Positive symplectic integrators for predator-prey dynamics

Fasma Diele; Carmela Marangi

doi:10.3934/dcdsb.2017185

Article Contents

2018, Volume 23, Issue 7: 2661-2678. Doi: 10.3934/dcdsb.2017185

This issue Previous Article A comparison of boundary correction methods for Strang splitting Next Article Effect of perturbation in the numerical solution of fractional differential equations

Positive symplectic integrators for predator-prey dynamics

Fasma Diele^, and
Carmela Marangi

Istituto per Applicazioni del Calcolo M.Picone, CNR, Bari, via Amendola 122/D, Italy

* Corresponding author
* Corresponding author

Received: September 30, 2016

Revised: April 30, 2017

Early access: June 2017

Published: July 2018

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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.

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Mathematics Subject Classification: Primary: 37M15; Secondary: 65P10.

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Figure 1. On the left: positive first-order schemes (7) and (8) compared with the symplectic Euler (SE) method, its explicit variant (EVSE) applied to the LV system at $T=8.3$, with $u_0 = 0.2$, $v_0 = 1.1$ and $\Delta t = 1.1$. Parameters: $a =b =1$. On the right: numerical accuracy of Poisson integrators at $T=10$, including Strang splitting (SS) and Yoshida composition (YC), applied to the LV system with $\Delta t = 1/k$, for $k = 3,\dots,8$. Parameters: $a = b = 0.5$. Initial values: $u_0 = v_0 = 0.2$

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Figure 2. Positive symplectic Euler (17) compared with the explicit Euler method applied to the Z-controlled LV dynamics (21) with $u_0 =v_0 = 40$ and $\Delta t = 0.1$. Parameters: $\alpha=\delta=0.6$, $\beta=\gamma=0.01$, $u_d=100$, $\lambda=1.4$. Phase space portrait (left), predator function versus time (right)

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Figure 3. Plots of the concentration profiles of $u(x,t)$ (right) and $v(x,t)$ (left) with positive Lie Splitting (solid line) and nonstandard positive method (dashed line) at $t=100$. Step sizes $h=0.4$ and $\Delta t=0.32$ for positive Lie Splitting. Refinements are obtained with $h=0.4,0.8,0.08$ and $\Delta t=0.32,0.8,0.032$ for the nonstandard positive approximations

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Figure 4. Prey densities approximation with IMEX (left), IMSP (center) and $\Phi^{(RM)}$ (right) schemes for different temporal step size: $\Delta t = 1/3, 1/24, 1/384$ (left and center columns), $\Delta t = 1, 1/3, 1/24$ (right column)

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