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Pattern formation in a predator-mediated coexistence model with prey-taxis. (English) Zbl 07208008
Summary: Can foraging by predators or a repulsive prey defense mechanism upset predator-mediated coexistence? This paper investigates one scenario involving a prey-taxis by a prey species. We study a system of three populations involving two competing species with a common predator. All three populations are mobile via random dispersal within a bounded spatial domain \(\Omega\), but the predator’s movement is influenced by one prey’s gradient representing a repulsive effect on the predator. We prove existence of positive solutions, and investigate pattern formation through bifurcation analysis and numerical simulation.
Reviewer: Reviewer (Berlin)
MSC:
35K59 Quasilinear parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
Software:
FEniCS; SyFi
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[1] B. E. Ainseba; M. Bendahmane; A. Noussair, A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear Anal. Real World Appl., 9, 2086-2105 (2008) · Zbl 1156.35404
[2] N. D. Alikakos, \(L^p\) bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4, 827-868 (1979) · Zbl 0421.35009
[3] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson and B. Kehlet, et al., The FEniCS project, version 1.5, Archive Numerical Software, 3.
[4] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993, 9-126. · Zbl 0810.35037
[5] H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ: Reaction-diffusion systems, Differential Integral Equations, 3, 13-75 (1990) · Zbl 0729.35062
[6] B. Ayuso; L. D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47, 1391-1420 (2009) · Zbl 1205.65308
[7] M. Bendahmane, Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis, Netw. Heterog. Media, 3, 863-879 (2008) · Zbl 1160.35438
[8] M. Burger; J.-F. Pietschmann, Flow characteristics in a crowded transport model, Nonlinearity, 29, 3528-3550 (2016) · Zbl 1349.35152
[9] G. Caristi, K. P. Rybakowski and T. Wessolek, Persistence and spatial patterns in a onepredator-two-prey Lotka-Volterra model with diffusion, Ann. Mat. Pura Appl. (4), 161 (1992), 345-377. · Zbl 0758.92015
[10] J. M. Chase, P. A. Abrams, J. P. Grover, S. Diehl and P. Chesson, et al., The interaction between predation and competition: A review and synthesis, Ecology Lett., 5 (2002), 302-315.
[11] A. Chertock; A. Kurganov; X. Wang; Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5, 51-95 (2012) · Zbl 1398.92033
[12] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34, 1701-1745 (2014) · Zbl 1277.35002
[13] N. Cramer; R. May, Interspecific competition, predation and species diversity: A comment, J. Theoretical Biology, 34, 289-293 (1972)
[14] D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Mathematics & Applications, 69, Springer, Heidelberg, 2012. · Zbl 1231.65209
[15] W. Feng, Coexistence, stability, and limiting behavior in a one-predator-two-prey model, J. Math. Anal. Appl., 179, 592-609 (1993) · Zbl 0846.35067
[16] C. Gai; Q. Wang; J. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35, 1239-1284 (2015) · Zbl 1327.92050
[17] T. Hillen; K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58, 183-217 (2009) · Zbl 1161.92003
[18] D. Horstmann et al., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. · Zbl 1071.35001
[19] S. Hsu, Predator-mediated coexistence and extinction, Math. Biosci., 54, 231-248 (1981) · Zbl 0456.92020
[20] G. E. Hutchinson, The paradox of the plankton, Amer. Naturalist, 95, 137-145 (1961)
[21] H.-Y. Jin; Z.-A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260, 162-196 (2016) · Zbl 1323.35001
[22] P. Kareiva; G. Odell, Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search, Amer. Naturalist, 130, 233-270 (1987)
[23] P. Korman; A. W. Leung, A general monotone scheme for elliptic systems with applications to ecological models, Proc. Roy. Soc. Edinburgh Sect. A, 102, 315-325 (1986) · Zbl 0606.35034
[24] A. Kurganov; M. Lukáčová-Medvid’ová, Numerical study of two-species chemotaxis models, Discrete Contin. Dyn. Syst. Ser. B, 19, 131-152 (2014) · Zbl 1282.92011
[25] N. Lakos, Existence of steady-state solutions for a one-predator-two-prey system, SIAM J. Math. Anal., 21, 647-659 (1990) · Zbl 0705.92019
[26] J. Lee; T. Hillen; M. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn., 3, 551-573 (2009) · Zbl 1315.92064
[27] C. Li and Z. Hong, Global existence of classical solutions to a three-species predator-prey model with two prey-taxes, J. Appl. Math., 2012, 12pp. · Zbl 1241.34061
[28] Y. Li; K. Lin; C. Mu, Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system, Electronic J. Differential Equations, 2015, 1-13 (2015) · Zbl 1322.35058
[29] A. Logg, K.-A. Mardal and G. N. Wells, Automated Solution of Differential Equations by the Finite Element Method. The FEniCS Book, Lecture Notes in Computational Science and Engineering, 84, Springer, Heidelberg, 2012. · Zbl 1247.65105
[30] R. M. May, Stability in multispecies community models, Math. Biosci., 12, 59-79 (1971) · Zbl 0224.92006
[31] J. Murray, Mathematical Biology. I: An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. · Zbl 1006.92001
[32] R. T. Paine, Food web complexity and species diversity, Amer. Naturalist, 100, 65-75 (1966)
[33] J. Parrish; S. Saila, Interspecific competition, predation and species diversity, J. Theoretical Biology, 27, 207-220 (1970)
[34] J. I. Tello; D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26, 2129-2162 (2016) · Zbl 1349.92133
[35] J. I. Tello; D. Wrzosek, Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459, 1233-1250 (2018) · Zbl 1381.92087
[36] S. Vȧge, G. Bratbak, J. Egge, M. Heldal and A. Larsen, et al., Simple models combining competition, defence and resource availability have broad implications in pelagic microbial food webs, Ecology Lett., 21 (2018), 1440-1452.
[37] X. Wang; W. Wang; G. Zhang, Global bifurcation of solutions for a predator-prey model with prey-taxis, Math. Methods Appl. Sci., 38, 431-443 (2015) · Zbl 1307.92333
[38] X. Wang; X. Zou, Pattern formation of a predator-prey model with the cost of anti-predator behaviors, Math. Biosci. Eng., 15, 775-805 (2018) · Zbl 1406.92530
[39] Z. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 13pp. · Zbl 1163.37383
[40] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248, 2889-2905 (2010) · Zbl 1190.92004
[41] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal., 59, 1293-1310 (2004) · Zbl 1065.35072
[42] S. Wu; J. Shi; B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations, 260, 5847-5874 (2016) · Zbl 1335.35131
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