Predator-Prey Linear Coupling with Hybrid Species

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 $\lambda$. Based on standard logarithmic transformations, we derive a novel $\lambda$-\textit{invariant} Hamiltonian resulting in two coupled first-order ODEs for ``hybrid-species'', \textit{albeit} with one being \textit{linear}; we thus derive a new exact, closed-form, single quadrature solution valid for any value of $\lambda$ and the system's energy. In the particular case $\lambda = 1$ the ODE system completely uncouples and a new, exact, energy-only dependent simple quadrature solution is derived. In the case $\lambda \neq 1$ 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 $\lambda$ and orbital energy is also derived; two fundamental properties of the period are established; for $\lambda = 1$ the period is expressed in terms of a universal energy function and shown to be the shortest.