2022/12/29 by Jean-Luc Boulnois, Boulnois, Jean-Luc
Medicine · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Biological sciences #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE) #Spectroscopy and Quantum Chemical Studies
paper · pdf · doi:10.48550/arxiv.2301.00673
openalex publication_date 2022/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical two-species non-linear Predator-Prey system, often used in population dynamics modeling, is expressed in terms of a single positive coupling parameter λ. Based on standard logarithmic transformations, we derive a novel λ-invariant Hamiltonian resulting in two coupled first-order ODEs for ``hybrid-species'', albeit with one being linear; we thus derive a new exact, closed-form, single quadrature solution valid for any value of λ and the system's energy. In the particular case λ= 1 the ODE system completely uncouples and a new, exact, energy-only dependent simple quadrature solution is derived. In the case λ≠ 1 an accurate practical approximation uncoupling the non-linear system is proposed and solutions are provided in terms of explicit quadratures together with high energy asymptotic solutions. A novel, exact, closed-form expression of the system's oscillation period valid for any value of λ and orbital energy is also derived; two fundamental properties of the period are established; for λ= 1 the period is expressed in terms of a universal energy function and shown to be the shortest.