A family of planar Lotka--Volterra systems with invariant algebraic curves of arbitrary degree

We introduce a new family of planar Lotka–Volterra systems admitting explicit invariant algebraic curves of arbitrarily high degree.

💡 Research Summary

This paper introduces and thoroughly analyzes a novel family of planar Lotka-Volterra differential systems characterized by the remarkable property of possessing explicit invariant algebraic curves of arbitrarily high degree. The work makes significant contributions to the theory of integrable systems and addresses classical questions concerning the relationship between high-degree invariant curves and integrability.

The authors define two primary families of systems parameterized by an integer n ∈ ℕ and a constant b ∈ K (where K is ℝ or ℂ). The first and main system is: ẋ = x(1 - x), ẏ = y(n + bx - y). The central technical achievement is the explicit construction, for any given n, of a polynomial F(x, y) of degree n (utilizing Pochhammer symbols in its coefficients) that defines an invariant algebraic curve {F(x,y)=0} for this system. The authors provide a rigorous proof (Proposition 1) that this curve is indeed invariant by verifying the condition P∂F/∂x + Q∂F/∂y = KF, with the cofactor K = n - nx - y. A similar construction (Proposition 5) yields an invariant curve for a related system where the second equation is ẏ = y(n + bx + y).

The paper then investigates the integrability of these systems. By identifying four distinct invariant algebraic curves for the main system (including the lines x=0, y=0, 1-x=0, and the high-degree curve F=0), the authors apply Darboux’s theory. They demonstrate that these curves can be combined to form a Darboux first integral, H(x, y) = y^λ₂ (1-x)^((n+b)λ₂) / F^λ₂, thereby proving the system is Darboux integrable (Theorem 2).

A particularly nuanced and important finding concerns the nature of this first integral. The analysis reveals a critical dependence on the parameter b. If b is a rational number, one can select the exponent λ₂ to be an integer such that the first integral H becomes a rational function. Consequently, in this case, the system admits a rational first integral. However, if b is irrational, no choice of λ₂ can transform H into a rational function, meaning the system, despite being Darboux integrable, does not possess a rational first integral. This result provides a clear dichotomy and deepens the understanding of the subtle differences between Darboux integrability and the existence of rational first integrals.

The research is contextualized within the framework of a classical problem posed by Poincaré, which inquires whether a polynomial differential system with an invariant algebraic curve of sufficiently high degree must necessarily have a rational first integral. The family presented here serves as a new and explicit example contributing to this discourse. It offers a concrete countable family of systems where high-degree invariant curves exist explicitly, and their integrability properties are fully classified based on a simple arithmetic condition on the parameter b. The work thus provides valuable insights into the construction of invariant curves and the delicate structure of integrability in planar polynomial systems.