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

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.
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. doi: 10.1137/0517094.

[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. doi: 10.1016/S0196-8858(82)80009-2.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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. doi: 10.1046/j.1365-2656.1998.00226.x.

[9]

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

[10]

R. E. Ricklefs, The Economy of Nature,, Freeman and Company, (2010).

[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. doi: 10.1088/0951-7715/27/1/87.

[12]

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

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. doi: 10.1137/0517094.

[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. doi: 10.1016/S0196-8858(82)80009-2.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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. doi: 10.1046/j.1365-2656.1998.00226.x.

[9]

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

[10]

R. E. Ricklefs, The Economy of Nature,, Freeman and Company, (2010).

[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. doi: 10.1088/0951-7715/27/1/87.

[12]

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

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