Volterra System Research Articles

Permanence and extinction in competitive Lotka–Volterra systems with delays

In this paper, a class of competitive Lotka–Volterra systems are considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. Under the assumption that all proper subsystems are permanent, it is shown that the asymptotic behaviour of the system is determined by the relationship between an equilibrium and a nullcline plane of the corresponding autonomous system: if the equilibrium is below the plane then the system is permanent; if the equilibrium is above the plane then this species will go extinct in an exponential rate while the other species will survive. Similar asymptotic behaviour is also retained under an alternative assumption.

CHARACTERIZING SPECIES DISTRIBUTIONS BY PRODUCTIVITY AND MORTALITY RATES IN MULTISPECIES MODELS

Abstract The temporal change in species composition is one dimension that may be used to characterize ecosystems. Multispecies interactions and species abundance (or biomass) distributions can be analyzed by an N‐species Lotka–Volterra (LV) system of equations. We derive a statistical approximation to the stationary probability distribution of relative species biomass for a system of interacting species. The distribution is parameterized by the mean and variance of the LV species interaction coefficients and can exhibit a variety of shapes. Given this distribution, one can describe the state of the system, and subsequent perturbations within the system, in terms of shifts in the resulting stable distribution. Theoretical distributions with known properties were compared to distributions estimated from simulations and good agreement was found. Our analytical result could be used to develop “ecosystem indicators” for describing changes in the state of species within an ecosystem.

Controllability of SISO Volterra Models via Diffusive Representation

AbstractThe problem under consideration is the controllability of a wide class of convolution Volterra systems, namely the class of “diffusive” systems, for which there exists an input-output state realization whose state evolves in the so-called diffusive representation space. We first show that this universal state variable is approximately controllable, and then deduce that such Volterra systems always possess suitable controllability properties, stated and proved. Then, we show how to solve the optimal null control problem in an LQ sense. A numerical example finally highlights these results.

Stability in distribution of competitive Lotka–Volterra system with Markovian switching

In this paper, we consider a stochastic Lotka–Volterra competitive system d x i ( t ) = x i ( t ) { [ b i ( ξ ( t ) ) - ∑ j = 1 n a ij ( ξ ( t ) ) x j ( t ) ] d t + σ i ( ( ξ ( t ) d w i ( t ) } , where w i ( t) ( i = 1, 2, … , n) are independent standard Brownian motions and ξ(·) is Markov chain taking values in a finite space M = { 1 , 2 , … , m } . Global attractivity, upper boundedness and other properties are obtained. In addition, asymptotically stable in distribution as the main result of our paper is derived under some conditions. We gave a numerical simulation for invariant distribution of an example by using the Monte Carlo simulation method at the end.

Unbounded delay stochastic functional Kolmogorov-type system

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, we perturb the functional Kolmogorov-type systeminto the stochastic functional differential equationWe show that different environmental noise structures have different effects on the population system with unbounded delay. Under two classes of different environmental noise perturbations, we establish existence theorems of the global positive solution to the unbounded delay stochastic functional Kolmogorov-type system. As the desired results for population dynamics, we also examine asymptotic boundedness, including the moment boundedness, stochastically ultimate boundedness and the moment average boundedness in time. To illustrate our idea more clearly, as a special case we also discuss a Lotka–Volterra system with unbounded delay.

Traveling waves for a Lotka–Volterra competition system with diffusion

The paper addresses the existence of traveling waves for a two species competitive Lotka–Volterra reaction–diffusion system with discrete delays {∂∂tu1(x,t)=d1∂2∂x2u1(x,t)+r1u1(x,t)[1−a1u1(x,t)−b1u1(x,t−τ1)−c1u2(x,t−τ2)],∂∂tu2(x,t)=d2∂2∂x2u2(x,t)+r2u2(x,t)[1−a2u2(x,t)−b2u2(x,t−τ3)−c2u1(x,t−τ4)]. Our results can extend some existing ones for two species diffusion competition Lotka–Volterra systems.

The cyclicity and period function of a class of quadratic reversible Lotka–Volterra system of genus one

In this paper, we investigate a class of quadratic reversible Lotka–Volterra system of genus one z˙=iz+z2+bz¯2 with b=3/5. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. Moreover, we prove that the period function of its period trajectories is monotone increasing.

Convergence to equilibrium in competitive Lotka–Volterra and chemostat systems

We study a generalized system of ODE's modeling a finite number of biological populations in a competitive interaction. We adapt the techniques in Jabin and Raoul [8] and Champagnat and Jabin (2010) [2] to prove the convergence to a unique stable equilibrium.

A Bäcklund transformation between two integrable discrete hungry systems

The discrete hungry Toda (dhToda) equation and the discrete hungry Lotka–Volterra (dhLV) system are known as integrable discrete hungry systems. In this Letter, through finding the LR transformations associated with the dhToda equation and the dhLV system, we present a Bäcklund transformation between these integrable systems.

Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines

In this article, we study the Lotka–Volterra planar quadratic differential systems. We denote by LV systems all systems which can be brought to a Lotka–Volterra system by an affine transformation and time homotheties. All these systems possess invariant straight lines. We classify the family of LV systems according to their geometric properties encoded in the configurations of invariant straight lines which these systems possess. We obtain a total of 65 such configurations which are distinguished, roughly speaking, by the multiplicity of their invariant lines and by the multiplicities of the singularities of the systems located on these lines. We determine an algebraic subvariety of \({\mathbb{R}^{12}}\) which contains all these systems and we find the bifurcation diagram of the configurations of LV systems within this algebraic subvariety, in terms of polynomial invariants with respect to the group action of affine transformations and time homotheties. This geometric classification will serve as a basis for the full topological classification of LV systems.

Backstepping for nonsmooth MIMO nonlinear Volterra systems with noninvertible input-output maps

We prove the global controllability for a class of nonlinear MIMO Volterra systems of the triangular form as well as for their bounded perturbations. In contrast to the related preceeding work [12], we replace the condition of C1 smoothness, which was essentially used before, with that of local Lipschitzness. Furthermore, we remove the assumption of the invertibility of the input-output interconnections, which was also essential in these preceeding results. In order to solve the problem, we revise the backstepping procedure proposed in these works, and combine it with another method of constructing discontinuous feedbacks proposed for the so-called “generalized triangular form” in the case of ODE [17, 13, 19].

Numerical Integration and Synchronization for the 3‐Dimensional Metriplectic Volterra System

The main purpose of this paper is to study the metriplectic system associated to 3‐dimensional Volterra model. For this system we investigate the stability problem and numerical integration via Kahan′s integrator. Finally, the synchronization problem for two coupled metriplectic Volterra systems is discussed.

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka–Volterra systems.

PARAFAC-based channel estimation and data recovery in nonlinear MIMO spread spectrum communication systems

In this paper, a new tensorial modeling is first proposed for nonlinear multiple-input multiple-output (MIMO) direct sequence spread spectrum communication systems. The channel is modeled as an instantaneous MIMO Volterra system. Then, a direct data approach for joint blind channel estimation and data recovery is developed using the parallel factor (PARAFAC) decomposition of a third-order tensor composed of received signals, exploiting space, time and code diversities. A blind channel estimation method based on the PARAFAC decomposition of a fifth-order tensor composed of covariances of the received signals is also proposed, considering phase shift keying (PSK) modulated transmitted signals. The proposed estimation algorithms are evaluated by simulating a nonlinear uplink MIMO radio over fiber (ROF) communication system.

Microscopic, mesoscopic and macroscopic descriptions of complex systems

In the present paper the general approach that relates microscopic models with those at the mesoscopic scale and then the macroscopic scale is reviewed. The models at the micro-scale level are defined by a large number of interacting entities (particles, agents, cells, individuals, etc), and is in terms of a Markov jump process and related linear evolution equations. The intermediate models refer to the meso-scale level of description of test entities and are given in terms of bilinear Boltzmann-type equations. Mathematical relationships between these three possible descriptions are presented and explicit error estimates are given. The general framework is applied to propose the microscopic and mesoscopic models that correspond to very well known models in biomathematics: the Verhulst logistic equation and the Lotka–Volterra system of equations. The asymptotic time behaviour for the mesoscopic model corresponding to the Verhulst logistic equation is defined. The mesoscopic model corresponding to the Verhulst equation is modified to a new mesoscopic model of DNA denaturation.

An integrating factor matrix method to find first integrals

In this paper we develop an integrating factor matrix method to derive conditions for the existence of first integrals. We use this novel method to obtain first integrals, along with the conditions for their existence, for two- and three-dimensional Lotka–Volterra systems with constant terms. The results are compared to previous results obtained by other methods.

On permanence of all subsystems of competitive Lotka–Volterra systems with delays

In this paper, permanence for a class of competitive Lotka–Volterra systems is considered that have distributed delays and constant coefficients on interaction terms and have time dependent growth rate vectors with an asymptotic average. A computable necessary and sufficient condition is found for the permanence of all subsystems of the system and its small perturbations on the interaction matrix. This is a generalization from systems without delays to delayed systems of Ahmad and Lazer’s work on total permanence (S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006) S47–S67). In addition to Ahmad and Lazer’s example showing that permanence does not imply total permanence, another example of permanent system is given having a non-permanent subsystem. As a particular case, a necessary and sufficient condition is given for all subsystems of the corresponding autonomous system to be permanent. As this condition does not rely on the delays, it actually shows the equivalence between such permanence of systems with delays and that of corresponding systems without delays. Moreover, this permanence property is still retained by systems as a small perturbation of the original system.

Cross rules and non-Abelian lattice equations for the discrete and confluent non-scalar ε-algorithms

In this paper, we give the cross rules of the discrete and confluent vector, topological and matrix ε-algorithms. Then, from the rules of these confluent algorithms, we derive non-Abelian lattice equations, in particular some extensions of the Lotka–Volterra system, in the style of the equation related to the confluent form of the scalar ε-algorithm.

On the long time behavior of non-autonomous Lotka–Volterra models with diffusion via the sub-supertrajectory method

In this paper we study in detail the geometrical structure of global pullback and forwards attractors associated to non-autonomous Lotka–Volterra systems in all the three cases of competition, symbiosis or prey–predator. In particular, under some conditions on the parameters, we prove the existence of a unique nondegenerate global solution for these models, which attracts any other complete bounded trajectory. Thus, we generalize the existence of a unique strictly positive stable (stationary) solution from the autonomous case and we extend to Lotka–Volterra systems the result for scalar logistic equations. To this end we present the sub-supertrajectory tool as a generalization of the now classical sub-supersolution method. In particular, we also conclude pullback and forwards permanence for the above models.

Polymorphic evolution sequence and evolutionary branching

We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence (TSS), is called polymorphic evolution sequence (PES). Then we introduce a scaling of the size of mutations and we study the PES in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. This phenomenon corresponds to the situation where the population, initially essentially single modal, is driven by the selective forces to divide into two separate subpopulations. To this end we finely analyze the asymptotic behavior of three-dimensional competitive Lotka–Volterra systems.