# American Institute of Mathematical Sciences

September  2018, 23(7): 2661-2678. doi: 10.3934/dcdsb.2017185

## Positive symplectic integrators for predator-prey dynamics

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

* Corresponding author

Received  October 2016 Revised  May 2017 Published  July 2017

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.

Citation: Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185
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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: $u_0 = v_0 = 0.2$
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)
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
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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