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Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case. (English) Zbl 1169.35328
Summary: We prove the non-existence of non-constant positive steady state solutions of two reaction-diffusion predator-prey models with Holling type-II functional response when the interaction between the predator and the prey is strong. The result implies that the global bifurcating branches of steady state solutions are bounded loops.

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B32 Bifurcations in context of PDEs
92C15 Developmental biology, pattern formation
92C40 Biochemistry, molecular biology
92D25 Population dynamics (general)
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[1] Blat, J.; Brown, K.J., Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. math. anal., 17, 1339-1353, (1986) · Zbl 0613.35008
[2] Cantrell, R.S.; Cosner, C., Spatial ecology via reaction – diffusion equation, Wiley ser. math. comput. biol., (2003), John Wiley & Sons · Zbl 1059.92051
[3] De Mottoni, P.; Rothe, F., Convergence to homogeneous equilibrium state for generalized volterra – lotka systems with diffusion, SIAM J. appl. math., 37, 648-663, (1979) · Zbl 0425.35055
[4] Du, Y.H.; Lou, Y., S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator – prey model, J. differential equations, 144, 390-440, (1998) · Zbl 0970.35030
[5] Du, Y.H.; Lou, Y., Qualitative behavior of positive solutions of a predator – prey model: effects of saturation, Proc. roy. soc. Edinburgh sect. A, 131, 321-349, (2001) · Zbl 0980.35028
[6] Du, Y.H.; Shi, J.P., Some recent results on diffusive predator – prey models in spatially heterogeneous environment, (), 95-135 · Zbl 1100.35041
[7] Du, Y.H.; Shi, J.P., Allee effect and bistability in a spatially heterogeneous predator – prey model, Trans. amer. math. soc., 359, 4557-4593, (2007) · Zbl 1189.35337
[8] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equation of second order, Classics math., (2001), Springer-Verlag Berlin, Reprint of the 1998 edition · Zbl 0691.35001
[9] Hastings, A., Global stability of two-species systems, J. math. biol., 5, 399-403, (1977) · Zbl 0382.92008
[10] Henry, D., Geometric theory of semilinear parabolic equations, Lecture notes in math., vol. 840, (1981), Springer-Verlag Berlin, New York · Zbl 0456.35001
[11] Holling, C.S., Some characteristics of simple types of predation and parasitism, Canad. entomologist, 91, 385-398, (1959)
[12] S.B. Hsu, J.P. Shi, Relaxation oscillator profile of limit cycle in predator – prey system, Discrete Contin. Dyn. Syst. Ser. B, doi:10.3934/dcdsb.2009.11.xx
[13] Huffaker, C.B., Experimental studies on predation: dispersion factors and predator – prey oscillations, Hilgardia, 27, 343-383, (1958)
[14] Jang, J.; Ni, W.M.; Tang, M.X., Global bifurcation and structure of Turing patterns in the 1-D lengyel – epstein model, J. dynam. differential equations, 16, 297-320, (2004) · Zbl 1072.35091
[15] J.Y. Jin, J.P. Shi, J.J. Wei, F.Q. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, submitted for publication · Zbl 1288.35051
[16] Kareiva, P., Habitat fragmentation and the stability of predator – prey interactions, Nature, 326, 388-390, (1987)
[17] Kareiva, P.; Odell, G., Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search, Amer. natur., 130, 233-270, (1987)
[18] Ko, W.; Ryu, K., Qualitative analysis of a predator – prey model with Holling type II functional response incorporating a prey refuge, J. differential equations, 231, 534-550, (2006) · Zbl 1387.35588
[19] Leung, A., Limiting behaviour for a prey – predator model with diffusion and crowding effects, J. math. biol., 6, 87-93, (1978) · Zbl 0386.92011
[20] Levin, S.A.; Segel, L.A., An hypothesis for the origin of planktonic patchiness, Nature, 259, 659, (1976)
[21] Lieberman, G.M., Bounds for the steady-state sel’kov model for arbitrary p in any number of dimensions, SIAM J. math. anal., 36, 1400-1406, (2005) · Zbl 1112.35062
[22] 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
[23] Lou, Y.; Ni, W.M., Diffusion, self-diffusion and cross-diffusion, J. differential equations, 131, 79-131, (1996) · Zbl 0867.35032
[24] Medvinsky, A.B.; Petrovskii, S.V.; Tikhonova, I.A.; Malchow, H.; Li, B.-L., Spatiotemporal complexity of plankton and fish dynamics, SIAM rev., 44, 311-370, (2002) · Zbl 1001.92050
[25] Mimura, M.; Murray, J.D., On a diffusive prey – predator model which exhibits patchiness, J. theoret. biol., 75, 249-262, (1978)
[26] Murdoch, W.W.; Briggs, C.J.; Nisbert, R.M., Consumer – resource dynamics, Monogr. population biol., vol. 36, (2003), Princeton University Press
[27] Murray, J.D., Mathematical biology. I. an introduction, Interdiscip. appl. math., Interdiscip. appl. math., vol. 18, (2002), Springer-Verlag New York, II. Spatial Models and Biomedical Applications
[28] Nishiura, Y., Global structure of bifurcating solutions of some reaction – diffusion systems, SIAM J. math. anal., 13, 555-593, (1982) · Zbl 0501.35010
[29] Okubo, A.; Levin, S., Diffusion and ecological problems: modern perspectives, Interdiscip. appl. math., vol. 14, (2001), Springer-Verlag New York · Zbl 1027.92022
[30] Peng, R., Qualitative analysis of steady states to the sel’kov model, J. differential equations, 241, 386-398, (2007) · Zbl 1210.35079
[31] Peng, R.; Shi, J.P.; Wang, M.X., Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. appl. math., 67, 1479-1503, (2007) · Zbl 1210.35268
[32] Peng, R.; Shi, J.P.; Wang, M.X., On stationary patterns of a reaction – diffusion model with autocatalysis and saturation law, Nonlinearity, 21, 1471-1488, (2008) · Zbl 1148.35094
[33] Peng, R.; Wang, M.X., Positive steady-states of the holling – tanner prey – predator model with diffusion, Proc. roy. soc. Edinburgh sect. A, 135, 16, 149-164, (2005) · Zbl 1144.35409
[34] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[35] Rosenzweig, M.L.; MacArthur, R., Graphical representation and stability conditions of predator – prey interactions, Amer. natur., 97, 209-223, (1963)
[36] Rothe, F., Convergence to the equilibrium state in the volterra – lotka diffusion equations, J. math. biol., 3, 319-324, (1976) · Zbl 0355.92013
[37] J.P. Shi, Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models, Front. Math. China, doi:10.1007/s11464-009-0026-4, in press · Zbl 1176.35021
[38] Shi, J.P.; Wang, X.F., On global bifurcation for quasilinear elliptic systems on bounded domains, J. differential equations, 246, 2788-2812, (2009) · Zbl 1165.35358
[39] Yi, F.Q.; Wei, J.J.; Shi, J.P., Diffusion-driven instability and bifurcation in the lengyel – epstein system, Nonlinear anal. real world appl., 9, 1038-1051, (2008) · Zbl 1146.35384
[40] Yi, F.Q.; Wei, J.J.; Shi, J.P., Global asymptotical behavior of the lengyel – epstein reaction – diffusion system, Appl. math. lett., 22, 52-55, (2009) · Zbl 1163.35422
[41] Yi, F.Q.; Wei, J.J.; Shi, J.P., Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator – prey system, J. differential equations, 246, 1944-1977, (2009) · Zbl 1203.35030
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