Analytical detection of stationary turing pattern in a predator-prey system with generalist predator

Subrata Dey¹ ; Malay Banerjee¹ ; Saktipada Ghorai¹

¹ Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, Uttar Pradesh, India

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 33.

Résumé

A prey-predator model with Holling type-II functional response and a generalist predator exhibits complex dynamics in response to parameter variation. Generalist predators implicitly exploiting multiple food resources reduce predation pressure on their focal prey species that causes it to become more stable compared to a prey-predator system with specialist predator. In the temporal system, bistability and tristability are observed along with various global and local bifurcations. Existence of homogeneous and heterogeneous positive steady state solutions are shown to exist for suitable ranges of parameter values in the corresponding spatio-temporal diffusive system. Weakly nonlinear analysis, using multi-scale perturbation technique, is employed to derive amplitude equation for the stationary patterns near the Turing bifurcation threshold. The analytical results of the amplitude equations are validated using exhaustive numerical simulations. We also identify bifurcation of multiple stable stationary patch solutions as well as dynamic pattern solution for parameter values in the Turing and Turing-Hopf regions.

Détail

DOI : 10.1051/mmnp/2022032

Bibliographie

[1] M. Banerjee, N. Mukherjee, S. Ghorai Detection of turing patterns in a three species food chain model via amplitude equation Commun. Nonlinear Sci. Numer. Simul. 2019 219 236

[2] M. Banerjee, S. Petrovskii Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system Theor. Ecol. 2011 37 53

[3] M. Banerjee, V. Volpert Spatio-temporal pattern formation in Rosenzweig—Macarthur model: effect of nonlocal interactions Ecol. Complex. 2017 2 10

[4] M. Baurmann, T. Gross, U. Feudel Instabilities in spatially extended predator—prey systems: spatio-temporal patterns in the neighborhood of Turing—Hopf bifurcations J. Theor. Biol. 2007 220 229

[5] M. Baurmann, T. Gross, U. Feudel Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations J. Theor. Biol. 2007 220 229

[6] I. Boudjema, S. Djilali Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application Nonlinear Stud. 2018 665 687

[7] B. Bozzini, G. Gambino, D. Lacitignola, S. Lupo, M. Sammartino, I. Sgura Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth Comput. Math. Appl. 2015 1948 1969

[8] B.I. Camara Waves analysis and spatiotemporal pattern formation of an ecosystem model Nonlinear Anal.: Real World Appl. 2011 2511 2528

[9] M.C. Cross, P.C. Hohenberg Pattern formation outside of equilibrium Rev. Modern Phys. 1993 851

[10] S. Djilali, S. Bentout Spatiotemporal patterns in a diffusive predator-prey model with prey social behavior Acta Appl. Math. 2020 125 143

[11] A. Ducrots, M. Langlais A singular reaction—diffusion system modelling prey–predator interactions: Invasion and coextinction waves J. Differ. Equ. 2012 502 532

[12] W.F. Paquin-Lefebvre, Nagata, M.J. Ward Pattern formation and oscillatory dynamics in a two-dimensional coupled bulk-surface reaction-diffusion system SIAM J. Appl. Dyn. Syst. 2019 1334 1390

[13] G.F. Gause , The Struggle for Existence. Williams and Wilkins, Baltimore, Maryland (1934).

[14] M. Giaquinta, G. Modica Local existence for quasilinear parabolic systems under nonlinear boundary conditions Ann. Matemat. Pura Appl. 1987 41 59

[15] A. Gierer, H. Meinhardt A theory of biological pattern formation Kybernetik 1972 30 39

[16] R. Han, B. Dai Cross-diffusion-driven Turing instability and weakly nonlinear analysis of Turing patterns in a unidirectional consumer-resource system Boundary Value Probl. 2017 1 33

[17] R. Han, B. Dai Cross-diffusion induced Turing instability and amplitude equation for a toxic-phytoplankton—zooplankton model with nonmonotonic functional response Int. J. Bifurc. Chaos 2017 1750088

[18] C. Huffaker Experimental studies on predation: dispersion factors and predator-prey oscillations Hilgardia 1958 343 383

[19] H.B. Keller , Numerical solution of bifurcation and nonlinear eigenvalue problems. In “Applications of Bifurcation Theory” (ed. Rabinowitz ), Academic Press, 359–384 (1977)

[20] C.A. Klausmeier Regular and irregular patterns in semiarid vegetation Science 1999 1826 1828

[21] S. Kondo, R. Asai A reaction—diffusion wave on the skin of the marine angelfish Pomacanthus Nature 1995 765 768

[22] V. Lakshmikantham , S. Leela and A.A. Martynyuk , Practical stability of nonlinear systems. World Scientific (1990).

[23] E.V. Leeuwen, V.A.A. Jansen, P.W. Bright How population dynamics shape the functional response in a one-predator– two-prey system Ecol. Soc. Am. 2007 1571 1581

[24] S.A. Levin, L.A. Segel Hypothesis for origin of planktonic patchiness Nature 1976 659 659

[25] Z. Ling, L. Zhang, Z. Lin Turing pattern formation in a predator—prey system with cross diffusion Appl. Math. Model. 2014 5022 5032

[26] M. Lio X. tang and C. Xu, Stability and instability analysis for a ratio-dependent predator–prey system with diffusion effect Nonlinear Anal.: Real World Appl. 2011 1616 1626

[27] A.J. Lotka Undamped oscillations derived from the law of mass action J. Am. Chem. Soc. 1920 1595 1599

[28] H. Malchow , Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulation. Chapman and Hall/CRC (2007).

[29] M. Manjun, G. Meiyan, R. Carretero-González Pattern formation for a two-dimensional reaction-diffusion model with chemotaxis. J Math. Anal. Appl. 2019 1883 1909

[30] A. Marasco, A. Luorio, F. Cartení, G. Bonanomi, D.M. Tartakovsky, S. Mazzoleni, F. Giannino Vegetation pattern formation due to interactions between water availability and toxicity in plant—soil feedback Bull. Math. Biol. 2014 2866 2883

[31] A. Medvinsky, S. Petrovskii Spatiotemporal complexity of plankton and fish dynamics SIAM Rev. 2002 311 370

[32] H.D. Mittelmann A pseudo-arclength continuation method for nonlinear eigenvalue problems SIAM J. Numer. Anal. 1986 1007 1016

[33] J.D. Murray , vol. 3 of Mathematical biology II: spatial models and biomedical applications. Springer, New York (2001).

[34] D. Nishiura, Y. Ueyama A skeleton structure of self-replicating dynamics Physica D 1999 73 104

[35] F.L. Ochoa, J.D. Murray A non-linear analysis for spatial structure in a reaction-diffusion model Bull. Math. Biol. 1983 917 930

[36] S. Pal, S. Ghorai, M. Banerjee Analysis of a prey–predator model with non-local interaction in the prey population Bull. Math. Biol. 2018 906 925

[37] C.V. Pao , Nonlinear parabolic and elliptic equations. Springer Science Business Media (2012).

[38] L. Perko , Differential Equations and Dynamical Systems. Springer-Verlag, New York (2000).

[39] S.V. Petrovskii, H. Malchow A minimal model of pattern formation in a prey-predator system Math. Comput. Modell. 1999 49 63

[40] L.A. Segel, J.L. Jackson Dissipative structure: an explanation and an ecological example J. Theor. Biol. 1972 545 559

[41] J. Sherratt, S. Matthew Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models J. R. Soc. Interface 2008 483 505

[42] J.A. Sherratt, B.T. Eagan, M.A. Lewis Oscillations and chaos behind predator—prey invasion: mathematical artifact or ecological reality? Philos. Trans. Royal Soc. London. Ser. B 1997 21 38

[43] J. Smoller , vol. 258 of Shock waves and reaction-diffusion equations. Springer Science Business Media (2012).

[44] P.D. Spencer, J.S. Collie A simple predator–prey model of exploited marine fish populations incorporating alternative prey ICES J. Marine Sci. 1995 615 628

[45] A.M. Turing The chemical basis of morphogenesis Phil. Trans. Royal Soc. 1952 37 72

[46] E. Venturino, S. Petrovskii Spatiotemporal behavior of a prey—predator system with a group defense for prey Ecol. Complex. 2013 37 47

[47] V. Volterra , Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. C. Ferrari (1926).

[48] W. Walter Differential inequalities and maximum principles: theory, new methods and applications Nonlinear Anal.: Theory, Methods Appl. 1997 4695 4711

[49] W. Wang, Q.X. Liu, Z. Jin Spatiotemporal complexity of a ratio-dependent predator-prey system Phys. Rev. E 2007 051913

[50] L. Zhang, J. Liu, M. Banerjee Hopf and steady state bifurcation analysis in a ratio-dependent predator–prey model Commun. Nonlinear Sci. Numer. Simulat 2017 52 73