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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. [ABSTRACT FROM AUTHOR]

Abstract: We consider explicit symplectic partitioned Runge–Kutta (ESPRK) methods for the numerical integration of non-autonomous dynamical systems. It is known that, in general, the accuracy of a numerical method can diminish considerably whenever an explicit time dependence enters the differential equations and the order reduction can depend on the way the time is treated. In the present paper, we demonstrate that explicit symplectic partitioned Runge–Kutta–Nyström (ESPRKN) methods specifically designed for second order differential equations , undergo an order reduction when , independently of the way the time is approximated. Furthermore, by means of symmetric quadrature formulae of appropriate order, we propose a different but still equivalent formulation of the original non-autonomous problem that treats the time as two added coordinates of an enlarged differential system. In so doing, the order reduction is avoided as confirmed by the presented numerical tests. [Copyright &y& Elsevier]

Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitzplus- Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application. [ABSTRACT FROM AUTHOR]