Species Lotka-Volterra System Research Articles

Propagating direction in the two species Lotka-Volterra competition-diffusion system

The sign of the traveling wave speed in the two species Lotka-Volterra competition-diffusion system under strong competition provides important information about the competitive advantage of the species. To study the sign problem, we propose a mini-max characterization of the coefficients in the reaction terms when the wave speed is zero. Applying our characterization, some explicit conditions under which the speed sign can be determined are obtained.

Global Attractivity for Three Species Lotka-Volterra Systems with Delays

This paper studies a type of non-autonomous three-species Lotka-Volterra(L-V) predator-prey-competitive systems with time delays is discussed. Several sufficient conditions on the global attractivity of the system are derived by employing the Lyapunov function method.

Properties of Traveling Wave Fronts for Three Species Lotka–Volterra System

The purpose of this paper is to investigate properties of traveling wave fronts for three species Lotka–Volterra system: the asymptotic behavior and uniqueness. Applying the Ikehara’s theorem, we determine the exponential rates of traveling wave fronts at the negative infinity. We further investigate the uniqueness of traveling wave fronts with the help of the sliding method.

Balance simplices of 3-species May-Leonard systems

We investigate the existence of a two-dimensional invariant manifold that attracts all nonzero orbits in 3 species Lotka-Volterra systems with identical linear growth rates. This manifold, which we call the balance simplex, is the common boundary of the basin of repulsion of the origin and the basin of repulsion of infinity. The balance simplex is linked to ecological models where there is ‘growth when rare’ and competition for finite resources. By including alternative food sources for predators we cater for predator-prey type models. In the case that the model is competitive, the balance simplex coincides with the carrying simplex which is an unordered manifold (no two points may be ordered componentwise), but for non-competitive models the balance simplex need not be unordered. The balance simplex of our models contains all limit sets and is the graph of a piecewise analytic function over the unit probability simplex.

The Impact of Immigration on a Stability Analysis of Lotka-Volterra System

The aim of this paper is to investigate three species Lotka-Volterra system with the immigration and harvesting effect. This work verifies the parametric conditions for the existence of the non-negative equilibrium points and its asymptotic stability. First, the sign stability of the system is analyzed through signed digraphs associated with the Jacobian matrix. Second, the asymptotic stability of the system is obtained by attaining the suitable parameters which are satisfying the Routh-Hurwitz stability criterion. In this model, the focus is mainly on the role of the immigration effect of every species. The paper is concluded with the immigration effect that supports to stabilize the unstable system and how the numerical result provides the guarantee to achieve the theoretical results.

An N-barrier maximum principle for autonomous systems of <inline-formula><tex-math id="M1">\begin{document}$n$\end{document}</tex-math></inline-formula> species and its application to problems arising from population dynamics

The main contribution of the N-barrier maximum principle is that it provides rather generic a priori upper and lower bounds for the linear combination of the components of a vector-valued solution. We show that the N-barrier maximum principle (NBMP, C.-C. Chen and L.-C. Hung (2016)) remains true for $n$ $(n>2)$ species. In addition, a stronger lower bound in NBMP is given by employing an improved tangent line method. As an application of NBMP, we establish a nonexistence result for traveling wave solutions to the four species Lotka-Volterra system.

On Solutions and Heteroclinic Orbits of Some Lotka-Volterra Systems

In this work, a principle for getting heteroclinic orbit of a dynamical system has been proposed when the solution is known in a compact form. The proposed principle has been tested through its application to a three species Lotka-Volterra system, which may appear as a mathematical model of human pathogen system. The domain in parameter space involve in the model, and the region of initial condition for the existence of heteroclinic orbit have been derived.

Traveling waves in a three species competition-cooperation system

This paper studies the traveling wave solutions to a three species competition cooperation system, which is derived from a spatially averaged and temporally delayed Lotka Volterra system. The existence of the traveling waves is investigated via a new type of monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain two species Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions.

Dynamics of a Two Species Competitive System with Pure Delays

A class of non-autonomous two species Lotka-Volterra competitive system with pure discrete time delays is discussed. Some sufficient conditions on the boundedness, permanence, periodic solution and global attractivity of the system are established by means of the comparison method and Liapunov functional.

Partial extinction, permanence and global attractivity in nonautonomous n-species Lotka-Volterra competitive systems with impulses

In this paper, the qualitative properties of general nonautonomous Lotka-Volterra n-species competitive systems with impulsive effects are studied. Some new criteria on the permanence, extinction and global attractivity of partial species are established by used the methods of inequalities estimate and Liapunov functions. As applications, nonautonomous two species Lotka-Volterra systems with impulses are discussed.

Permanence for Lotka–Volterra systems with multiple delays

In this paper, we consider a class of n species Lotka–Volterra systems with time delays and establish sufficient conditions which ensure the systems to be permanent. We improve and extend the known conditions of the permanence in Lu and Lu (2008) [4], Lu et al. (2010) [5] and Nakata and Muroya (2010) [7]. Moreover, we give the conditions of the permanence for a class of three species Lotka–Volterra predator–prey systems which need no restriction on the size of time delays and improve the result in [2]. Some examples for comparison with the previous results are given to illustrate the main results.

Convergent and divergent solutions of a discrete nonautonomous Lotka-Volterra model

In this paper, a discrete nonautonomous $m$-species Lotka-Volterra system is investigated. By using fixed point theorems, a set of simple and easily verifiable conditions are given for the existence of convergent or divergent positive solutions.

Effects of ecological differentiation on Lotka–Volterra systems for species with behavioral adaptation and variable growth rates

We study the properties of a n 2-dimensional Lotka–Volterra system describing competing species that include behaviorally adaptive abilities. We indicate as behavioral adaptation a mechanism, based on a kind of learning, which is not viewed in the evolutionary sense but is intended to occur over shorter time scales. We consider a competitive adaptive n species Lotka–Volterra system, n ⩾ 3, in which one species is made ecologically differentiated with respect to the others by carrying capacity and intrinsic growth rate. The symmetry properties of the system and the existence of a certain class of invariant subspaces allow the introduction of a 7-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the complete system. The reduced model is effective also in describing the time-dependent regimes for a large range of parameter values. The case in which one species has a strong ecological advantage (i.e. with a carrying capacity higher than the others), but with a varying growth rate, has been analyzed in detail, and time-dependent behaviors have been investigated in the case of adaptive competition among four species. Relevant questions, as species survival/exclusion, are addressed focusing on the role of adaptation. Interesting forms of species coexistence are found (i.e. competitive stable equilibria, periodic oscillations, strange attractors).

The necessary and sufficient conditions for the existence of periodic orbits in a Lotka–Volterra system

By extending Darboux method to three dimension, we present necessary and sufficient conditions for the existence of periodic orbits in three species Lotka–Volterra systems with the same intrinsic growth rates. Therefore, all the published sufficient or necessary conditions for the existence of periodic orbits of the system are included in our results. Furthermore, we prove the stability of periodic orbits. Hopf bifurcation is shown for the emergence of periodic orbits and new phenomenon is presented: at critical values, each equilibrium are surrounded by either equilibria or periodic orbits.

Limit cycles in periodically perturbed population systems

A perturbation method is proposed to calculate approximately the limit cycle type non-equilibrium steady-state resulting from periodic perturbation of coefficients of stable population systems; the two species Lotka-Volterra competition system is explicitly studied and the results are formulated for general multi-species population systems. Avoidance of competitive or other types of exclusion of species in a periodic environment is indicated.

Persistence and extinction in three species Lotka-Volterra competitive systems

Extinction in a three species Lotka-Volterra competitive system is classified in terms of the model parameters. Necessary and sufficient conditions are given for one and two species extinction; these are, alternatively, conditions for two and one species persistence. Persistence of the system is studied assuming all pairwise interactions between species are known. An intransitive species arrangement is the only case of persistence where pairwise interactions are, by themselves, sufficient to govern persistence. No persistent arrangement can contain a pair of species that interacts in an unstable manner.

A Liapunov function for competition communities

An algorithmically defined Liapunov function is proposed for competition systems. Although it is shown that this function is not useful for determinations of neighborhood asymptotic stability, the function does show the range and degree of global asymptotic stability. For competition systems that are disturbed by internal and external factors, the Liapunov function can be interpreted to give, at least, the relative probabilities of the extinction of single species, and it may also be interpreted to give the complete distribution function for the system. Some examples are given for two species Lotka-Volterra systems. An analytical approximation is produced that gives the Liapunov functions at points of single species extinction.