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Protection zone in a modified Lotka-Volterra model
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
2. | Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 |
References:
[1] |
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
doi: 10.2307/3866. |
[2] |
R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the coexistence of competing species, J. Math. Biol., 37 (1998), 103-145.
doi: 10.1007/s002850050122. |
[3] |
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[4] |
R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence, and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[5] |
W. Y. Chen and M. X. Wang, Qualitative analysis of predator-prey model with Beddington-DeAngelis functional response and diffudion, Math. Comput. Modelling, 42 (2005), 31-44.
doi: 10.1016/j.mcm.2005.05.013. |
[6] |
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. |
[7] |
D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.
doi: 10.2307/1936298. |
[8] |
Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.
doi: 10.1016/j.jde.2007.10.005. |
[9] |
Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.
doi: 10.1016/j.jde.2008.11.007. |
[10] |
Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013. |
[11] |
Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially hererogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.
doi: 10.1090/S0002-9947-07-04262-6. |
[12] |
S. Geritz and M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, J. Theoret. Biol., 314 (2012), 106-108.
doi: 10.1016/j.jtbi.2012.08.030. |
[13] |
G. H. Guo and J. H. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646.
doi: 10.1016/j.na.2009.09.003. |
[14] |
X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, preprint, arXiv:1505.06625 |
[15] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[17] |
C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[18] |
K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009.
doi: 10.1016/j.jde.2011.01.026. |
[19] |
Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245.
doi: 10.1016/j.nonrwa.2012.06.001. |
show all references
References:
[1] |
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
doi: 10.2307/3866. |
[2] |
R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the coexistence of competing species, J. Math. Biol., 37 (1998), 103-145.
doi: 10.1007/s002850050122. |
[3] |
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[4] |
R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence, and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[5] |
W. Y. Chen and M. X. Wang, Qualitative analysis of predator-prey model with Beddington-DeAngelis functional response and diffudion, Math. Comput. Modelling, 42 (2005), 31-44.
doi: 10.1016/j.mcm.2005.05.013. |
[6] |
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. |
[7] |
D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.
doi: 10.2307/1936298. |
[8] |
Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.
doi: 10.1016/j.jde.2007.10.005. |
[9] |
Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.
doi: 10.1016/j.jde.2008.11.007. |
[10] |
Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013. |
[11] |
Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially hererogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.
doi: 10.1090/S0002-9947-07-04262-6. |
[12] |
S. Geritz and M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, J. Theoret. Biol., 314 (2012), 106-108.
doi: 10.1016/j.jtbi.2012.08.030. |
[13] |
G. H. Guo and J. H. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646.
doi: 10.1016/j.na.2009.09.003. |
[14] |
X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, preprint, arXiv:1505.06625 |
[15] |
Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
Y. Lou and W. M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[17] |
C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[18] |
K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009.
doi: 10.1016/j.jde.2011.01.026. |
[19] |
Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245.
doi: 10.1016/j.nonrwa.2012.06.001. |
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