Darboux polynomials for Lotka-Volterra systems in three dimensions

On the application of the Lindstedt-Poincaré method to the Lotka-Volterra system

We apply the Lindstedt-Poincaré method to the Lotka-Volterra model and discuss alternative implementations of the approach. By means of an efficient systematic algorithm we obtain an unprecedented number of perturbation corrections for the two dynamical variables and the frequency. They enable us to estimate the radius of convergence of the perturbation series for the frequency as a function of the only model parameter. The method is suitable for the treatment of systems with any number of dynamical variables.

Algebraic Integrability of Lotka-Volterra equations in three dimensions

We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions for the algebraic integrability of the corresponding systems. We also show that the conditions are sufficient.

Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results

This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka-Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtaine…

Discrete-time analysis for the integrable discrete Toda equations and the discrete Lotka-Volterra system

The discrete autonomous/non-autonomous Toda equations and the discrete Lotka-Volterra system are important integrable discrete systems in fields such as mathematical physics, mathematical biology and statistical physics. They also have applications to numerical linear algebra. In this paper, we first simultaneously obtain their general solutions. Then, we show the asymptotic behavior of the solutions for any initial values as the discrete-time variables go to infinity. Our two main techniques for understanding the distinct integrable systems are to introduce two types of discrete-time variable…

The extended Lotka-Volterra lattice and affine Jacobi varieties of spectral curves

Based on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka-Volterra lattice. We investigate the integrable structure of the representative for the gauge equivalence class of matrices, which is isomorphic to the affine Jacobi variety, and make use it to discuss the solvability of the model.

Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems

Given a constant skew-symmetric matrix A, it is a difficult open problem whether the associated Lotka-Volterra system is integrable or not. We solve this problem in the special case when A is a Toepliz matrix where all off-diagonal entries are plus or minus one. In this case, the associated Lotka-Volterra system turns out to be a reduction of Liouville integrable systems, whose integrability was shown by Bogoyavlenskij and Itoh. We prove that the reduced systems are also Liouville integrable and that they are also non-commutative integrable by constructing a set of independent first integrals,…

Lie and conditional symmetries of the three-component diffusive Lotka - Volterra system

Lie and Q-conditional symmetries of the classical three-component diffusive Lotka - Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component non-linear reaction-diffusion system are constructed for the first time. An example of non-Lie symmetry reduction for solving a biologically motivated problem is presented.

The matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability…

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 Darb…

由时滞离散 Lotka–Volterra 方程超离散化产生的箱球系统时滞类比

A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation

研究通过对可积的时滞离散 Lotka–Volterra 方程超离散化，构造箱球系统的时滞类比，并分类、数值发现及解析刻画其新型异常孤子。

Generalized Volterra lattices: binary Darboux transformations and self-consistent sources

We study two families of (matrix versions of) generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions.