Generalized Lotka-Volterra Systems Research Articles

Periodic Solutions for n-Species Lotka-Volterra Competitive Systems with Pure Delays

We study a class of periodic general n-species competitive Lotka-Volterra systems with pure delays. Based on the continuation theorem of the coincidence degree theory and Lyapunov functional, some new sufficient conditions on the existence and global attractivity of positive periodic solutions for the n-species competitive Lotka-Volterra systems are established. As an application, we also examine some special cases of the system, which have been studied extensively in the literature.

Melnikov Functions in Quadratic Perturbations of Generalized Lotka–Volterra Systems

We present a detailed analysis of Melnikov functions which arise in quadratic perturbations of generalized Lotka---Volterra vector fields with the first integral x?ys(1 ? x ? y). That analysis was sketched in ?ola?dek (J Differ Equ 109:223---273, 1994). In particular, we prove that the maximal number of limit cycles in the generic case equals 2 and in the Hamiltonian triangle case, this number is 3.

Generalized Lotka—Volterra systems connected with simple Lie algebras

We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.

The Jungle Universe: coupled cosmological models in a Lotka–Volterra framework

In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann–Lemaître universes is a special case of generalized Lotka–Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka–Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann–Lemaître cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserves the structure of the dynamical equations. Using the standard tools of Lotka–Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.

Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems

In this paper we study the limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of a differential system in R 3 \mathbb {R}^3 . The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using analysis techniques and the averaging theory of the second order, explicit conditions are given for the existence of one or two limit cycles bifurcating from such a zero-Hopf equilibrium. This result is applied to study three-dimensional generalized Lotka-Volterra systems in a paper by Bobieński and Żołądek (2005). The necessary and sufficient conditions for the existence of a non-isolated zero-Hopf equilibrium of this system are given, and it is shown that two limit cycles can be bifurcated from the non-isolated zero-Hopf equilibrium under a general small perturbation of three-dimensional generalized Lotka-Volterra systems.

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

Geometry of carrying simplices of 3-species competitive Lotka–Volterra systems

We investigate the existence, uniqueness and Gaussian curvature of the invariant carrying simplices of 3 species autonomous totally competitive Lotka–Volterra systems. Explicit examples are given where the carrying simplex is convex or concave, but also where the curvature is not single-signed. Our method monitors the curvature of an evolving surface that converges uniformly to the carrying simplex, and generally relies on establishing that the Gaussian image of the evolving surface is confined to an invariant cone. We also discuss the relationship between the curvature of the carrying simplex near an interior fixed point and its Split Lyapunov stability. Finally we comment on extensions to general Lotka–Volterra systems that are not competitive.

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.

Extinction in neutrally stable stochastic Lotka-Volterra models

Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. Here, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact of intrinsic noise on the survival of species is dramatic: It destroys coexistence of interacting species on a time scale proportional to the population size. We introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower-dimensional effective dynamics. This is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations. The extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations.

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.

Dynamical Principles of Emotion-Cognition Interaction: Mathematical Images of Mental Disorders

The key contribution of this work is to introduce a mathematical framework to understand self-organized dynamics in the brain that can explain certain aspects of itinerant behavior. Specifically, we introduce a model based upon the coupling of generalized Lotka-Volterra systems. This coupling is based upon competition for common resources. The system can be regarded as a normal or canonical form for any distributed system that shows self-organized dynamics that entail winnerless competition. Crucially, we will show that some of the fundamental instabilities that arise in these coupled systems are remarkably similar to endogenous activity seen in the brain (using EEG and fMRI). Furthermore, by changing a small subset of the system's parameters we can produce bifurcations and metastable sequential dynamics changing, which bear a remarkable similarity to pathological brain states seen in psychiatry. In what follows, we will consider the coupling of two macroscopic modes of brain activity, which, in a purely descriptive fashion, we will label as cognitive and emotional modes. Our aim is to examine the dynamical structures that emerge when coupling these two modes and relate them tentatively to brain activity in normal and non-normal states.

On the period function in a class of generalized Lotka–Volterra systems

In this note, motivated by the recent results of Wang et al. [Wang et al., Local bifurcations of critical periods in a generalized 2D LV system, Appl. Math. Comput. 214 (2009) 17–25], we study the behaviour of the period function of the center at the point (1,1) of the planar differential system u ′ = u p ( 1 - v q ) , v ′ = μ v q ( u p - 1 ) , where p , q , μ ∈ R with pq > 0 and μ > 0 . Our aim is twofold. Firstly, we determine regions in the parameter space for which the corresponding system has a center with a monotonic period function. Secondly, by taking advantage of the results of Wang et al., we show some properties of the bifurcation diagram of the period function and we make some comments for further research. The differential system under consideration is a generalization proposed by Farkas and Noszticzius of the Lotka–Volterra model.

Non-equilibrium thermodynamics of ecosystems: Entropic analysis of stability and diversity

The aim of the paper is first to develop a non-equilibrium thermodynamic model of a multi-species ecosystem. The key to this is the expression of entropy-production (change of entropy) which is equivalent to a well-known expression of Liapunov function used in many problems of ecological and chemical reaction systems. On the basis of the expression of entropy-production we have made an entropic (thermodynamic) analysis of ecological stability (both global and local) including the study of interrelation between ecological diversity and stability. As illustrative examples we have discussed some basic problems of classical and generalized Lotka–Volterra systems.

The period function of the generalized Lotka–Volterra centers

The present paper deals with the period function of the quadratic centers. In the literature different terminologies are used to classify these centers, but essentially there are four families: Hamiltonian, reversible Q 3 R , codimension four Q 4 and generalized Lotka–Volterra systems Q 3 LV . Chicone [C. Chicone, Review in MathSciNet, Ref. 94h:58072] conjectured that the reversible centers have at most two critical periods, and that the centers of the three other families have a monotonic period function. With regard to the second part of this conjecture, only the monotonicity of the Hamiltonian and Q 4 families [W.A. Coppel, L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993) 1357–1365; Y. Zhao, The monotonicity of period function for codimension four quadratic system Q 4 , J. Differential Equations 185 (2002) 370–387] has been proved. Concerning the Q 3 LV family, no substantial progress has been made since the middle 80s, when several authors showed independently the monotonicity of the classical Lotka–Volterra centers [F. Rothe, The periods of the Volterra–Lokta system, J. Reine Angew. Math. 355 (1985) 129–138; R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984) 1–31; J. Waldvogel, The period in the Lotka–Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986) 178–184]. By means of the first period constant one can easily conclude that the period function of the centers in the Q 3 LV family is monotone increasing near the inner boundary of its period annulus (i.e., the center itself). Thus, according to Chicone's conjecture, it should be also monotone increasing near the outer boundary, which in the Poincaré disc is a polycycle. In this paper we show that this is true. In addition we prove that, except for a zero measure subset of the parameter plane, there is no bifurcation of critical periods from the outer boundary. Finally we show that the period function is globally (i.e., in the whole period annulus) monotone increasing in two other cases different from the classical one.

The three-dimensional generalized Lotka–Volterra systems

This work is devoted to the study of periodic solutions of generalized Lotka–Volterra systems in three dimensions. The group A3 of isometries of the regular tetrahedron acts on the twelve-dimensional space of such systems. We find four components (modulo the action of A3) of the center variety, i.e. the semi-algebraic subset consisting of systems with a family of periodic solutions (which degenerate at the center). We focus attention on the component which consists of systems X0 with an invariant plane L0 transversal to the coordinate axes and with a (partial) first integral on L0. Under the assumption of normal hyperbolicity of X0 along L0 we begin study of limit cycles appearing for a perturbed system (in the Lotka–Volterra class). The linearization of the problem leads to the problem of zeroes of certain Poincare–Melnikov integrals of a new type. We determine the asymptotic behavior of these integrals near two critical values of the partial first integral of the restricted system X0|L0. We also study the situation near the intersection of different components of the center variety.

Adaptive dynamics of Lotka–Volterra systems with trade-offs: the role of interspecific parameter dependence in branching

We determine the adaptive dynamics of a general Lotka–Volterra system containing an intraspecific parameter dependency – in the form of an explicit functional trade-off between evolving parameters – and interspecific parameter dependencies – arising from modelling species interactions. We develop expressions for the fitness of a mutant strategy in a multi-species resident environment, the position of the singular strategy in such systems and the non-mixed second-order partial derivatives of the mutant fitness. These expressions can be used to determine the evolutionary behaviour of the system. The type of behaviour expected depends on the curvature of the trade-off function and can be interpreted in a biologically intuitive manner using the rate of acceleration/deceleration of the costs implicit in the trade-off function. We show that for evolutionary branching to occur we require that one (or both) of the traded-off parameters includes an interspecific parameter dependency and that the trade-off function has weakly accelerating costs. This could have important implications for understanding the type of mechanisms that cause speciation. The general theory is motivated by using adaptive dynamics to examine evolution in a predator–prey system. The applicability of the general theory as a tool for examining specific systems is highlighted by calculating the evolutionary behaviour in a three species (prey–predator–predator) system.

Coexistence, permanence and resilience in biological reaction-diffusion systems

Reaction-diffusion systems are widely used to describe spatio-temporal phenomena in a variety of scientific fields, including population ecology. In this paper, I demonstrate that existing results for coexistence and permanence of general Lotka-Volterra systems with absorbing boundaries can be applied in a complementary manner to address a variety of boundary conditions, including the insulating problem. Furthermore, the condition is applicable even to systems containing positive feedback mechanisms in the dynamics. A single (vector) inequality, the first iterate condition, is derived which serves as a sufficient condition for coexistence, permanence and resilience. Additionally, I demonstrate that this inequality condition is but the first in a series of conditions that can be used to describe the behaviour of such systems. Finally, I provide a comparison between the iterate conditions and an alternative test for solution resiliency.

Theoretical analysis and simulations of the generalized Lotka-Volterra model.

The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables w(i), i=1,...,N. The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments), and a time dependent saturation term (due to the finite size of the economy). The w(i)'s turn out to exhibit a power-law distribution of the form P(w) approximately w(-1-alpha). It is shown analytically that the exponent alpha can be expressed as a function of one parameter, which is the ratio between the constant drift component (social security) and the fluctuating component (investments). This result provides a link between the lower and upper cutoffs of this distribution, namely, between the resources available to the poorest and those available to the richest in a given society. The value of alpha is found to be insensitive to variations in the saturation term, which represent the expansion or contraction of the economy. The results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power-law distribution with 1<alpha<2.

Lyapunov Functions for Lotka-Volterra Systems: An Overview and Problems

The biggest advantage of Lyapunov's function method is that it is "direct", as "increase along orbits", hence one does not need to solve the equation explicitly. The main goal of this paper is to give the short overview of the existing two types: fitness-like and entropy-like Lyapunov functions for generalized Lotka-Volterra systems and to discuss correspondence between the structure of the interaction matrix and these Lyapunov functions. Special attention will be given in case the interaction matrix A = D1BD2 where D1 and D2 are diagonal matrix. B is symmetrical matrix and to the set of diagonal-symmetrizable matrix.

Hamiltonian structure and Darboux theorem for families of generalized Lotka–Volterra systems

This work is devoted to the establishment of a Poisson structure for a format of equations known as generalized Lotka–Volterra systems. These equations, which include the classical Lotka–Volterra systems as a particular case, have been deeply studied in the literature. They have been shown to constitute a whole hierarchy of systems, the characterization of which is made in the context of simple algebra. Our main result is to show that this algebraic structure is completely translatable into the Poisson domain. Important Poisson structures features, such as the symplectic foliation and the Darboux canonical representation, rise as a result of rather simple matrix manipulations.