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Stationary patterns caused by cross-diffusion for a three-species prey-predator model. (English) Zbl 1121.92069

Summary: We study a system arising from a three-component prey-predator model with prey- dependent and ratio-dependent functional responses, where cross-diffusion is included in such a way that the predator chases the prey and the prey runs away from the predator. We prove that cross-diffusion can generate stationary patterns (nonconstant positive steady states).

MSC:

92D40 Ecology
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B32 Bifurcations in context of PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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[1] Hsu, S.B.; Hwang, T.W.; Kuang, Y., A ratio-dependent food chain model and its applications to biological control, Math. biosci., 181, 55-83, (2003) · Zbl 1036.92033
[2] Ebert, D.; Lipstich, M.; Mangin, K.L., The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites, Am. natural., 156, 459-477, (2000)
[3] Hwang, T.W.; Kuang, Y., Deterministic extinction effect of parasites on host populations, J. math. biol., 46, 17-30, (2003) · Zbl 1015.92042
[4] Wang, M.X., Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196, 172-192, (2004) · Zbl 1081.35025
[5] Pang, P.Y.H.; Wang, M.X., Strategy and stationary pattern in a three-species predator-prey model, J. differential equations, 200, 2, 245-273, (2004) · Zbl 1106.35016
[6] Ni, W.M., Diffusion, cross-diffusion and their spike-layer steady states, Notices amer. math. soc., 45, 1, 9-18, (1998) · Zbl 0917.35047
[7] Okubo, A., Diffusion and ecological problems: mathematical models, (1980), Springer-Verlag Berlin · Zbl 0422.92025
[8] Amann, H., Dynamic theory of quasilinear parabolic equations II: reaction-diffusion systems, Diff. int. eq., 3, 13-75, (1990) · Zbl 0729.35062
[9] Choi, Y.S.; Lui, R.; Yamada, Y., Existence of global solutions for the shigesada-Kawasaki-teramoto model with strongly coupled cross-diffusion, Discrete contin. dyn. syst., 10, 3, 719-730, (2004) · Zbl 1047.35054
[10] Turing, A., The chemical basis of morphogenesis, Philos. trans. royal soc. B, 237, 37-72, (1952) · Zbl 1403.92034
[11] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Kybernetik, 12, 30-39, (1972)
[12] Iron, D.; Ward, M.J.; Wei, J.C., The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150, 1-2, 25-62, (2001) · Zbl 0983.35020
[13] Wei, J.C.; Winter, M., Existence and stability of multiple-spot solutions for the gray-Scott model in \(\mathbb{R}^2\), Phys. D, 176, 3-4, 147-180, (2003) · Zbl 1014.37036
[14] Yanagida, E., Mini-maximizer for reaction-diffusion systems with skew-gradient structure, J. of differential equations, 179, 1, 311-335, (2002) · Zbl 0993.35047
[15] Davidson, F.A.; Rynne, B.P., A priori bounds and global existence of solutions of the steady-state sel’kov model, Proc. roy. soc. Edinburgh A, 130, 507-516, (2000) · Zbl 0960.35026
[16] Wang, M.X., Non-constant positive steady states of the sel’kov model, J. differential equations, 190, 2, 600-620, (2003) · Zbl 1163.35362
[17] Lin, C.S.; Ni, W.M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis systems, J. differential equations, 72, 1-27, (1988) · Zbl 0676.35030
[18] Wang, X.F.; Wu, Y.P., Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quarterly appl. math., LX, 3, 505-531, (2002) · Zbl 1039.35131
[19] Wu, Y.P., Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. of the royal soc. of Edinburgh, 132A, 1493-1511, (2002) · Zbl 1054.34089
[20] Lou, Y.; Martinez, S.; Ni, W.M., On 3×3 Lotka-Volterra competition systems with cross-diffusion, Discrete contin. dyn. syst., 6, 1, 175-190, (2000) · Zbl 1008.92035
[21] Lou, Y.; Ni, W.M., Diffusion, self-diffusion and cross-diffusion, J. differential equations, 131, 79-131, (1996) · Zbl 0867.35032
[22] Lou, Y.; Ni, W.M., Diffusion vs. cross-diffusion: an elliptic approach, J. differential equations, 154, 157-190, (1999) · Zbl 0934.35040
[23] X.F. Chen, W.M. Ni, Y.W. Qi and M.X. Wang, A strongly coupled predator-prey system with non-monotonic functional response, Preprint.
[24] Du, Y.H.; Lou, Y., Qualitative behavior of positive solutions of a predator-prey model: effects of saturation, Proc. roy. soc. Edinburgh, 131A, 321-349, (2001) · Zbl 0980.35028
[25] Ikeda, T.; Mimura, M., An interfacial approach to regional segregation of two competing species mediated by a predator, J. math. biol., 31, 215-240, (1993) · Zbl 0774.92023
[26] Kan-on, Y., Existence and instability of Neumann layer solutions for a 3-component Lotka-Volterra model with diffusion, J. math. anal. appl., 243, 357-372, (2000) · Zbl 0963.35012
[27] Pang, P.Y.H.; Wang, M.X., Non-constant positive steady states of a predator-prey system with nonmonotonic functional response and diffusion, Proc. London math. soc., 88, 1, 135-157, (2004) · Zbl 1134.35373
[28] Pang, P.Y.H.; Wang, M.X., Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. roy. soc. Edinburgh A, 133, 4, 919-942, (2003) · Zbl 1059.92056
[29] Wang, M.X., Stationary patterns of strongly coupled predator-prey models, J. math. anal. appl., 292, 2, 484-505, (2004) · Zbl 1160.35325
[30] Brown, K.J.; Davidson, F.A., Global bifurcation in the Brusselator system, Nonlinear analysis, TMA, 24, 1713-1725, (1995) · Zbl 0829.35010
[31] ()
[32] Cross, M.C.; Hohenberg, P.S., Pattern formation outside of equilibrium, Rev. modern phys., 65, 851-1112, (1993) · Zbl 1371.37001
[33] Ikeda, T.; Nishiura, Y., Pattern selection for two breathers, SIAM J. appl. math., 54, 1, 195-230, (1994) · Zbl 0791.35063
[34] Leach, J.A.; Wei, J.C., Pattern formation in a simple chemical system with general orders of autocatalysis and decay. I. stability analysis, Physica. D, 180, 3-4, 185-209, (2003) · Zbl 1027.80010
[35] Mimura, M.; Nishiura, Y., Pattern formation in coupled reaction-diffusion systems, Japan J. indust. appl. math., 12, 3, 385-424, (1995) · Zbl 0849.35053
[36] Castets, V.; Dulos, E.; Boissonade, J.; DeKepper, P., Experimental evidence of a sustained Turing-type equilibrium chemical pattern, Phys. rev. lett., 64, 2953-2956, (1990)
[37] Ouyang, Q.; Li, R.; Li, G.; Swinney, H.L., Dependence of Turing pattern wavelength on diffusion rate, Notices J. chem. phys., 102, 1, 2551-2555, (1995)
[38] Nirenberg, L., Topics in nonlinear functional analysis, (2001), American Mathematical Society Providence, RI · Zbl 0992.47023
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