Lie symmetries and invariants of the Lotka–Volterra system

Abstract

In this paper we use the Lie symmetry method for finding rational and transcendental symmetry transformations and invariants for the 3D Lotka–Volterra system.

Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems

“…Additionally we also prove that when a = b = −1 the system has two independent first integrals. We note that the existence of first integrals for system (2) imply the existence of invariants for system (1). Here an invariant is a first integral depending on the time.…”

“…So the study of the existence of first integrals is an important subject in the qualitative theory of differential equations. Many different methods have been used for studying the existence of first integrals of non-linear differential systems based on: Noether symmetries [6], the Darboux theory of integrability [8,17], the Lie symmetries [1,23], the Painlevé analysis [3], the use of Lax pairs [12], the direct method [9,10], the linear compatibility analysis method [24], the Carlemann embedding procedure [7,2], the quasimonomial formalism [4], etc.…”

Rings of constants of four-variable Lotka-Volterra systems

“…We call a polynomial ∈ R strict if it is homogeneous and not divisible by the variables 1 . Every nonzero homogeneous polynomial ∈ R has the unique representation = X α , where X α is a monomial and is strict.…”

“…Hence 2 = 1 C 1 . A simple induction shows that the coefficient of the monomial in the expansion of is equal to 1 …”

“…Lotka-Volterra systems have a variety of applications in such branches of science as population biology, laser physics and plasma physics (for more details we refer the reader for example to [1][2][3] and their references). Moreover, they play a significant role in derivation theory.…”