Numerical convergence of a Telegraph Predator-Prey system

Kariston Stevan Luiz; Juniormar Organista; Eliandro Rodrigues Cirilo; Neyva Maria Lopes Romeiro; Paulo Laerte Natti

doi:10.5433/1679-0375.2022v43n1Espp51

Numerical convergence of a Telegraph Predator-Prey system

Authors

Kariston Stevan Luiz Londrina State University - UEL https://orcid.org/0000-0002-4904-144X
Juniormar Organista University of São Paulo – USP - São Carlos https://orcid.org/0000-0002-3627-6901
Eliandro Rodrigues Cirilo Londrina State University - UEL https://orcid.org/0000-0001-7530-1770
Neyva Maria Lopes Romeiro Londrina State University - UEL https://orcid.org/0000-0002-3249-3490
Paulo Laerte Natti Londrina State University - UEL https://orcid.org/0000-0002-5988-2621

DOI:

https://doi.org/10.5433/1679-0375.2022v43n1Espp51

Keywords:

Reactive-Diffusive-Telegraph system, Maxwell-Cattaneo delay, discretization consistency, Von Neumann stability, numerical experimentation

Abstract

Numerical convergence of a Telegraph Predator-Prey system is studied. This partial differential equation (PDE) system can describe various biological systems with reactive, diffusive, and delay effects. Initially, the PDE system was discretized by the Finite Differences method. Then, a system of equations in a time-explicit form and in a space-implicit form was obtained. The consistency of the Telegraph Predator-Prey system discretization was verified. Von Neumann stability conditions were calculated for a Predator-Prey system with reactive terms and for a Delayed Telegraph system. On the other hand, for our Telegraph Predator-Prey system, it was not possible to obtain the von Neumann conditions analytically. In this context, numerical experiments were carried out and it was verified that the mesh refinement and the model parameters, reactive constants, diffusion coefficients and delay constants, determine the stability/instability conditions of the discretized equations. The results of numerical experiments were presented.

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Author Biographies

Kariston Stevan Luiz, Londrina State University - UEL

Master by the Program of Applied and Computational Mathematics-PGMAC/UEL

Juniormar Organista, University of São Paulo – USP - São Carlos

PhD student in Applied Mathematics from ICMC-USP. Master by the Program of Applied and Computational Mathematics-PGMAC/UEL

Eliandro Rodrigues Cirilo, Londrina State University - UEL

Prof. Dr., Mathematics Depto., UEL, Londrina, PR, Brazil

Neyva Maria Lopes Romeiro, Londrina State University - UEL

Prof. Dr., Mathematics Depto., UEL, Londrina, PR, Brazil.

Paulo Laerte Natti, Londrina State University - UEL

Prof. Dr., Department of Mathematics, UEL, Londrina, PR, Brazil

References

ABIDEMI, A.; ZAINIDDIN, Z. M.; AZIZ, N. A. B. Impact of control interventions on COVID-19 population dynamics in Malaysia: a mathematical study. The European Physical Journal Plus, Heidelberg, v. 136, n. 2, p. 1-35, 2021. DOI: https://doi.org/10.1140/epjp/s13360-021-01205-5. DOI: https://doi.org/10.1140/epjp/s13360-021-01205-5

ASH, T.; BENTO, A. M.; KAFFINE, D.; RAO, A.; BENTO, A. I. Disease-economy trade-offs under alternative epidemic control strategies. Nature Communications, London, v. 13, p. 3319, 2022. DOI: https://doi.org/10.1038/s41467-022-30642-8. DOI: https://doi.org/10.1038/s41467-022-30642-8

ATANGANA, A. On the stability and convergence of the time-fractional variable order telegraph equation. Journal of Computational Physics, San Diego, v. 293, p. 104-114, 2015. DOI: https://doi.org/10.1016/j.jcp.2014.12.043

AYALA, Y. S. S. Global existence and exponential stability for a coupled wave system. Journal of Mathematical Sciences: Advances and Applications, Allahabad, v. 16, p. 29-46, 2012.

BAZYKIN, A. D. Nonlinear dynamics of interacting populations. Amsterdam: World Scientific Publishing, 1998. DOI: https://doi.org/10.1142/2284

BEARUP, D.; PETROVSKAYA, N. B.; PETROVSKII, S. Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration. Mathematical Biosciences, London, v. 263, p. 143–160, 2015. DOI: https://doi.org/10.1016/j.mbs.2015.02.008. DOI: https://doi.org/10.1016/j.mbs.2015.02.008

BOCIU, L.; LASIECKA, I. Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping. Journal of Differential Equations, Amsterdam, v. 249, p. 654-683, 2010. DOI: https://doi.org/10.1016/j.jde.2010.03.009. DOI: https://doi.org/10.1016/j.jde.2010.03.009

BURDEN, R. L.; and FAIRES, J. D. Análise numérica. São Paulo: Cengage Learning, 2008.

CATTANEO, C.R. Sur une forme de l’équation de la chaleur éliminant le paradoxe de l’une propagation instantanée. Comptes Rendus, Grenoble, v. 247, n. 4, p. 431–433, 1958.

CAVALCANTI, M. M.; CAVALCANTI, V. N. D.; FERREIRA, J. Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Mathematical Methods in the Applied Sciences, Stuttgart, v. 24, p. 1043-1053, 2001. DOI: https://doi.org/10.1002/mma.250

CAVALCANTI, M. M.; CAVALCANTI, V. N. D.; SORIANO, J. A. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electronic Journal of Differential Equations, San Marcos, v. 44, p. 1-14, 2002. DOI: https://doi.org/10.57262/die/1356123377

CIRILO, E. R.; BARBA, A. N. D.; NATTI, P. L.; ROMEIRO, N. M. L. A numerical model based on the curvilinear coordinate system for the MAC method simplified. Semina. Ciências Exatas e Tecnológicas, Londrina, v. 39, n. 2, p. 87-98, 2018. DOI: https://doi.org/10.5433/1679-0375.2018v39n2p87. DOI: https://doi.org/10.5433/1679-0375.2018v39n2p87

CIRILO, E. R.; PETROVSKII, S. V.; ROMEIRO, N. M. L.; NATTI, P. L. Investigation into the critical domain problem for the reaction-telegraph equation using advanced numerical algorithms.

International Journal of Applied and Computational Mathematics, New Delhi, v. 5, p. 1-15, 2019.

CIRILO, E. R.; NATTI, P. L.; ROMEIRO, N. M. L.; CANDEZANO, M. A. C.; POLO, J. M. P. One study of COVID-19 spreading at the United States - Brazil - Colombia. Trends in Computational and Applied Mathematics, São Carlos, v. 22, p. 435-452, 2021a. DOI: https://doi.org/10.5540/tcam.2021.022.03.00435

CIRILO, E. R.; NATTI, P. L.; GODOI, P. H. V.; LERMA, A. A.; MATIAS, V. P.; ROMEIRO, N. M. L. COVID-19 in Londrina-PR: SEIR Model with Parameter Optimization. SEMINA: Ciências Exatas e Tecnológicas, Londrina, v. 42, n. 1Supl, p. 45-54, 2021b. DOI: https://doi.org/10.5433/1679-0375.2021v42n1Suplp45. DOI: https://doi.org/10.5433/1679-0375.2021v42n1Suplp45

CRANK, J.; NICOLSON, P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proceedings of the Cambridge Philosophical Society, Cambridge, v. 43, p. 50-67, 1947. DOI: https://doi.org/10.1017/S0305004100023197

CUMINATO, J. A.; MENEGUETTE, M. Discretização de equações diferenciais parciais: técnicas de diferenças finitas. Rio de Janeiro: SBMAC, 1999.

DE ROOS, A. M.; MC CAULEY, E.; WILSON, W. G. Mobility versus densirty-limited predador-prey dynamics on diferent spacial scales. Proceedings of the Royal Society of London B: Biological Sciences, London, v. 246, n. 1316, p. 117–122, 1991. DOI: https://doi.org/10.1098/rspb.1991.0132. DOI: https://doi.org/10.1098/rspb.1991.0132

EL-AZAB, M.S.; EL-GAMEL, M. A numerical algorithm for the solution of telegraph equations. Applied Mathematics and Computation, New York, v. 190, p. 757-764, 2007. DOI: https://doi.org/10.1016/j.amc.2007.01.091. DOI: https://doi.org/10.1016/j.amc.2007.01.091

FERREIRA, V. G.; CUMINATO, J. A.; TOMÉ, M. F.; FORTUNA, A. O.; MANGIAVACCHI, N.; CASTELO, A.; NONATO, L. G. Análise e implementação de modelos de turbulência K-epsilon para simulação de escoamentos imcompreessíveis envolvendo superfícies livres e rígidas. Trends in Computational and Applied Mathematics, São Carlos, v. 2, p. 81-90, 2001. DOI: https://doi.org/10.5540/tema.2001.02.01.0081. DOI: https://doi.org/10.5540/tema.2001.02.01.0081

FORTUNA, A. O. Técnicas computacionais para dinâmica dos fluidos. São Paulo: EDUSP, 2012.

GREENHALGH, D.; KHAN, Q. J. A.; PETTI, J. S. An eco-epidemiological predator–prey model where predators distinguish between susceptible and infected prey. Mathematical Methods in the Applied Sciences, London, v. 40, n. 1, p. 146–166, 2017. DOI: https://doi.org/10.1002/mma.3974. DOI: https://doi.org/10.1002/mma.3974

GRIEBEL, M.; DORNSEIFER, T.; NEUNHOEFFER, T. Numerical simulation in fluid dynamics: a practical introduction. Philadelphia: Society for Industrial and Applied Mathematics, 1998. DOI: https://doi.org/10.1137/1.9780898719703

HIRSCH, C. Numerical computation of internal and external flows. Oxford: John Wiley & Sons, 2007.

HOLLING, C. S. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. The Canadian Entomologist, Cambridge, v. 91, n. 5, p. 293–320, 1959a. DOI: https://doi.org/10.4039/ent91293-5. DOI: https://doi.org/10.4039/Ent91293-5

HOLLING, C. S. Some characteristics of simple types of predation and parasitism. The Canadian Entomologist, Cambridge, v. 91, n. 7, p. 385–398, 1959b. DOI: https://doi.org/10.4039/Ent91385-7. DOI: https://doi.org/10.4039/Ent91385-7

HUFFAKER, C. B. Experimental studies on predation: Dispersion factors and predator-prey oscillations. Hilgardia, Berkeley, v. 27, n. 14, p. 343–383, 1958. DOI: https://doi.org/10.3733/hilg.v27n14p343. DOI: https://doi.org/10.3733/hilg.v27n14p343

ISAACSON, E.; KELLER, H. B. Analysis of numerical methods. New York, Dover Publications, 1994.

KISHORE, N.; KAHN, R.; MARTINEZ, P. P.; DE SALAZAR, P. M.; MAHMUD, A. S.; BUCKEE, C. O. Lockdowns result in changes in human mobility which may impact the epidemiologic dynamics of SARS-CoV-2. Scientific Reports, London, v. 11, p. 6995, 2021. DOI: https://doi.org/10.1038/s41598-021-86297-w. DOI: https://doi.org/10.1038/s41598-021-86297-w

LAX, P. D.; RICHTMYER, R. D. Survey of the stability of linear finite difference equations. Communications on Pure and Applied Mathematics, New York, v. 9, p. 267-293, 1956. DOI: https://doi.org/10.1002/cpa.3160090206. DOI: https://doi.org/10.1002/cpa.3160090206

LEWIS, M.A.; PETROVSKII, S. V.; POTTS, J. R. The mathematics behind biological invasions, Switzerland: Springer International Publishing, 2016. (Interdisciplinary Applied Mathematics, v. 44). DOI: https://doi.org/10.1007/978-3-319-32043-4

LI, H.; LI, Y.; YANG, W. Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion. Nonlinear Analysis: Real World Applications, Oxford, v. 27, p. 261–282, 2016. DOI: https://doi.org/10.1016/j.nonrwa.2015.07.010. DOI: https://doi.org/10.1016/j.nonrwa.2015.07.010

LI, S.; YUAN, S.; WANG, H. Disease transmission dynamics of an epidemiological predator-prey system in open advective environments. Discrete and Continuous Dynamical Systems - B, Springfield, v. 28, p. 1480-1502, 2023. DOI: https://doi.org/10.3934/dcdsb.2022131. DOI: https://doi.org/10.3934/dcdsb.2022131

LOTKA, A. J. Elements of physical biology. Baltimore: World Scientific, 1925.

MENDEZ, V.; FEDOTOV, S.; HORSTHEMKE, W. Reaction-transport systems: mesoscopic foundations, fronts and spatial instabilities. Berlin: Springer-Verlag, 2010. DOI: https://doi.org/10.1007/978-3-642-11443-4

MICKENS, R. E.; JORDAN, P. M. A Positivity-preserving nonstandard finite difference scheme for the damped wave equation. Numerical Methods for Partial Differential Equations, Grenoble, v. 20, p. 639-649, 2003. DOI: https://doi.org/10.1002/num.20003

MUSTAFA, M. I. Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations. Nonlinear Analysis Real World Application, London, v. 13, p. 452-463, 2012. DOI: https://doi.org/10.1016/j.nonrwa.2011.08.002. DOI: https://doi.org/10.1016/j.nonrwa.2011.08.002

NATTI, P. L.; ROMEIRO, N. M. L.; CIRILO, E. R.; NATTI, E. R. T.; OLIVEIRA, C. F.; SOBRINHO, A. S. O.; KITA, C. M. Modelagem matemática e estabilidade de sistemas predador-presa. In: GOMES, I. A. (org.). A produção do conhecimento nas ciências exatas e da terra 2. Ponta Grossa: Atena Editora, 2019. v. 2, p. 162-177. DOI: https://doi.org/10.22533/at.ed.39519040416. DOI: https://doi.org/10.22533/at.ed.39519040416

OLIVEIRA, C. F; NATTI, P. L.; CIRILO, E. R.; ROMEIRO, N. M. L.; NATTI, E. R. T. Numerical stability of solitons waves through splices in quadratic optical media. Acta Scientiarum. Technology, Maringá, v. 42, p. e46881, 2019. DOI: https://doi.org/10.4025/actascitechnol.v42i1.46881. DOI: https://doi.org/10.4025/actascitechnol.v42i1.46881

PAUL, A.; LAURILA, T.; VUORINEN, V.; DIVINSKI, S. V. Thermodynamics, diffusion and the kirkendall effect in solids. London: Springer, 2014. DOI: https://doi.org/10.1007/978-3-319-07461-0

PAUL, A.; REJA, S.; KUNDU, S.; BHATTACHARYA, S. COVID-19 pandemic models revisited with a new proposal: Plenty of epidemiological models outcast the simple population dynamics solution. Chaos, Solitons and Fractals, Oxford, v. 144, p. 110697, 2021. DOI: https://doi.org/10.1016/j.chaos.2021.110697. DOI: https://doi.org/10.1016/j.chaos.2021.110697

ROMEIRO, N. M. L.; BELINELLI, E. O.; Maganin, J.; NATTI, P.L.; CIRILO, E. R. Numerical study of different methods applied to the one-dimensional transient heat equation. REMAT: Revista Eletrônica da Matemática, Caxias do Sul, v. 7, p. e3012, 2021.DOI: https://doi.org/10.35819/remat2021v7i1id4767. DOI: https://doi.org/10.35819/remat2021v7i1id4767

SAITA, T. M.; NATTI, P. L.; CIRILO, E. R.; ROMEIRO, N. M. L.; CANDEZANO, M. A. C.; ACUNA, R. A. B.; MORENO, L. C. G. Proposals for Sewage Management at Luruaco Lake, Colombia. Environmental Engineering Science, Larchmont, v. 38, n. 12, p. 1140-1148, 2021. DOI: https://doi.org/10.1089/ees.2020.0401. DOI: https://doi.org/10.1089/ees.2020.0401

SANGAY, J. C. A. Aplicação do método de complementaridade mista para problemas parabólicos não lineares. 2015. Dissertação (Mestrado) - Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, 2015.

SHAOYONG, L. The asymptotic theory of semilinear perturbed telegraph equation and its application. Applied Mathematics and Mechanics, Shanghai, v. 18, p. 657-662, 1997. DOI: https://doi.org/10.1007/BF00127013. DOI: https://doi.org/10.1007/BF00127013

STRIKWERDA, J. C. Finite difference schemes and partial differential equations. Philadelphia: SIAM, 2004. DOI: https://doi.org/10.1137/1.9780898717938

TANSKY, M. Switching effect in prey-predator system. Journal of Theoretical Biology, Amsterdam, v. 70, n. 3, p. 263-271, 1978. DOI: https://doi.org/10.1016/0022-5193(78)90376-4. DOI: https://doi.org/10.1016/0022-5193(78)90376-4

TILLES, P. F. C.; PETROVSKII, S. V.; NATTI, P. L. A random walk description of individual animal movement accounting for periods of rest. Royal Society Open Science, London, v. 3, p. 160566, 2016. DOI: https://doi.org/10.1098/rsos.160566. DOI: https://doi.org/10.1098/rsos.160566

TILLES, P. F. C.; PETROVSKII, S. V.; NATTI, P. L. A random acceleration model of individual animal movement allowing for diffusive, superdiffusive and superballistic regimes. Scientific Reports, London, v. 7, p. 14364, 2017. DOI: https://doi.org/10.1038/s41598-017-14511-9. DOI: https://doi.org/10.1038/s41598-017-14511-9

VOLTERRA, V. Fluctuations in the abundance of a species considered mathematically. Nature, London, v. 118, p. 558–560, 1926. DOI: https://doi.org/10.1038/118558a0. DOI: https://doi.org/10.1038/118558a0