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Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. (English) Zbl 1228.35037
The dynamics of a predator-prey reaction-diffusion system with the functional response of Holling type II and Allee effect in the prey population is investigated. The global existence of its solutions and in various situations their global asymptotic behavior is determined. It is shown that a large amount of predators initially always drive both population into extinction, which is characteristic for predator-prey systems with Allee effect. Energy estimates are used for obtaining a priori bounds for the dynamic and steady state solutions. On this base nonexistence of nonconstant positive steady state solutions is shown for identifying the ranges of parameters of spatial pattern formation. With the aid of global bifurcation theory developed in [J. Shi and X. Wang, J. Differ. Equations 246, No. 7, 2788–2812 (2009; Zbl 1165.35358)], the existence of nonconstant steady state solutions with certain eigenmodes is obtained. Also following to [F. Yi, J. Wei and J. Shi, J. Differ. Equations 246, No. 5, 1944–1977 (2009; Zbl 1203.35030)] the existence of spatially nonhomogeneous time-periodic orbits is proved together with their bifurcation analysis.
MSC:
35B32Bifurcation (PDE)
35K57Reaction-diffusion equations
35B36Pattern formation in solutions of PDE
92D25Population dynamics (general)
35B10Periodic solutions of PDE
References:
[1]Alikakos, N. D.: An application of the invariance principle to reaction-diffusion equations, J. differential equations 33, 201-225 (1979) · Zbl 0386.34046 · doi:10.1016/0022-0396(79)90088-3
[2]Allee, W. C.: Animal aggregations: A study in general sociology, (1931)
[3]Boukal, S. D.; Sabelis, W. M.; Berec, L.: How predator functional responses and allee effects in prey affect the paradox of enrichment and population collapses, Theoret. popul. Biol. 72, 136-147 (2007)
[4]Cantrell, R. S.; Cosner, C.; Hutson, V.: Permanence in ecological systems with spatial heterogeneity, Proc. roy. Soc. Edinburgh sect. A 123, 533-559 (1993) · Zbl 0796.92026 · doi:10.1017/S0308210500025877
[5]Cantrell, R. S.; Cosner, C.: Spatial ecology via reaction-diffusion equations, Wiley ser. Math. comput. Biol. (2003)
[6]Chafee, N.; Infante, E. F.: A bifurcation problem for a nonlinear partial differential equation of parabolic type, Appl. anal. 4, 17-37 (1974/1975) · Zbl 0296.35046 · doi:10.1080/00036817408839081
[7]Chen, X. -Y.; Poláčik, P.: Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, J. dynam. Differential equations 7, No. 1, 73-107 (1995) · Zbl 0822.35073 · doi:10.1007/BF02218815
[8]Conway, E.; Hoff, D.; Smoller, J.: Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. math. 35, No. 1, 1-16 (1978) · Zbl 0383.35035 · doi:10.1137/0135001
[9]Courchamp, F.; Berec, L.; Gascoigne, J.: Allee effects in ecology and conservation, (2008)
[10]Crandall, M. G.; Rabinowitz, P. H.: Bifurcation from simple eigenvalues, J. funct. Anal. 8, 321-340 (1971) · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[11]Du, Y. -H.; Shi, J. -P.: Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Fields inst. Commun. 48, 95-135 (2006) · Zbl 1100.35041
[12]Du, Y. -H.; Shi, J. -P.: A diffusive predator-prey model with a protection zone, J. differential equations 229, No. 1, 63-91 (2006) · Zbl 1142.35022 · doi:10.1016/j.jde.2006.01.013
[13]Du, Y. -H.; Shi, J. -P.: Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. amer. Math. soc. 359, No. 9, 4557-4593 (2007) · Zbl 1189.35337 · doi:10.1090/S0002-9947-07-04262-6
[14]Hale, J. K.: Asymptotic behavior of dissipative systems, Math. surveys monogr. 25 (1988) · Zbl 0642.58013
[15]Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. -H.: Theory and applications of Hopf bifurcation, (1981)
[16]Henry, D.: Geometric theory of semilinear parabolic equations, Lecture notes in math. 840 (1981) · Zbl 0456.35001
[17]Holling, C. S.: The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canad. entomol. 91, 293-320 (1959)
[18]Hollis, S. L.; Martin, R. H.; Pierre, M.: Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. anal. 18, No. 3, 744-761 (1987) · Zbl 0655.35045 · doi:10.1137/0518057
[19]Huang, J. -H.; Lu, G.; Ruan, S. -G.: Existence of traveling wave solutions in a diffusive predator-prey model, J. math. Biol. 46, No. 2, 132-152 (2003) · Zbl 1018.92026 · doi:10.1007/s00285-002-0171-9
[20]Jiang, J. -F.; Liang, X.; Zhao, X. -Q.: Saddle-point behavior for monotone semiflows and reaction-diffusion models, J. differential equations 203, No. 2, 313-330 (2004) · Zbl 1063.35083 · doi:10.1016/j.jde.2004.05.002
[21]Jiang, J. -F.; Shi, J. -P.: Dynamics of a reaction-diffusion system of autocatalytic chemical reaction, Discrete contin. Dyn. syst. 21, No. 1, 245-258 (2008) · Zbl 1145.37343 · doi:10.3934/dcds.2008.21.245 · doi:http://aimsciences.org/journals/pdfs.jsp?paperID=3166{&}mode=abstract
[22]Jiang, J. -F.; Shi, J. -P.: Bistability dynamics in some structured ecological models, Spatial ecology (2009)
[23]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, Rocky Mountain J. Math. (2011), in press.
[24]Kazarinov, N.; Den Driessche, P. Van: A model predator-prey systems with functional response, Math. biosci. 39, 125-134 (1978) · Zbl 0382.92007 · doi:10.1016/0025-5564(78)90031-7
[25]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)
[26]Lewis, M. A.; Karevia, P.: Allee dynamics and the spread of invading organisms, Theoret. popul. Biol. 43, 141-158 (1993) · Zbl 0769.92025 · doi:10.1006/tpbi.1993.1007
[27]Lotka, A. J.: Elements of physical biology, (1925) · Zbl 51.0416.06
[28]Lou, Y.; Ni, W. -M.: Diffusion, self-diffusion and cross-diffusion, J. differential equations 131, 79-131 (1996) · Zbl 0867.35032 · doi:10.1006/jdeq.1996.0157
[29]Lou, Y.; Ni, W. -M.: Diffusion vs cross-diffusion: an elliptic approach, J. differential equations 154, 157-190 (1999) · Zbl 0934.35040 · doi:10.1006/jdeq.1998.3559
[30]Medvinsky, A. B.; Petrovskii, S. V.; Tikhonova, I. A.; Malchow, H.; Li, B. -L.: Spatiotemporal complexity of plankton and fish dynamics, SIAM rev. 44, No. 3, 311-370 (2002) · Zbl 1001.92050 · doi:10.1137/S0036144502404442
[31]Mischaikow, K.; Smith, H.; Thieme, H. R.: Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. amer. Math. soc. 347, No. 5, 1669-1685 (1995) · Zbl 0829.34037 · doi:10.2307/2154964
[32]Morozov, A.; Petrovskii, S.; Li, B. -L.: Bifurcations and chaos in a predator-prey system with the allee effect, Proc. R. Soc. lond. Ser. B 271, 1407-1414 (2004)
[33]Morozov, A.; Petrovskii, S.; Li, B. -L.: Spatiotemporal complexity of patchy invasion in a predator prey system with the allee effect, J. theoret. Biol. 238, 18-35 (2006)
[34]Nisbet, R. M.; Gurney, W. S. C.: Modelling fluctuating populations, (1982) · Zbl 0593.92013
[35]Owen, M. R.; Lewis, M. A.: How predation can slow, stop or reverse a prey invasion, Bull. math. Biol. 63, 655-684 (2001)
[36]Pao, C. -V.: Nonlinear parabolic and elliptic equations, (1992)
[37]Peng, R.; Shi, J. -P.; Wang, M. -X.: Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. math. 65, 1479-1503 (2007) · Zbl 1210.35268 · doi:10.1137/05064624X
[38]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 · doi:10.1088/0951-7715/21/7/006
[39]Petrovskii, S. V.; Morozov, A. Y.; Venturino, E.: Allee effect makes possible patchy invasion in a predator prey system, Ecol. lett. 5, 345-352 (2002)
[40]Petrovskii, S. V.; Morozov, A. Y.; Li, B. -L.: Regimes of biological invasion in a predator prey system with the allee effect, Bull. math. Biol. 67, 637-661 (2005)
[41]Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[42]Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS reg. Conf. ser. Math. 65 (1984) · Zbl 0609.58002
[43]Rosenzweig, L. M.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science 171, No. 3969, 385-387 (1971)
[44]Sherratt, J. A.; Eagan, B. T.; Lewis, M. A.: Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?, Philos. trans. R. soc. Lond. ser. B 352, 21-38 (1997)
[45]Shi, J. -P.: Solution set of semilinear elliptic equations: global bifurcation and exact multiplicity, (2011)
[46]Shi, J. -P.: Semilinear Neumann boundary value problems on a rectangle, Trans. amer. Math. soc. 354, No. 8, 3117-3154 (2002) · Zbl 0992.35031 · doi:10.1090/S0002-9947-02-03007-6
[47]Shi, J. -P.; Shivaji, R.: Persistence in reaction diffusion models with weak allee effect, J. math. Biol. 52, No. 6, 807-829 (2006) · Zbl 1110.92055 · doi:10.1007/s00285-006-0373-7
[48]Shi, J. -P.; Wang, X. -F.: On the global bifurcation for quasilinear elliptic systems on bounded domains, J. differential equations 246, No. 7, 2788-2812 (2009) · Zbl 1165.35358 · doi:10.1016/j.jde.2008.09.009
[49]Smoller, J.: Shock waves and reaction-diffusion equations, Grundlehren math. Wiss. 258 (1994) · Zbl 0807.35002
[50]Taylor, C. M.; Hastings, A.: Allee effects in biological invasions, Ecol. lett. 8, 895-908 (2005)
[51]Van Voorn, G. A. K.; Hemerik, L.; Boer, M. P.; Kooi, B. W.: Heteroclinic orbits indicate overexploitation in predator prey systems with a strong allee effect, Math. biosci. 209, 451-469 (2007) · Zbl 1126.92062 · doi:10.1016/j.mbs.2007.02.006
[52]Volterra, V.: Fluctuations in the abundance of species, considered mathematically, Nature 118, 558 (1926) · Zbl 52.0453.03 · doi:10.1038/118558a0
[53]Wang, J. -F.; Shi, J. -P.; Wei, J. -J.: Predator-prey system with strong allee effect in prey, J. math. Biol. 3, No. 3, 291-331 (2011) · Zbl 1232.92076 · doi:10.1007/s00285-010-0332-1
[54]J.-F. Wang, J.-P. Shi, J.-J. Wei, Nonexistence of periodic orbits for predator-prey system with strong Allee effect in prey population, submitted for publication.
[55]Wang, M. -X.: Non-constant positive steady states of the sel’kov model, J. differential equations 190, No. 2, 600-620 (2003) · Zbl 1163.35362 · doi:10.1016/S0022-0396(02)00100-6
[56]Wang, M. -E.; Kot, M.: Speeds of invasion in a model with strong or weak allee effects, Math. biosci. 171, 83-97 (2001) · Zbl 0978.92033 · doi:10.1016/S0025-5564(01)00048-7
[57]Smoller, J.; Wasserman, A.: Global bifurcation of steady-state solutions, J. differential equations 39, No. 2, 269-290 (1981) · Zbl 0425.34028 · doi:10.1016/0022-0396(81)90077-2
[58]Wu, J. -H.: Theory and applications of partial functional-differential equations, (1996)
[59]Ye, Q. -X.; Li, Z. -Y.: Introduction to reaction-diffusion equations, (1994)
[60]Yi, F. -Q.; Wei, J. -J.; Shi, J. -P.: Bifurcation analysis of a diffusive predator-prey system with Holling type-II function response, J. differential equations 246, 1944-1977 (2009)