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
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.  Google Scholar

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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.  Google Scholar

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S. BlanesF. CasasP. 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.  Google Scholar

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S. BlanesF. DieleC. 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.  Google Scholar

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F. DieleI. 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.  Google Scholar

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F. DieleC. 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.  Google Scholar

[7]

F. DieleC. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[13]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations Springer-Verlag, Berlin, 2006.  Google Scholar

[14]

E. HansenF. 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.  Google Scholar

[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.  Google Scholar

[16]

D. LacitignolaF. DieleC. 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.  Google Scholar

[17]

J. Martinez-Linares, Phase space formulation of population dynamics in ecology, preprint, arXiv: q-bio.PE/1304.2324v. Google Scholar

[18]

A. B. MedvinskyS. V. PetrovskiiI. A. TikhonovaH. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.  Google Scholar

[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.   Google Scholar

[20]

J. H. MerkinD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 doi: 10.1007/978-3-319-15585-2_2.  Google Scholar

[24]

E. H. TwizellY. WangW. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

S. BlanesF. CasasP. 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.  Google Scholar

[4]

S. BlanesF. DieleC. 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.  Google Scholar

[5]

F. DieleI. 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.  Google Scholar

[6]

F. DieleC. 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.  Google Scholar

[7]

F. DieleC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations Springer-Verlag, Berlin, 2006.  Google Scholar

[14]

E. HansenF. 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.  Google Scholar

[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.  Google Scholar

[16]

D. LacitignolaF. DieleC. 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.  Google Scholar

[17]

J. Martinez-Linares, Phase space formulation of population dynamics in ecology, preprint, arXiv: q-bio.PE/1304.2324v. Google Scholar

[18]

A. B. MedvinskyS. V. PetrovskiiI. A. TikhonovaH. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Review, 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.  Google Scholar

[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.   Google Scholar

[20]

J. H. MerkinD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 doi: 10.1007/978-3-319-15585-2_2.  Google Scholar

[24]

E. H. TwizellY. WangW. 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.  Google Scholar

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: $u_0 = v_0 = 0.2$
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)
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
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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