Abstract
For the May–Leonard asymmetric system, which is a quadratic system of the Lotka–Volterra type depending on six parameters, we first look for subfamilies admitting invariant algebraic surfaces of degree two. Then for some such subfamilies we construct first integrals of the Darboux type, identifying the systems with one first integral or with two independent first integrals.
Highlights
An important class of mathematical models describing different phenomena in biology, ecology and chemistry are the so-called Lotka–Volterra systems, which are written in the form n ẋi = xi ( ∑ aij x j + bi )(i = 1, . . . , n). (1) j =1They were introduced independently by Lotka and Volterra in the 1920s to model the interaction among species, see [1,2], and continue being intensively investigated
To find the integrals we look for invariant surfaces of the form (12) with h000 = 0 using the procedure described at the beginning of Section 3
We have found some Darboux first integrals of the May–Leonard system (4) which are constructed using Darboux polynomials of degree one and two
Summary
There is a separatix cycle F formed by orbits connecting E1 , E2 and E3 on the boundary of R3+ and every orbit in R3+ , except of the equilibrium point C, has F as ω-limit It was shown in [3] that in the degenerate case α + β = 2, the cycle F becomes a triangle on the invariant plane x + y + z = 1, all orbits inside the triangle are closed and every orbit in the interior of R3+ has one of these closed orbits as an ω-limit. In this paper we study integrability of the asymmetric May–Leonard model (4). The set of systems with first integrals is much larger for system (4) than for the classical May–Leonard system (2)
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