaviors of positive solutions are respectively studied as the cross‐diffusion coefficient tends to infinity and the interaction rate tends to zero. Finally, we compare our results with those of the Lotka‐Volterra predator and competition systems.     

The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka–Volterra Systems

In this paper, a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n-dimensional Lotka–Volterra system. The submodels are the following systems: competition (cooperative or predator–prey) chain system and competition (cooperative or predator–prey) model between one and multispecies. The criteria in this paper are in explicit forms of the parameters and thus are easily verifiable.     

A Lotka-Volterra type food chain model with stage structure and time delays

A three-species Lotka-Volterra type food chain model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators (immature top predators) do not have the ability to feed on prey (predator). By using some comparison arguments, we first discuss the permanence of the model. By means of an iterative technique, a set of easily verifiable sufficient conditions are established for the global attractivity of the nonnegative equilibria of the model.