메뉴 건너뛰기




Volumn 246, Issue 5, 2009, Pages 1944-1977

Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system

Author keywords

Diffusive predator prey system; Global bifurcation; Holling type II functional response; Hopf bifurcation; Spatially non homogeneous periodic orbits; Steady state bifurcation

Indexed keywords


EID: 58749112528     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.10.024     Document Type: Article
Times cited : (520)

References (50)
  • 1
    • 0007280433 scopus 로고
    • Stable limit cycles in predator-prey populations
    • Albrecht F., Gatzke H., Wax N., and May R.M. Stable limit cycles in predator-prey populations. Science 181 (1973) 1073-1074
    • (1973) Science , vol.181 , pp. 1073-1074
    • Albrecht, F.1    Gatzke, H.2    Wax, N.3    May, R.M.4
  • 2
    • 0041868333 scopus 로고
    • Lyapunov functions for a generalized Gause-type model
    • Ardito A., and Ricciardi P. Lyapunov functions for a generalized Gause-type model. J. Math. Biol. 33 8 (1995) 816-828
    • (1995) J. Math. Biol. , vol.33 , Issue.8 , pp. 816-828
    • Ardito, A.1    Ricciardi, P.2
  • 3
    • 0001747368 scopus 로고
    • Uniqueness of a limit cycle for a predator-prey system
    • Cheng K.-S. Uniqueness of a limit cycle for a predator-prey system. SIAM J. Math. Anal. 12 4 (1981) 541-548
    • (1981) SIAM J. Math. Anal. , vol.12 , Issue.4 , pp. 541-548
    • Cheng, K.-S.1
  • 4
    • 0027047080 scopus 로고
    • The spatial dynamics of host-parasitoid systems
    • Comins H.N., Hassell M.P., and May R.M. The spatial dynamics of host-parasitoid systems. J. Animal Ecol. 61 (1992) 735-748
    • (1992) J. Animal Ecol. , vol.61 , pp. 735-748
    • Comins, H.N.1    Hassell, M.P.2    May, R.M.3
  • 5
    • 0001842649 scopus 로고
    • Large time behavior of solutions of systems of nonlinear reaction-diffusion equations
    • Conway E., Hoff D., and Smoller J. Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35 (1978) 1-16
    • (1978) SIAM J. Appl. Math. , vol.35 , pp. 1-16
    • Conway, E.1    Hoff, D.2    Smoller, J.3
  • 8
    • 0018737036 scopus 로고
    • Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems
    • de Motoni P., and Rothe F. Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems. SIAM J. Appl. Math. 37 (1979) 648-663
    • (1979) SIAM J. Appl. Math. , vol.37 , pp. 648-663
    • de Motoni, P.1    Rothe, F.2
  • 9
    • 0001674774 scopus 로고    scopus 로고
    • S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model
    • Du Y.-H., and Lou Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model. J. Differential Equations 144 2 (1998) 390-440
    • (1998) J. Differential Equations , vol.144 , Issue.2 , pp. 390-440
    • Du, Y.-H.1    Lou, Y.2
  • 10
    • 23044527512 scopus 로고    scopus 로고
    • Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation
    • Du Y.-H., and Lou Y. Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation. Proc. Roy. Soc. Edinburgh Sect. A 131 2 (2001) 321-349
    • (2001) Proc. Roy. Soc. Edinburgh Sect. A , vol.131 , Issue.2 , pp. 321-349
    • Du, Y.-H.1    Lou, Y.2
  • 11
    • 36148989064 scopus 로고    scopus 로고
    • Some recent results on diffusive predator-prey models in spatially heterogeneous environment
    • Nonlinear Dynamics and Evolution Equations, Amer. Math. Soc., Providence, RI
    • Du Y.-H., and Shi J.-P. Some recent results on diffusive predator-prey models in spatially heterogeneous environment. Nonlinear Dynamics and Evolution Equations. Fields Inst. Commun. vol. 48 (2006), Amer. Math. Soc., Providence, RI 95-135
    • (2006) Fields Inst. Commun. , vol.48 , pp. 95-135
    • Du, Y.-H.1    Shi, J.-P.2
  • 12
    • 33746733702 scopus 로고    scopus 로고
    • A diffusive predator-prey model with a protection zone
    • Du Y.-H., and Shi J.-P. A diffusive predator-prey model with a protection zone. J. Differential Equations 229 1 (2006) 63-91
    • (2006) J. Differential Equations , vol.229 , Issue.1 , pp. 63-91
    • Du, Y.-H.1    Shi, J.-P.2
  • 13
    • 36148940895 scopus 로고    scopus 로고
    • Allee effect and bistability in a spatially heterogeneous predator-prey model
    • Du Y.-H., and Shi J.-P. Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans. Amer. Math. Soc. 359 9 (2007) 4557-4593
    • (2007) Trans. Amer. Math. Soc. , vol.359 , Issue.9 , pp. 4557-4593
    • Du, Y.-H.1    Shi, J.-P.2
  • 15
    • 0042456559 scopus 로고    scopus 로고
    • Elliptic Partial Differential Equations
    • Amer. Math. Soc., Providence, RI
    • Han Q., and Lin F.H. Elliptic Partial Differential Equations. Courant Lect. Notes Math. vol. 1 (1997), Amer. Math. Soc., Providence, RI
    • (1997) Courant Lect. Notes Math. , vol.1
    • Han, Q.1    Lin, F.H.2
  • 17
    • 0026268962 scopus 로고
    • Spatial structure and chaos in insect population dynamics
    • Hassell M.P., Comins H.N., and May R.M. Spatial structure and chaos in insect population dynamics. Nature 353 (1991) 255-258
    • (1991) Nature , vol.353 , pp. 255-258
    • Hassell, M.P.1    Comins, H.N.2    May, R.M.3
  • 18
    • 0003304963 scopus 로고
    • Geometric Theory of Semilinear Parabolic Equations
    • Springer-Verlag, Berlin, New York
    • Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. vol. 840 (1981), Springer-Verlag, Berlin, New York
    • (1981) Lecture Notes in Math. , vol.840
    • Henry, D.1
  • 19
    • 84880026161 scopus 로고
    • Some characteristics of simple types of predation and parasitism
    • Holling C.S. Some characteristics of simple types of predation and parasitism. Canadian Entomologist 91 (1959) 385-398
    • (1959) Canadian Entomologist , vol.91 , pp. 385-398
    • Holling, C.S.1
  • 20
    • 0018102678 scopus 로고
    • On global stability of a predator-prey system
    • Hsu S.-B. On global stability of a predator-prey system. Math. Biosci. 39 1-2 (1978) 1-10
    • (1978) Math. Biosci. , vol.39 , Issue.1-2 , pp. 1-10
    • Hsu, S.-B.1
  • 21
    • 27844511040 scopus 로고    scopus 로고
    • A survey of constructing Lyapunov functions for mathematical models in population biology
    • Hsu S.-B. A survey of constructing Lyapunov functions for mathematical models in population biology. Taiwanese J. Math. 9 2 (2005) 151-173
    • (2005) Taiwanese J. Math. , vol.9 , Issue.2 , pp. 151-173
    • Hsu, S.-B.1
  • 22
    • 58749092741 scopus 로고    scopus 로고
    • Ordinary Differential Equations with Applications
    • World Scientific, Hackensack, NJ
    • Hsu S.-B. Ordinary Differential Equations with Applications. Ser. Appl. Math. vol. 16 (2006), World Scientific, Hackensack, NJ
    • (2006) Ser. Appl. Math. , vol.16
    • Hsu, S.-B.1
  • 24
    • 58749084942 scopus 로고    scopus 로고
    • S.-B. Hsu, J.-P. Shi, Relaxation oscillator profile of limit cycle in predator-prey system, 2008, submitted for publication
    • S.-B. Hsu, J.-P. Shi, Relaxation oscillator profile of limit cycle in predator-prey system, 2008, submitted for publication
  • 25
    • 0000809487 scopus 로고
    • Experimental studies on predation: Dispersion factors and predator-prey oscillations
    • Huffaker C.B. Experimental studies on predation: Dispersion factors and predator-prey oscillations. Hilgardia 27 (1958) 343-383
    • (1958) Hilgardia , vol.27 , pp. 343-383
    • Huffaker, C.B.1
  • 26
    • 9244262277 scopus 로고    scopus 로고
    • Bifurcation Theory: An Introduction with Applications to PDEs
    • Springer-Verlag, New York
    • Kielhófer H. Bifurcation Theory: An Introduction with Applications to PDEs. Appl. Math. Sci. vol. 156 (2004), Springer-Verlag, New York
    • (2004) Appl. Math. Sci. , vol.156
    • Kielhófer, H.1
  • 27
    • 33749640702 scopus 로고    scopus 로고
    • Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge
    • Ko W., and Ryu K. Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. J. Differential Equations 231 (2006) 534-550
    • (2006) J. Differential Equations , vol.231 , pp. 534-550
    • Ko, W.1    Ryu, K.2
  • 28
    • 0002861760 scopus 로고
    • Uniqueness of limit cycles in Gause-type models of predator-prey systems
    • Kuang Y., and Freedman H.I. Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math. Biosci. 88 1 (1988) 67-84
    • (1988) Math. Biosci. , vol.88 , Issue.1 , pp. 67-84
    • Kuang, Y.1    Freedman, H.I.2
  • 29
    • 0000378831 scopus 로고
    • Population dynamic models in heterogeneous environments
    • Levin S.A. Population dynamic models in heterogeneous environments. Annu. Rev. Ecol. Syst. 7 (1976) 1287-1311
    • (1976) Annu. Rev. Ecol. Syst. , vol.7 , pp. 1287-1311
    • Levin, S.A.1
  • 30
    • 0002542873 scopus 로고
    • Large amplitude stationary solutions to a chemotaxis system
    • Lin C.-S., Ni W.-M., and Takagi I. Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72 1 (1988) 1-27
    • (1988) J. Differential Equations , vol.72 , Issue.1 , pp. 1-27
    • Lin, C.-S.1    Ni, W.-M.2    Takagi, I.3
  • 31
    • 34548415794 scopus 로고    scopus 로고
    • Imperfect transcritical and pitchfork bifurcations
    • Liu P., Shi J.-P., and Wang Y.-W. Imperfect transcritical and pitchfork bifurcations. J. Funct. Anal. 251 2 (2007) 573-600
    • (2007) J. Funct. Anal. , vol.251 , Issue.2 , pp. 573-600
    • Liu, P.1    Shi, J.-P.2    Wang, Y.-W.3
  • 32
    • 0030579038 scopus 로고    scopus 로고
    • Diffusion, self-diffusion and cross-diffusion
    • Lou Y., and Ni W.-M. Diffusion, self-diffusion and cross-diffusion. J. Differential Equations 131 (1996) 79-131
    • (1996) J. Differential Equations , vol.131 , pp. 79-131
    • Lou, Y.1    Ni, W.-M.2
  • 34
    • 0039645998 scopus 로고
    • Limit cycles in predator-prey communities
    • May R.M. Limit cycles in predator-prey communities. Science 177 (1972) 900-902
    • (1972) Science , vol.177 , pp. 900-902
    • May, R.M.1
  • 37
    • 35348966287 scopus 로고    scopus 로고
    • Stationary pattern of a ratio-dependent food chain model with diffusion
    • Peng R., Shi J.-P., and Wang M.-X. Stationary pattern of a ratio-dependent food chain model with diffusion. SIAM J. Appl. Math. 67 5 (2007) 1479-1503
    • (2007) SIAM J. Appl. Math. , vol.67 , Issue.5 , pp. 1479-1503
    • Peng, R.1    Shi, J.-P.2    Wang, M.-X.3
  • 38
    • 44949117505 scopus 로고    scopus 로고
    • On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law
    • Peng R., Shi J.-P., and Wang M.-X. On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law. Nonlinearity 21 7 (2008) 1471-1488
    • (2008) Nonlinearity , vol.21 , Issue.7 , pp. 1471-1488
    • Peng, R.1    Shi, J.-P.2    Wang, M.-X.3
  • 39
    • 36448974335 scopus 로고
    • Some global results for nonlinear eigenvalue problems
    • Rabinowitz P.H. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971) 487-513
    • (1971) J. Funct. Anal. , vol.7 , pp. 487-513
    • Rabinowitz, P.H.1
  • 40
    • 0015243940 scopus 로고
    • Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time
    • Rosenzweig M.L. Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science 171 3969 (1971) 385-387
    • (1971) Science , vol.171 , Issue.3969 , pp. 385-387
    • Rosenzweig, M.L.1
  • 41
    • 0001524523 scopus 로고
    • Graphical representation and stability conditions of predator-prey interactions
    • Rosenzweig M.L., and MacArthur R. Graphical representation and stability conditions of predator-prey interactions. Amer. Natur. 97 (1963) 209-223
    • (1963) Amer. Natur. , vol.97 , pp. 209-223
    • Rosenzweig, M.L.1    MacArthur, R.2
  • 42
    • 34548687100 scopus 로고    scopus 로고
    • Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis
    • Ruan S.-G. Diffusion-driven instability in the Gierer-Meinhardt model of morphogenesis. Natur. Resource Modeling 11 2 (1998) 131-142
    • (1998) Natur. Resource Modeling , vol.11 , Issue.2 , pp. 131-142
    • Ruan, S.-G.1
  • 43
    • 0015442349 scopus 로고
    • Dissipative structure: An explanation and an ecological example
    • Segel L.A., and Jackson J.L. Dissipative structure: An explanation and an ecological example. J. Theoret. Biol. 37 (1972) 545-559
    • (1972) J. Theoret. Biol. , vol.37 , pp. 545-559
    • Segel, L.A.1    Jackson, J.L.2
  • 44
    • 60349102594 scopus 로고    scopus 로고
    • J. Shi, X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations (2009), doi: 10.1016/j.jde.2008.09.009
    • J. Shi, X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations (2009), doi: 10.1016/j.jde.2008.09.009
  • 47
    • 33646864830 scopus 로고    scopus 로고
    • Symmetric functional-differential equations and neural networks with memory
    • Wu J.-H. Symmetric functional-differential equations and neural networks with memory. Trans. Amer. Math. Soc. 350 12 (1998) 4799-4838
    • (1998) Trans. Amer. Math. Soc. , vol.350 , Issue.12 , pp. 4799-4838
    • Wu, J.-H.1
  • 48
    • 38949167125 scopus 로고    scopus 로고
    • Diffusion-driven instability and bifurcation in the Lengyel-Epstein system
    • Yi F.-Q., Wei J.-J., and Shi J.-P. Diffusion-driven instability and bifurcation in the Lengyel-Epstein system. Nonlinear Anal. Real World Appl. 9 3 (2008) 1038-1051
    • (2008) Nonlinear Anal. Real World Appl. , vol.9 , Issue.3 , pp. 1038-1051
    • Yi, F.-Q.1    Wei, J.-J.2    Shi, J.-P.3
  • 49
    • 55649115788 scopus 로고    scopus 로고
    • Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system
    • Yi F.-Q., Wei J.-J., and Shi J.-P. Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system. Appl. Math. Lett. 22 1 (2009) 52-55
    • (2009) Appl. Math. Lett. , vol.22 , Issue.1 , pp. 52-55
    • Yi, F.-Q.1    Wei, J.-J.2    Shi, J.-P.3
  • 50
    • 84953472772 scopus 로고
    • Proof of the uniqueness theorem of limit cycles of generalized Lienard equations
    • Zhang Z.-F. Proof of the uniqueness theorem of limit cycles of generalized Lienard equations. Appl. Anal. 23 1-2 (1986) 63-76
    • (1986) Appl. Anal. , vol.23 , Issue.1-2 , pp. 63-76
    • Zhang, Z.-F.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.