Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Positive symplectic integrators for predator-prey dynamics

Page numbers are going to be assigned later 2017      doi:10.3934/dcdsb.2017185

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Fasma Diele - Istituto per Applicazioni del Calcolo M.Picone, CNR, Bari, via Amendola 122/D, Italy (email)
Carmela Marangi - Istituto per Applicazioni del Calcolo M.Picone, CNR, Bari, via Amendola 122/D, Italy (email)

Abstract: We propose novel positive numerical integrators for approximating predator-prey models. The schemes are based on suitable symplectic procedures applied to the dynamical system written in terms of the log transformation of the original variables. Even if this approach is not new when dealing with Hamiltonian systems, it is of particular interest in population dynamics since the positivity of the approximation is ensured without any restriction on the temporal step size. When applied to separable $M$-systems, the resulting schemes are proved to be explicit, positive, Poisson maps. The approach is generalized to predator-prey dynamics which do not exhibit an $M$-system structure and successively to reaction-diffusion equations describing spatially extended dynamics. A classical polynomial Krylov approximation for the diffusive term joint with the proposed schemes for the reaction, allows us to propose numerical schemes which are explicit when applied to well established ecological models for predator-prey dynamics. Numerical simulations show that the considered approach provides results which outperform the numerical approximations found in recent literature.

Keywords:  Positive numerical integration, symplectic integrators, Poisson systems, predator-prey dynamics, Rosenzweig-MacArthur model.
Mathematics Subject Classification:  Primary: 37M15 ; Secondary: 65P10.

Received: October 2016;      Revised: May 2017;      First Online: July 2017.