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2015, 12(4): 699-715. doi: 10.3934/mbe.2015.12.699

Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions

Lin Wang 1, , James Watmough 2, and  Fang Yu 2,

1. 

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3
2. 

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada

Received  March 2014 Revised  December 2014 Published  April 2015

In this paper, we study a diffusive plant-herbivore system with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability of spatially homogeneous steady states is established. We also derive conditions ensuring the occurrence of Hopf bifurcation and steady state bifurcation. Interesting transient spatio-temporal behaviors including oscillations in one or both of space and time are observed through numerical simulations.
Keywords: Plant-herbivore interaction, transient dynamics., Hopf bifurcation, diffusion, steady state bifurcation.
Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 92B0.
Citation: Lin Wang, James Watmough, Fang Yu. Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions. Mathematical Biosciences & Engineering, 2015, 12 (4) : 699-715. doi: 10.3934/mbe.2015.12.699
References:
[1]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some system of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.  Google Scholar

[2]

E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math., 3 (1982), 288-334. doi: 10.1016/S0196-8858(82)80009-2.  Google Scholar

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L. R. Fox, Defense and dynamics in plant-herbivore systems, Amer. Zool., 21 (1981), 853-864. doi: 10.1093/icb/21.4.853.  Google Scholar

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B. D. Hassard, N. D. KazavinoJ and Y. H. Wan, Theory and Applications of the Hopf Bifurcation, Cambridge University Press, Cambridge, MA, 1981.  Google Scholar

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L. R. Ginzburg, Assuming reproduction to be a function of consumption raises doubts about some popular predator-prey models, J. of Animal Ecology, 67 (1998), 325-327. doi: 10.1046/j.1365-2656.1998.00226.x.  Google Scholar

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L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

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Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 27 (2014), 87-104. doi: 10.1088/0951-7715/27/1/87.  Google Scholar

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Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345. doi: 10.1137/0521018.  Google Scholar

show all references

References:
[1]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some system of elliptic equations, SIAM J. Math. Anal., 17 (1986), 1339-1353. doi: 10.1137/0517094.  Google Scholar

[2]

E. Conway, R. Gardner and J. Smoller, Stability and bifurcation of steady-state solutions for predator-prey equations, Adv. in Appl. Math., 3 (1982), 288-334. doi: 10.1016/S0196-8858(82)80009-2.  Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

[5]

E. N. Dancer, On positive solutions of some pairs of differential equations II, J. Diff. Equat., 60 (1985), 236-258. doi: 10.1016/0022-0396(85)90115-9.  Google Scholar

[6]

L. R. Fox, Defense and dynamics in plant-herbivore systems, Amer. Zool., 21 (1981), 853-864. doi: 10.1093/icb/21.4.853.  Google Scholar

[7]

B. D. Hassard, N. D. KazavinoJ and Y. H. Wan, Theory and Applications of the Hopf Bifurcation, Cambridge University Press, Cambridge, MA, 1981.  Google Scholar

[8]

L. R. Ginzburg, Assuming reproduction to be a function of consumption raises doubts about some popular predator-prey models, J. of Animal Ecology, 67 (1998), 325-327. doi: 10.1046/j.1365-2656.1998.00226.x.  Google Scholar

[9]

L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar

[10]

R. E. Ricklefs, The Economy of Nature, Freeman and Company, New York, 2010. Google Scholar

[11]

Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 27 (2014), 87-104. doi: 10.1088/0951-7715/27/1/87.  Google Scholar

[12]

Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. Math. Anal., 21 (1990), 327-345. doi: 10.1137/0521018.  Google Scholar

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