Volterra System Research Articles

New criteria on persistence in mean and extinction for stochastic competitive Lotka–Volterra systems with regime switching

This paper concerns persistence in mean and extinction for stochastic competitive Lotka–Volterra systems with regime switching. By using some novel stochastic analysis techniques, sufficient criteria for the partial permanence and partial extinction are established. Some novel sufficient conditions on persistence in mean and extinction are also obtained. Nontrivial examples are provided to illustrate our results.

Optimal harvesting strategies for stochastic competitive Lotka–Volterra ecosystems

This work develops optimal harvest strategies for Lotka–Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. To better reflect reality, a continuous-time Markov chain is used to model the random environment. The underlying systems are thus controlled regime-switching diffusions that belong to the class of singular control problems. Starting with a model having multiple species, we construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. Moreover, in some cases, the lower bounds provide a good approximation of the value functions.

Global attractivity and extinction for Lotka–Volterra systems with infinite delay and feedback controls

The paper deals with a multiple species Lotka–Volterra model with infinite distributed delays and feedback controls, for which we assume a weak form of diagonal dominance of the instantaneous negative intra-specific terms over the infinite delay effect in both the population variables and controls. General sufficient conditions for the existence and attractivity of a saturated equilibrium are established. When the saturated equilibrium is on the boundary of , sharper criteria for the extinction of all or part of the populations are given. While the literature usually treats the case of competitive systems only, here no restrictions on the signs of the intra- and inter-specific delayed terms are imposed. Moreover, our technique does not require the construction of Lyapunov functionals.

Stability of planar waves in a Lotka–Volterra system

This paper is concerned with the large time behavior of disturbed planar fronts in the Lotka–Volterra system in Rn(n⩾2). We first show that the large time behavior of the disturbed fronts can be approximated by that of the mean curvature flow with a drift term for all large time up to t=+∞. And then we prove that the planar front is asymptotically stable in L∞(Rn) under ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases.

Local well-posedness for a Lotka–Volterra system in Besov spaces

This paper is devoted to study local existence, uniqueness, and continuous dependence upon the initial data of mild solutions for a diffusive non-autonomous Lotka–Volterra system. Initial data are taken in the Besov space Bp,q,Nσ.

Global asymptotic stability of stochastic Lotka–Volterra systems with infinite delays

Over the past few decades, differential equations with time delays have received great attention owing to their extensive applications as models in a variety of scientific areas, such as population dynamics, epidemiology and biology. One of the most interesting topics associated with these delay population systems, especially Lotka–Volterra (or LV for short) systems, is the investigation of the global asymptotic stability (or global attractivity) of the positive solution. There is an extensive literature dealing with the global asymptotic stability of LV systems with delays (see, e.g. Gopalsamy, 1992; Kuang, 1993; Kuang & Smith, 1993; Lu & Takeuchi, 1994; Wang & Zhang, 1995; Bereketoglu & Gyori, 1997; He & Gopalsamy, 1997; Ahmad & Lazer, 2000; Teng & Yu, 2000; Saito, 2002; Teng, 2002; Zhao et al., 2004; Chen, 2005, 2006; Faria, 2009; Hu et al., 2011; Zhang & Teng, 2011 and the references therein), but mainly in the deterministic case. Since population systems are inevitably subject to environmental fluctuations (e.g. variation in intensity of sunlight, temperature, water level, etc.; see, e.g. Renshaw, 1991; May, 2001), it is important and necessary to investigate the global asymptotic stability of stochastic LV models. So far, to the best of our knowledge, there are only a few papers (Jiang et al., 2008; Li & Mao, 2009; Liu & Wang, 2013a,b; Yao & Liu, 2014) which studied the global asymptotic stability of stochastic population systems. Jiang et al. (2008) studied the following stochastic logistic model:

Almost periodic solutions of nonautonomous Lotka–Volterra competitive systems with dominated delays

In this paper, a class of nonautonomous Lotka–Volterra type multispecies competitive systems with delays is studied. By employing Lyapunov functional, some sufficient conditions to guarantee the existence of almost periodic solutions for the Lotka–Volterra system are obtained.

IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics

We examine spatially explicit models described by reaction–diffusion partial differential equations for the study of predator–prey population dynamics. The numerical methods we propose are based on the coupling of a finite difference/element spatial discretization and a suitable partitioned Runge–Kutta scheme for the approximation in time. The RK scheme here implemented uses an implicit scheme for the stiff diffusive term and a partitioned RK symplectic scheme for the reaction term (IMSP schemes). We revisit some results provided in the literature for the classical Lotka–Volterra system and the Rosenzweig–MacArthur model. We then extend the approach to metapopulation dynamics in order to numerically investigate the effect of migration through a corridor connecting two habitat patches. Moreover, we analyze the synchronization properties of subpopulation dynamics, when the migration occurs through corridors of variable sizes.

Dynamics of stochastic delay Lotka–Volterra systems with impulsive toxicant input and Lévy noise in polluted environments

In this paper, two stochastic delay Lotka–Volterra systems (i.e., competition system and predator–prey system) with impulsive toxicant input and Lévy noise in polluted environments are proposed and investigated. Under some simple assumptions, sufficient and necessary criteria for stability in time average and extinction of each population are established. The thresholds between stability in time average and extinction of each model are obtained. Some recent results are improved and extended greatly.

Horseshoes in 3D equations with applications to Lotka–Volterra systems

We discuss a geometric configuration for a class of homeomorphisms in \({\mathbb{R}^3}\) producing the existence of infinitely many periodic points as well a complex dynamics due to the presence of a topological horseshoe. We also show that such a class of homeomorphisms appears in the classical Lotka–Volterra system.

Population Interactions in Ecology: A Rule-Based Approach to Modeling Ecosystems in a Mass-Conserving Framework

Mathematical biology/ecology teaching for undergraduates has generally relied on the Lotka--Volterra competition and predator-prey models to introduce students to population dynamics. Students are provided with an understanding of the application of dynamical system theory in simulating and understanding the behavior of the natural world, and they are provided with opportunities to practice phase plane analysis techniques such as determining the stability of equilibrium points and bifurcation analysis. This paper outlines a course in ecological modeling suitable for all students in the life sciences. The course is based on realistic ecological principles, such as using nutrient concentration to measure populations together with explicit resource availability to constrain population growth, and it considers simple Lotka--Volterra systems within this theoretical framework. An advantage of this approach is that the widely experimentally observed models of mixotrophy and mutualism can be naturally and simply ...

Investigation of transition to chaos for a Lotka-Volterra system with the seasonality factor using the dissipative Henon map

Conditions under which the classical Lotka–Volterra system with a seasonality factor exhibits chaotic behavior are investigated. The seasonality factor is introduced in such a way that the system can be investigated using the Henon map. The reduction of the system to Henon’s map allows one to calculate the period doubling bifurcations and determine the point of the transition to chaos.

About several classes of bi-orthogonal polynomials and discrete integrable systems

By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka–Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient–quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

Stability in Nonlinear Delay Volterra Integro-differential Systems

We employ Lyapunov functionals to the system of Volterra integro-dierenti al equations of the form

A system dynamics approach to technology interaction: From asymptotic to cyclic behaviour

This paper is an extension and elaboration of previous research on the simulation of three competing technologies that interact. A modified version of the three-technology system is investigated, and some initial system dynamics results are reported illustrating the progression from asymptotic to cyclic behaviour. Technology is considered in this research as a result of innovation, a rate-dependent process that may include several non-linearities due to interaction with the environment and social context. Using bibliometrics as a research data source is an interesting way to trace technology growth patterns very effectively. In this research, the existence of cyclic behaviour in two real life technologies is illustrated using bibliometrics. In this paper, a technology system consisting of three interacting technologies is treated and modelled in a coupled manner where the interacting dynamics is described by the Lotka–Volterra system of differential equations. The effect of interaction between the technologies and the period of cyclic behaviour is illustrated parametrically. Furthermore, the possible uncertain diffusion as well as interaction effect for two of the technologies is also addressed in this research using a Monte Carlo multivariate simulation technique and a system dynamics approach. The research method is exploratory and case based.

Phase portraits of quadratic Lotka–Volterra systems with a Darboux invariant in the Poincaré disc

We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant.

First integrals of Darboux type for a family of 3-dimensional Lotka–Volterra systems

We provide all the first integrals of Darboux type for the system studied by Leach and Miritzis (2006) [7] on the classical model of competition between three species considered by May and Leonard (1975) [12].

Liouvillian integrability of gravitating static isothermal fluid spheres

We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = −1 and n = −3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = −5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka–Volterra quadratic polynomial differential system in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^2$\end{document}R2, and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka–Volterra system possesses Liouvillian first integrals for n = −1, −3, −5, which descend from the existence of invariant algebraic curves of degree one, and for n = −6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka–Volterra system, and consequently of the second order ODE for the mass function must be non-Liouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely.

Global dynamics for Lotka–Volterra systems with infinite delay and patch structure

We study some aspects of the global dynamics of an n-dimensional Lotka–Volterra system with infinite delay and patch structure, such as extinction, persistence, existence and global asymptotic stability of a positive equilibrium. Both the cases of an irreducible and reducible linear community matrix are considered, and no restriction on the signs of the intra- and inter-specific delayed terms is imposed. Although the system is not cooperative, our approach often uses comparative results with an auxiliary cooperative system. Some models in recent literature are generalized, and results improved.

On generalized Volterra systems

We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The algorithm uses the root system of a complex simple Lie algebra. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka–Volterra system. We present in detail all such systems in the cases of A 3 , A 4 and we also give some examples from higher dimensions. We classify all possible Lotka–Volterra systems that arise via this algorithm in the A n case.