Hybrid Species and Closed-Form Solutions of the Lotka-Volterra Equations |Biomedgrid

The classical Lotka-Volterra system of two coupled non-linear ordinary differential equations is expressed in terms of a single positive coupling parameter λ , ratio of the respective natural growth and decay rates of the prey and predator populations. “Hybrid-species” are introduced resulting in a novel λ − invariant Hamiltonian of two coupled first-order ODE albeit with one being linear; a new exact, closed-form, single quadrature solution valid for any value of λ and the system's energy is derived. In the particular case λ = 1 the ODE system partially uncouples and new, exact, closed-form time-dependent solutions are derived for each individual species. In the case λ ≠ 1 an accurate practical approximation uncoupling the non-linear system is proposed; solutions are provided in terms of explicit quadratures together with analytical 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 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.