# American Institute of Mathematical Sciences

## 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, doi: 10.3934/dcdsb.2017185
##### References:
 [1] M. Beck and M. Gander, On the positivity of Poisson integrators for the Lotka-Volterra equations, BIT Numerical Mathematics, 55 (2015), 319-340. doi: 10.1007/s10543-014-0505-1. [2] S. Blanes and F. Casas, Splitting methods in the numerical integration of non-autonomous dynamical systems, Journal of Physics A: Mathematical and General, 39 (2006), 5405-5423. doi: 10.1088/0305-4470/39/19/S05. [3] S. Blanes, F. Casas, P. Chartier and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Mathematics of Computiation, 82 (2013), 1559-1576. doi: 10.1090/S0025-5718-2012-02657-3. [4] S. Blanes, F. Diele, C. Marangi and S. Ragni, Splitting and composition methods for explicit time dependence in separable dynamical systems, Journal of Computational and Applied Mathematics, 235 (2010), 646-659. doi: 10.1016/j.cam.2010.06.018. [5] F. Diele, I. Moret and S. Ragni, Error estimates for polynomial Krylov approximations to matrix functions, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1546-1565. doi: 10.1137/070688924. [6] F. Diele, C. Marangi and S. Ragni, Implicit-Symplectic Partitioned (IMSP) Runge-Kutta Schemes for Predator-Prey Dynamics, AIP Conference Proceedings, 1479 (2012), 1177-1180. doi: 10.1063/1.4756360. [7] F. Diele, C. Marangi and S. Ragni, IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics, Mathematics and Computers in Simulation, 100 (2014), 41-53. doi: 10.1016/j.matcom.2014.01.006. [8] L. Einkemmer and A. Ostermann, Overcoming order reduction in reaction-diffusion splitting. Part 1: Dirichlet boundary conditions, SIAM Journal on Scientific Computing, 37 (2015), A1577-A1592. doi: 10.1137/140994204. [9] L. Einkemmer and A. Ostermann, Overcoming order reduction in diffusion-reaction splitting. Part 2: Oblique boundary conditions, SIAM Journal on Scientific Computing, 38 (2016), A3741-A3757. doi: 10.1137/16M1056250. [10] M. R. Garvie, Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bulletin of Mathematical Biology, 69 (2007), 931-956. doi: 10.1007/s11538-006-9062-3. [11] M. R. Garvie and M. Golinski, Metapopulation dynamics for spatially extended predator-prey interactions, Ecological Complexity, 7 (2010), 55-59. doi: 10.1016/j.ecocom.2009.05.001. [12] M. R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator-prey interactions with the Holling type Ⅱ functional response, Numerische Mathematik, 107 (2007), 641-667. doi: 10.1007/s00211-007-0106-x. [13] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations Springer-Verlag, Berlin, 2006. [14] E. Hansen, F. Kramer and A. Ostermann, A second-order positivity preserving scheme for semilinear parabolic problems, Applied Numerical Mathematics, 62 (2012), 1428-1435. doi: 10.1016/j.apnum.2012.06.003. [15] T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, Journal of Computational and Applied Mathematics, 215 (2008), 182-195. doi: 10.1016/j.cam.2007.04.003. [16] D. Lacitignola, F. Diele, C. Marangi and A. Provenzale, On the dynamics of a generalized predator-prey system with Z-type control, Mathematical Biosciences, 280 (2016), 10-23. doi: 10.1016/j.mbs.2016.07.011. [17] J. Martinez-Linares, Phase space formulation of population dynamics in ecology, preprint, arXiv: q-bio. PE/1304. 2324v. [18] A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370. doi: 10.1137/S0036144502404442. [19] M. Mehdizadeh Khalsaraei and R. Shokri Jahandizi, Positivity-preserving nonstandard finite difference schemes for simulation of advection-diffusion reaction equation, Computational Methods for Differential Equations, 2 (2014), 256-267. [20] J. H. Merkin, D. J. Needham and S. K. Scott, The Development of Travelling Waves in a Simple Isothermal Chemical System Ⅰ. Quadratic Autocatalysis with Linear Decay, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 424 (1989), 187-209. doi: 10.1098/rspa.1989.0075. [21] R. E. Mickens, Nonstandard finite difference schemes for reaction-diffusion equations, Numerical Methods for Partial Differential Equations, 15 (1999), 201-214. doi: 10.1002/(SICI)1098-2426(199903)15:2<201::AID-NUM5>3.0.CO;2-H. [22] M. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209-223. doi: 10.1086/282272. [23] V. Thomée, On Positivity Preservation in Some Finite Element Methods for the Heat Equation. In: Dimov I. , Fidanova S. , Lirkov I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science, Springer, 8962 (2015), 13-24 [24] E. H. Twizell, Y. Wang, W. G. Price and F. Fakhr, Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system, Numerical Methods for Partial Differential Equations, 10 (1994), 435-454. doi: 10.1002/num.1690100404.

show all references

##### References:
 [1] M. Beck and M. Gander, On the positivity of Poisson integrators for the Lotka-Volterra equations, BIT Numerical Mathematics, 55 (2015), 319-340. doi: 10.1007/s10543-014-0505-1. [2] S. Blanes and F. Casas, Splitting methods in the numerical integration of non-autonomous dynamical systems, Journal of Physics A: Mathematical and General, 39 (2006), 5405-5423. doi: 10.1088/0305-4470/39/19/S05. [3] S. Blanes, F. Casas, P. Chartier and A. Murua, Optimized high-order splitting methods for some classes of parabolic equations, Mathematics of Computiation, 82 (2013), 1559-1576. doi: 10.1090/S0025-5718-2012-02657-3. [4] S. Blanes, F. Diele, C. Marangi and S. Ragni, Splitting and composition methods for explicit time dependence in separable dynamical systems, Journal of Computational and Applied Mathematics, 235 (2010), 646-659. doi: 10.1016/j.cam.2010.06.018. [5] F. Diele, I. Moret and S. Ragni, Error estimates for polynomial Krylov approximations to matrix functions, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1546-1565. doi: 10.1137/070688924. [6] F. Diele, C. Marangi and S. Ragni, Implicit-Symplectic Partitioned (IMSP) Runge-Kutta Schemes for Predator-Prey Dynamics, AIP Conference Proceedings, 1479 (2012), 1177-1180. doi: 10.1063/1.4756360. [7] F. Diele, C. Marangi and S. Ragni, IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics, Mathematics and Computers in Simulation, 100 (2014), 41-53. doi: 10.1016/j.matcom.2014.01.006. [8] L. Einkemmer and A. Ostermann, Overcoming order reduction in reaction-diffusion splitting. Part 1: Dirichlet boundary conditions, SIAM Journal on Scientific Computing, 37 (2015), A1577-A1592. doi: 10.1137/140994204. [9] L. Einkemmer and A. Ostermann, Overcoming order reduction in diffusion-reaction splitting. Part 2: Oblique boundary conditions, SIAM Journal on Scientific Computing, 38 (2016), A3741-A3757. doi: 10.1137/16M1056250. [10] M. R. Garvie, Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bulletin of Mathematical Biology, 69 (2007), 931-956. doi: 10.1007/s11538-006-9062-3. [11] M. R. Garvie and M. Golinski, Metapopulation dynamics for spatially extended predator-prey interactions, Ecological Complexity, 7 (2010), 55-59. doi: 10.1016/j.ecocom.2009.05.001. [12] M. R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator-prey interactions with the Holling type Ⅱ functional response, Numerische Mathematik, 107 (2007), 641-667. doi: 10.1007/s00211-007-0106-x. [13] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations Springer-Verlag, Berlin, 2006. [14] E. Hansen, F. Kramer and A. Ostermann, A second-order positivity preserving scheme for semilinear parabolic problems, Applied Numerical Mathematics, 62 (2012), 1428-1435. doi: 10.1016/j.apnum.2012.06.003. [15] T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, Journal of Computational and Applied Mathematics, 215 (2008), 182-195. doi: 10.1016/j.cam.2007.04.003. [16] D. Lacitignola, F. Diele, C. Marangi and A. Provenzale, On the dynamics of a generalized predator-prey system with Z-type control, Mathematical Biosciences, 280 (2016), 10-23. doi: 10.1016/j.mbs.2016.07.011. [17] J. Martinez-Linares, Phase space formulation of population dynamics in ecology, preprint, arXiv: q-bio. PE/1304. 2324v. [18] A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, H. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370. doi: 10.1137/S0036144502404442. [19] M. Mehdizadeh Khalsaraei and R. Shokri Jahandizi, Positivity-preserving nonstandard finite difference schemes for simulation of advection-diffusion reaction equation, Computational Methods for Differential Equations, 2 (2014), 256-267. [20] J. H. Merkin, D. J. Needham and S. K. Scott, The Development of Travelling Waves in a Simple Isothermal Chemical System Ⅰ. Quadratic Autocatalysis with Linear Decay, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 424 (1989), 187-209. doi: 10.1098/rspa.1989.0075. [21] R. E. Mickens, Nonstandard finite difference schemes for reaction-diffusion equations, Numerical Methods for Partial Differential Equations, 15 (1999), 201-214. doi: 10.1002/(SICI)1098-2426(199903)15:2<201::AID-NUM5>3.0.CO;2-H. [22] M. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209-223. doi: 10.1086/282272. [23] V. Thomée, On Positivity Preservation in Some Finite Element Methods for the Heat Equation. In: Dimov I. , Fidanova S. , Lirkov I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science, Springer, 8962 (2015), 13-24 [24] E. H. Twizell, Y. Wang, W. G. Price and F. Fakhr, Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system, Numerical Methods for Partial Differential Equations, 10 (1994), 435-454. doi: 10.1002/num.1690100404.
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)
 [1] Jinfeng Wang, Hongxia Fan. Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 909-918. doi: 10.3934/dcdsb.2016.21.909 [2] Wei Feng, Nicole Rocco, Michael Freeze, Xin Lu. Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1215-1230. doi: 10.3934/dcdss.2014.7.1215 [3] Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189 [4] Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026 [5] Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831 [6] H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2/3) : 221-251. doi: 10.3934/dcds.2007.18.221 [7] Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703 [8] Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-21. doi: 10.3934/dcdsb.2017209 [9] Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693 [10] Fei Xu, Ross Cressman, Vlastimil Křivan. Evolution of mobility in predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3397-3432. doi: 10.3934/dcdsb.2014.19.3397 [11] Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure & Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005 [12] Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455 [13] Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807 [14] Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265 [15] Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877 [16] Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857 [17] Verónica Anaya, Mostafa Bendahmane, Mauricio Sepúlveda. Mathematical and numerical analysis for Predator-prey system in a polluted environment. Networks & Heterogeneous Media, 2010, 5 (4) : 813-847. doi: 10.3934/nhm.2010.5.813 [18] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [19] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [20] Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

2016 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables