On Hybrid-Function Solutions of the Lotka–Volterra Equations

The classical Lotka–Volterra predator–prey system is often used in modeling species competition. The two-species nonlinear system is expressed in terms of a single positive coupling parameter λ. Based on a standard functional transformation, a novel λ-invariant Hamiltonian yields a system of two partially uncoupled first-order hybrid-function ODEs, albeit with one being linear. An exact single quadrature solution that is valid for any value of λ and the system’s energy is derived. In the particular case of λ=1, the ODE system completely uncouples. One ODE is autonomous. An exact analytic quadrature solution is derived that is predicated on the exact turning-point solutions. It is expressed in terms of the Lambert W function and must be evaluated numerically. Exact time-dependent solutions are presented for each individual species separately. In the case of λ≠1 an accurate practical approximation uncoupling the nonlinear system is proposed and solutions are provided in terms of explicit quadratures together with high-energy asymptotic solutions. An exact analytic expression for the system’s oscillation period that is valid for any value of λ and orbital energy is derived in terms of a dimensionless energy function.