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.     

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.     

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.     

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.     

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.     

Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.     

Global asymptotic stability of a Lotka–Volterra competition model with stochasticity in inter-specific competition

In the paper we first propose a two-species Lotka–Volterra competition model with the stochastic terms related to the inter-specific competition rates and the coexistence equilibrium of the deterministic model. Then we establish the global asymptotic stability of the coexistence equilibrium. Finally, we provide some discussions and numerical examples to illustrate our mathematical results.     

Global directed dynamics of a Lotka–Volterra competition-diffusion system

In this study, we consider a directed–diffusion system describing the interactions between two organisms in heterogeneous environment. We first establish a linearly stability of the co-existence (positive) steady state. Then we further present a classification on all possible long-time dynamical behaviors by appealing to the theory of monotone dynamical systems.     

Global analysis of an impulsive delayed Lotka–Volterra competition system

In this paper, a retarded impulsive n-species Lotka–Volterra competition system with feedback controls is studied. Some sufficient conditions are obtained to guarantee the global exponential stability and global asymptotic stability of a unique equilibrium for such a high-dimensional biological system. The problem considered in this paper is in many aspects more general and incorporates as special cases various problems which have been extensively studied in the literature. Moreover, applying the obtained results to some special cases, I derive some new criteria which generalize and greatly improve some well known results. A method is proposed to investigate biological systems subjected to the effect of both impulses and delays. The method is based on Banach fixed point theory and matrix’s spectral theory as well as Lyapunov function. Moreover, some novel analytic techniques are employed to study GAS and GES. It is believed that the method can be extended to other high-dimensional biological systems and complex neural networks. Finally, two examples show the feasibility of the results.     

Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model

A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients.     

Stability of Lotka–Volterra system

The stability of Lotka–Volterra system has been discussed by many authors for two and three species. In this paper, we will discussed the notion of stability for a Lotka–Volterra system with four species. Some criteria and results are given. Our technique depends on the Lyapunov–Razumikhin method. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

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.     

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.     

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.     

Robust cycles in Kolmogorov–Lotka–Volterra class of models with intraspecific co-operation

Using the Andronov–Hopf bifurcation theorem and the Poincaré–Bendixson Theorem, we explore robust cyclical possibilities in Kolmogorov–Lotka–Volterra class of models with positive intraspecific cooperation (in the form of social networks) in the prey population. We find that this additional feedback effect of intraspecific cooperation introduces nonlinearities which modify the cyclical outcomes of the model. We show that the cyclical outcomes are more robust than in the existing literature in this area due to introduction of such non-linearities. We also demonstrate the possibilities of multiple limit cycles under certain situations.     

Permanence of Stochastic Lotka–Volterra Systems

This paper proposes a new definition of permanence for stochastic population models, which overcomes some limitations and deficiency of the existing ones. Then, we explore the permanence of two-dimensional stochastic Lotka–Volterra systems in a general setting, which models several different interactions between two species such as cooperation, competition, and predation. Sharp sufficient criteria are established with the help of the Lyapunov direct method and some new techniques. This study reveals that the stochastic noises play an essential role in the permanence and characterize the systems being permanent or not.     

Dynamics of Some Three-Dimensional Lotka–Volterra Systems

We characterize the dynamics of the following two Lotka–Volterra differential systems:

$$\begin{aligned} \begin{array}{lll} \dot{x}=x(r+a y+b z), &{} &{} \dot{x}=x(r+ax+b y+c z),\\ \dot{y}=y(r-a x+c z), &{} \quad \text{ and }\quad \quad &{} \dot{y}=y(r+a x+dy+e z),\\ \dot{z}=z(r-b x-c y), &{} &{} \dot{z}=z(r+a x+d y+fz). \end{array} \end{aligned}$$

We analyze the biological meaning of the dynamics of these Lotka–Volterra systems