Spatiotemporal pattern formation in a prey–predator model with generalist predator

Kalyan Manna¹ ; Malay Banerjee¹

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. 6

Résumé

Generalist predators exploit multiple food sources and it is economical for them to reduce predation pressure on a particular prey species when their density level becomes comparatively less. As a result, a prey-predator system tends to become more stable in the presence of a generalist predator. In this article, we investigate the roles of both the diffusion and nonlocal prey consumption in shaping the population distributions for interacting generalist predator and its focal prey species. In this regard, we first derive the conditions associated with Turing instability through linear analysis. Then, we perform a weakly nonlinear analysis and derive a cubic Stuart-Landau equation governing amplitude of the resulting patterns near Turing bifurcation boundary. Further, we present a wide variety of numerical simulations to corroborate our analytical findings as well as to illustrate some other complex spatiotemporal dynamics. Interestingly, our study reveals the existence of traveling wave solutions connecting two spatially homogeneous coexistence steady states in Turing domain under the influence of temporal bistability phenomenon. Also, our investigation shows that nonlocal prey consumption acts as a stabilizing force for the system dynamics.

Détail

DOI : 10.1051/mmnp/2022007

Bibliographie

[1] J.L. Aragón, R.A. Barrio, T.E. Woolley, R.E. Baker, P.K. Maini Nonlinear effects on Turing patterns: time oscillations and chaos Phys. Rev. E 2012 026201

[2] C. Arancibia-Ibarra, M. Bode, J. Flores, G. Pettet, P. Van Heijster Turing patterns in a diffusive Holling-Tanner predator-prey model with an alternative food source for the predator Commun. Nonlinear Sci. Numer. Simul 2021 105802

[3] E.A. Autry, A. Bayliss, V.A. Volpert Biological control with nonlocal interactions Math. Biosci 2018 129 146

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

[5] M. Banerjee, V. Volpert Prey-predator model with a nonlocal consumption of prey Chaos 2016 083120

[6] M. Banerjee, V. Volpert Spatio-temporal pattern formation in Rosenzweig-MacArthur model: Effect of nonlocal interactions Ecol. Complex 2017 2 10

[7] 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

[8] E. Ben-Jacob, I. Cohen, H. Levine Cooperative self-organization of microorganisms Adv. Phys 2000 395 554

[9] R. Biggs, S.R. Carpenter, W.A. Brock Turning back from the brink: detecting an impeding regime shift in time to avert it Proc. Natl. Acad. Sci. USA 2009 826 831

[10] 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 69

[11] N.F. Britton Aggregation and the competitive exclusion principle J. Theor. Biol 1989 57 66

[12] R.S. Cantrell, C. Cosner Models for predator-prey systems at multiple scales SIAM Rev 1996 256 286

[13] S. Chakraborty The influence of generalist predators in spatially extended predator-prey systems Ecol. Complex 2015 50 60

[14] M.G. Clerc, D. Escaff, V.M. Kenkre Analytical studies of fronts, colonies, and patterns: combination of the Allee effect and nonlocal competition interactions Phys. Rev. E 2010 036210

[15] A. Enkegaard, H.F. Brødsgaard, D.L. Hansen Macrolophus caliginosus: Functional response to whiteflies and preference and switching capacity between whiteflies and spider mites Entomolog. Exper. Appl 2001 81 88

[16] A. Erbach, F. Lutscher, G. Seo Bistability and limit cycles in generalist predator-prey dynamics Ecol. Complex 2013 48 55

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

[18] R. Han, B. Dai, Y. Chen Pattern formation in a diffusive intraguild predation model with nonlocal interaction effects AIP Adv 2019 035046

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

[20] B.J. Kealy, D.J. Wollkind A nonlinear stability analysis of vegetative Turing pattern formation for an interaction-diffusion plant-surface water model system in an arid flat environment Bull. Math. Biol 2012 803 833

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

[22] M. Kot, Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001).

[23] N. Kumari Pattern formation in spatially extended tritrophic food chain model systems: Generalist versus specialist top predators ISRN Biomath 2013 198185

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

[25] X. Li, G. Hu, Z. Feng Pattern dynamics in a spatial predator-prey model with nonmonotonic response function Int. J. Bifurc. Chaos 2018 1850077

[26] L.S. Luckinbill Coexistence in laboratory populations of Paramecium aurelia and its predator Didinum nasutum Ecology 1973 1320 1327

[27] L.S. Luckinbill The effects of space and enrichment on a predator-prey system Ecology 1974 1142 1147

[28] M. Ma, M. Gao, R. Carretero-González Pattern formation for a two-dimensional reaction-diffusion model with chemotaxis J. Math. Anal. Appl 2019 1883 1909

[29] C. Magal, C. Cosner, S. Ruan Control of invasive hosts by generalist parasitoids Math. Med. Biol 2008 1 20

[30] K. Manna, M. Banerjee Stationary, non-stationary and invasive patterns for a prey-predator system with additive Allee effect in prey growth Ecol. Complex 2018 206 217

[31] K. Manna, S. Pal, M. Banerjee Analytical and numerical detection of traveling wave and wave-train solutions in a prey-predator model with weak Allee effect Nonlinear Dyn 2020 2989 3006

[32] K. Manna, V. Volpert, M. Banerjee Dynamics of a diffusive two-prey-one-predator model with nonlocal intra-specific competition for both the prey species Mathematics 2020 101

[33] K. Manna, V. Volpert, M. Banerjee Pattern formation in a three-species cyclic competition model Bull. Math. Biol 2021 52

[34] A.B. Medvinsky, S.V. Petrovskii, I.A. Tikhonova, H. Malchow, B.-L. Li Spatiotemporal complexity of plankton and fish dynamics SIAM Rev 2002 311 370

[35] S.M. Merchant, W. Nagata Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition Theor. Popul. Biol 2011 289 297

[36] A. Morozov, S. Petrovskii, B.-L. Li Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect J. Theor. Biol 2006 18 35

[37] W.W. Murdoch Switching in general predators: Experiments on predator specificity and stability of prey populations Ecol. Monogr 1969 335 354

[38] J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, Berlin, Germany (2003).

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

[40] S. Pal, M. Banerjee, S. Ghorai Effects of boundary conditions on pattern formation in a nonlocal prey-predator model Appl. Math. Model 2020 809 823

[41] S. Pal, S. Ghorai, M. Banerjee Effect of kernels on spatio-temporal patterns of a non-local prey-predator model Math. Biosci 2019 96 107

[42] S.V. Petrovskii, A.Y. Morozov, E. Venturino Allee effect makes possible patchy invasion in a predator-prey system Ecol. Lett 2002 345 352

[43] F. Rao, Y. Kang The complex dynamics of a diffusive prey-predator model with an Allee effect in prey Ecol. Complex 2016 123 44

[44] M. Rietkerk, S.C. Dekker, P.C. De Ruiter, J. Van De Koppel Self-organized patchiness and catastrophic shifts in ecosystems Science 2004 1926 1929

[45] V.W. Rodrigues, D.C. Mistro, L.A.D. Rodrigues Pattern formation and bistability in a generalist predator-prey model Mathematics 2020 20

[46] M. Scheffer, S. Carpenter, J.A. Foley, B. Walker Catastrophic shifts in ecosystems Nature 2001 591 596

[47] B.L. Segal, V.A. Volpert, A. Bayliss Pattern formation in a model of competing populations with nonlocal interactions Physica D 2013 12 22

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

[49] N. Shigesada and K. Kawasaki, Biological invasions: theory and practice. Oxford University Press, Oxford (1997).

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

[51] Y. Tanaka, H. Yamaoto, H. Ninomiya Mathematical approach to nonlocal interactions using a reaction-diffusion system Dev. Growth Differ 2017 388 95

[52] A. Turing The chemical basis of morphogenesis Philos. Trans. R. Soc. Lond. B 1952 37 72

[53] V.K. Vanag, I.R. Epstein Out-of-phase oscillatory Turing patterns in a bistable reaction-diffusion system Phys. Rev. E 2005 066212

[54] M. Van Baalen, V. Křivan, P.C.J. Van Rijn, M.W. Sabelis Alternative food, switching predators, and the persistence of predator-prey systems Am. Natural 2001 512 524

[55] E. Van Leeuwen, V.A.A. Jansen, P.W. Bright How population dynamics shape the functional response in a one-predator-two-prey system Ecology 2007 1571 81

[56] J.C. Van Lenteren, L. Hemerik, J.C. Lins, V.H.P. Bueno Functional responses of three neotropical mirid predators to eggs of Tuta absoluta on tomato Insects 2016 34