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Stationary patterns of strongly coupled prey–predator models. (English) Zbl 1160.35325

The author considers the strongly coupled system \[ -\text{div} (K(u)\nabla u)= G(u) \quad\text{in }\Omega, \qquad \frac {\partial u}{\partial n}= 0 \quad\text{on }\partial\Omega. \tag{1} \] The author establishes a priori upper and lower bounds for positive solutions of (1), and studies the non-existence of nonconstant positive solutions. Moreover the author considers the bifurcation and the global existence with respect to diffusion terms of nonconstant positive solutions. The analysis of the author shows that the cross-diffusions may be source for the stationary patterns.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
92D25 Population dynamics (general)
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[1] Brown, P.N., Decay to uniform states in ecological interactions, SIAM J. appl. math., 38, 22-37, (1980) · Zbl 0511.92019
[2] X.F. Chen, W.M. Ni, Y.W. Qi, M.X. Wang, Steady states of a strongly coupled prey – predator model, preprint · Zbl 1144.35470
[3] W.Y. Chen, R. Peng, Stationary patterns created by cross-diffusion for the competitor – competitor – mutualist model, J. Math. Anal. Appl., in press · Zbl 1060.35146
[4] Dancer, E.N., On uniqueness and stability for solutions of singularly perturbated predator – prey type equations with diffusions, J. differential equations, 102, 1-32, (1993) · Zbl 0817.35042
[5] Dancer, E.N., A counterexample of competing species equations, Differential integral equations, 9, 239-246, (1996) · Zbl 0842.35033
[6] Delgado, M.; Lopez-Gomez, J.; Suarez, A., On the symbiotic lotka – volterra model with diffusion and transport effects, J. differential equations, 160, 175-262, (2000) · Zbl 0948.35040
[7] Du, Y.H.; Lou, Y., Some uniqueness and exact multiplicity results for a predator – prey model, Trans. amer. math. soc., 349, 2443-2475, (1997) · Zbl 0965.35041
[8] Du, Y.H.; Lou, Y., S-shaped global solution curve and Hopf bifurcation of positive solutions to a predator – prey model, J. differential equations, 144, 390-440, (1998) · Zbl 0970.35030
[9] Du, Y.H.; Lou, Y., Qualitative behaviour of positive solutions of a predator – prey model: effects of saturation, Proc. roy. soc. Edinburgh sect. A, 131, 321-349, (2001) · Zbl 0980.35028
[10] Ermentrout, B., Strips or spots? nonlinear effects in bifurcation of reaction diffusion equation on the square, Proc. roy. soc. London, 434, 413-417, (1991) · Zbl 0727.92003
[11] Fife, P., Mathematical aspect of reacting and diffusing systems, Lecture notes in biomathematics, vol. 28, (1979), Spring-Verlag New York · Zbl 0403.92004
[12] Furter, J.E.; Lopez-Gomez, J., Diffusion mediated permanence problem for an heterogeneous lotka – volterra competition model, Proc. roy. soc. Edinburgh sect. A, 127, 281-336, (1997) · Zbl 0941.92022
[13] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Kybernetik, 12, 30-39, (1972)
[14] Gilpin, M.E., Spiral chaos in a predator – prey model, Amer. naturalist, 113, 306-308, (1979)
[15] 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
[16] 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
[17] Kan-on, Y.; Mimura, M., Singular perturbation approach to a 3-component reaction – diffusion system arising in population dynamics, SIAM J. math. anal., 29, 1519-1536, (1998) · Zbl 0920.35015
[18] Klebanoff, A.; Hastings, A., Chaos in one-predator, two-prey model: general results from bifurcation theory, Math. biosci., 122, 221-233, (1994) · Zbl 0802.92017
[19] Leng, A.; Clark, D., Bifurcation and large-time asymptotic behavior for a prey – predator reaction – diffusions with Dirichlet boundary data, J. differential equations, 35, 113-127, (1980) · Zbl 0427.35014
[20] Leng, A., Monotone schemes for semilinear elliptic systems related to ecology, Math. methods appl. sci., 4, 272-285, (1982) · Zbl 0493.35044
[21] Leng, A., Equilibria and stabilities for competing species reaction – diffusion equations with Dirichlet boundary data, J. math. anal. appl., 73, 204-218, (1980) · Zbl 0427.35011
[22] Leng, A., Stabilities for equilibria of competing-species reaction – diffusion equations with homogeneous Dirichlet condition, Funk. erv. (ser. interna.), 24, 201-210, (1981) · Zbl 0486.35044
[23] Li, L., Coexistence theorems of steady-states for predator – prey interacting systems, Trans. amer. math. soc., 305, 143-166, (1988) · Zbl 0655.35021
[24] Lin, C.S.; Ni, W.M.; Takagi, I., Large amplitude stationary solutions to a chemotais systems, J. differential equations, 72, 1-27, (1988) · Zbl 0676.35030
[25] Lou, Y.; Martinez, S.; Ni, W.M., On 3×3 lotka – volterra competition systems with cross-diffusion, Discrete contin. dynam. systems, 6, 175-190, (2000) · Zbl 1008.92035
[26] Lou, Y.; Ni, W.M., Diffusion, self-diffusion and cross-diffusion, J. differential equations, 131, 79-131, (1996) · Zbl 0867.35032
[27] Lou, Y.; Ni, W.M., Diffusion, vs. cross-diffusion: an elliptic approach, J. differential equations, 154, 157-190, (1999) · Zbl 0934.35040
[28] Lopez-Gomez, J.; Pardo San Gil, R., Coexistence in a simple food chain with diffusion, J. math. biol., 30, 655-668, (1992) · Zbl 0763.92010
[29] Mimura, M.; Nishiura, Y.; Tesei, A.; Tsujikawa, T., Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima math. J., 14, 425-496, (1984)
[30] Ni, W.M., Diffusion, cross-diffusion and their spike-layer steady states, Notices amer. math. soc., 45, 9-18, (1998) · Zbl 0917.35047
[31] Nirenberg, L., Topics in nonlinear functional analysis, (1974), Courant Institute of Mathematical Sciences New York · Zbl 0286.47037
[32] Okubo, A., Diffusion and ecological problems: mathematical models, (1980), Springer-Verlag Berlin · Zbl 0422.92025
[33] Pao, C.V., Coexistence and stability of a competing-diffusion system in population dynamics, J. math. anal. appl., 83, 54-76, (1981) · Zbl 0479.92013
[34] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Plenum New York · Zbl 0780.35044
[35] Pang, P.Y.H.; Wang, M.X., Non-constant positive steady states of a predator – prey system with non-monotonic functional response and diffusion, Proc. London math. soc., 88, 135-157, (2004) · Zbl 1134.35373
[36] Pang, P.Y.H.; Wang, M.X., Qualitative analysis of a ratio-dependent predator – prey system with diffusion, Proc. roy. soc. Edinburgh sect. A, 133, 919-942, (2003) · Zbl 1059.92056
[37] Peng, R.; Wang, M.X., Positive steady-state solutions of the noyes-field model for belousov – zhabotinskii reaction, Nonlinear anal., 56, 451-464, (2004) · Zbl 1055.35044
[38] Rabinowitz, P., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[39] Smoller, J., Shock waves and reaction – diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002
[40] Turing, A., The chemical basis of morphogenesis, Philos. trans. roy. soc. London ser. B, 237, 37-72, (1952) · Zbl 1403.92034
[41] Wang, M.X., Non-constant positive steady states of the sel’kov model, J. differential equations, 190, 600-620, (2003) · Zbl 1163.35362
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