Numerical solution of a spatio-temporal predator-prey model with infected prey

Raimund Bürger; Gerardo Chowell; Elvis Gavilán; Pep Mulet; Luis M. Villada; Raimund Bürger; Gerardo Chowell; Elvis Gavilán; Pep Mulet; Luis M. Villada

doi:10.3934/mbe.2019021

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 1: 438-473. doi: 10.3934/mbe.2019021

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Research article

Numerical solution of a spatio-temporal predator-prey model with infected prey

1.
CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2.
School of Public Health, Georgia State University, Atlanta, Georgia, USA
3.
Simon A. Levin Mathematical and Computational Modeling Sciences Center, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, USA
4.
Division of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, USA
5.
Departament de Matem`atiques, Universitat de Val`encia, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
6.
GIMNAP-Departamento de Matemáticas, Universidad del B´ıo-B´ıo,Casilla 5-C, Concepción, Chile

Received: 10 April 2018 Accepted: 07 September 2018 Published: 17 December 2018

A spatio-temporal eco-epidemiological model is formulated by combining an available non-spatial model for predator-prey dynamics with infected prey [D. Greenhalgh and M. Haque, Math. Meth. Appl. Sci., 30 (2007), 911–929] with a spatio-temporal susceptible-infective (SI)-type epidemic model of pattern formation due to diffusion [G.-Q. Sun, Nonlinear Dynamics, 69 (2012), 1097–1104]. It is assumed that predators exclusively eat infected prey, in agreement with the hypothesis that the infection weakens the prey, making it available for predation otherwise we assume that the predator has essentially no access to healthy prey of the same species. Furthermore, the movement of predators is described by a non-local convolution of the density of infected prey as proposed in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369–400]. The resulting convection-diffusion-reaction system of three partial differential equations for the densities of susceptible and infected prey and predators is solved by an efficient method that combines weighted essentially non-oscillatory (WENO) reconstructions and an implicit-explicit Runge-Kutta (IMEX-RK) method for time stepping. Numerical examples illustrate the formation of spatial patterns involving all three species.
- convection-di usion-reaction system,
- implicit-explicit Runge-Kutta scheme,
- non-local velocity,
- predator-prey model
Citation: Raimund Bürger, Gerardo Chowell, Elvis Gavilán, Pep Mulet, Luis M. Villada. Numerical solution of a spatio-temporal predator-prey model with infected prey[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 438-473. doi: 10.3934/mbe.2019021

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Abstract

A spatio-temporal eco-epidemiological model is formulated by combining an available non-spatial model for predator-prey dynamics with infected prey [D. Greenhalgh and M. Haque, Math. Meth. Appl. Sci., 30 (2007), 911–929] with a spatio-temporal susceptible-infective (SI)-type epidemic model of pattern formation due to diffusion [G.-Q. Sun, Nonlinear Dynamics, 69 (2012), 1097–1104]. It is assumed that predators exclusively eat infected prey, in agreement with the hypothesis that the infection weakens the prey, making it available for predation otherwise we assume that the predator has essentially no access to healthy prey of the same species. Furthermore, the movement of predators is described by a non-local convolution of the density of infected prey as proposed in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369–400]. The resulting convection-diffusion-reaction system of three partial differential equations for the densities of susceptible and infected prey and predators is solved by an efficient method that combines weighted essentially non-oscillatory (WENO) reconstructions and an implicit-explicit Runge-Kutta (IMEX-RK) method for time stepping. Numerical examples illustrate the formation of spatial patterns involving all three species.

References

[1]	L.J.S. Allen, B.M. Bolker, Y. Lou, and A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM J. Appl. Math., 67 (2007), 1283–1309.
[2]	J. Arino, Diseases in metapopulations. In Z. Ma, Y. Zhou, and J. Wu (Eds.), Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 2009, 64–122.
[3]	J. Arino, J.R. Davis, D. Hartley, R. Jordan, J.M. Miller, and P. van den Driessche, A multi-species epidemic model with spatial dynamics, Math. Med. Biol., 22 (2005), 129–142.
[4]	U. Ascher, S. Ruuth, and J. Spiteri, Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151–167.
[5]	S. Boscarino, R. Bürger, P. Mulet, G. Russo, and L.M. Villada, Linearly implicit IMEX Runge- Kutta methods for a class of degenerate convection-diffusion problems, SIAM J. Sci. Comput., 37 (2015), B305–B331.
[6]	S. Boscarino, F. Filbet, and G. Russo, High order semi-implicit schemes for time dependent partial differential equations, J. Sci. Comput., 68 (2016), 975–1001.
[7]	S. Boscarino, P.G. LeFloch, and G. Russo, High-order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput., 36 (2014), A377–A395.
[8]	S. Boscarino and G. Russo, On a class of uniformly accurate IMEX Runge-Kutta schemes and applications to hyperbolic systems with relaxation, SIAM J. Sci. Comput., 31 (2009), 1926–1945.
[9]	S. Boscarino and G. Russo, Flux-explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems, SIAM J. Numer. Anal., 51 (2013), 163–190.
[10]	F. Brauer and C. Kribs, Dynamical Systems for Biological Modeling: An Introduction, CRC Press, Boca Raton, FL, USA, 2016.
[11]	R. Bürger, G. Chowell, E. Gavilán, P. Mulet, and L.M. Villada, Numerical solution of a spatiotemporal gender-structured model for hantavirus infection in rodents, Math. Biosci. Eng., 15 (2018), 95–123.
[12]	R. Bürger, G. Chowell, P. Mulet, and L.M. Villada, Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile, Math. Biosci. Eng., 13 (2016), 43–65.
[13]	J. Chattopadhyay and O. Arino, A predator-prey model with disease in the prey, Nonlin. Anal., 36 (1999), 747–766.
[14]	J. Chattopadhyay and N. Bairagi, Pelicans at risk in Salton sea-an eco-epidemiological model, Ecol. Model., 136 (2001), 103–112.
[15]	J. Chattopadhyay, P.D.N. Srinivasu, and N. Bairagi, Pelicans at risk in Salton sea-an ecoepidemiological model-II, Ecol. Model., 167 (2003), 199-211.
[16]	R.M. Colombo and E. Rossi, Hyperbolic predators versus parabolic preys, Commun. Math. Sci., 13 (2015), 369–400.
[17]	M. Crouzeix, Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques, Numer. Math., 35 (1980), 257–276.
[18]	O. Diekmann, H. Heesterbeek, and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2012.
[19]	R. Donat and I. Higueras, On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms, Math. Comp., 80 (2011), 2097–2126.
[20]	L. Edelstein-Keshet, Mathematical Models in Biology, reprint, SIAM, 2005.
[21]	I.M. Foppa, A Historical Introduction to Mathematical Modeling of Infectious Diseases, Academic Press, London, UK, 2016.
[22]	D. Greenhalgh and M. Haque, A predator-prey model with disease in the prey species only, Math. Meth. Appl. Sci., 30 (2007), 911–929.
[23]	D. Greenhalgh, Q.J.A. Khan, and J.S. Pettigrew, An eco-epidemiological predator-prey modelmwhere predators distinguish between susceptible and infected prey, Math. Meth. Appl. Sci., 40 (2017), 146–166.
[24]	K.P. Hadeler and H.I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27 (1989), 609–631.
[25]	H.W. Hethcote, W. Wang, L. Han, and Z. Ma, A predator-prey model with infected prey, Theor. Population Biol., 66 (2004), 259–268.
[26]	G.S. Jiang and C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.
[27]	Y. Katznelson, An Introduction to Harmonic Analysis, Third Ed., Cambridge University Press, Cambridge, UK, 2004.
[28]	C.A. Kennedy and M.H. Carpenter, Additive Runge-Kutta schemes for convection-diffusionreaction equations, Appl. Numer. Math., 44 (2003), 139–181.
[29]	Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389–406.
[30]	K. Kundu and J. Chattopadhyay, A ratio-dependent eco-epidemiological model of the Salton Sea, Math. Meth. Appl. Sci., 29 (2006), 191–207.
[31]	P.H. Leslie and J.C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219–234.
[32]	W.M. Liu, H.W. Hethcote, and S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359–380.
[33]	W.M. Liu, S.A. Levin, and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187–204.
[34]	X.D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200–212.
[35]	H. Malchow, S.V. Petrovskii, and E. Venturino, Spatial Patterns in Ecology and Epidemiology: Theory, Models, and Simulation. Chapman & Hall/CRC, Boca Raton, FL, USA, 2008.
[36]	J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications. Third Edition. Springer, New York, 2003.
[37]	A. Okubo and S.A. Levin, Diffusion and Ecological Problems: Modern Perspectives. Second Edition, Springer-Verlag, New York, 2001.
[38]	L. Pareschi and G. Russo, Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25 (2005), 129–155.
[39]	E. Rossi and V. Schleper, Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions, ESAIM Math. Modelling Numer. Anal., 50 (2016), 475–497.
[40]	R. Sarkar, J. Chattopadhyay, and N. Bairagi, Effects of environmental fluctuation on an ecoepidemiological model of the Salton Sea, Environmetrics, 12 (2001), 289–300.
[41]	L. Sattenspiel, The Geographic Spread of Infectious Diseases: Models and Applications, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2009.
[42]	C.W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 83 (1988), 32–78.
[43]	S.W. Smith, Digital Signal Processing: A Practical Guide for Engineers and Scientists. Demystifying technology series: by engineers, for engineers. Newnes, 2003.
[44]	S.H. Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering, Second Edition, CRC Press, Boca Raton, FL, 2015.
[45]	G. Q. Sun, Pattern formation of an epidemic model with diffusion, Nonlinear Dynamics, 69 (2012), 1097–1104.
[46]	P. van den Driessche, Deterministic compartmental models: extensions of basic models. In F. Brauer, P. van den Driessche, and J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 2008, 147–157.
[47]	P. van den Driessche, Spatial structure: patch models. In F. Brauer, P. van den Driessche and J.Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 2008, 179–189.
[48]	E. Vynnycky and R.E. White, An Introduction to Infectious Disease Modelling, Oxford University Press, 2010.
[49]	J. Wu, Spatial structure: partial differential equations models. In F. Brauer, P. van den Driessche and J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 2008, 191–203.
[50]	P. Yu, Closed-form conditions of bifurcation points for general differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1467–1483.
[51]	Y.T. Zhang and C.W. Shu, ENO and WENO schemes. Chapter 5 in R. Abgrall and C.W. Shu (eds.), Handbook of Numerical Methods for Hyperbolic Problems Basic and Fundamental Issues. Handbook of Numerical Analysis vol. 17, North Holland (2016), 103–122.
[52]	X. Zhong, Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows, J. Comput. Phys., 128 (1996), 19–31.

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