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September  2015, 20(7): 2027-2038. doi: 10.3934/dcdsb.2015.20.2027

Protection zone in a modified Lotka-Volterra model

Xiao He 1, and  Sining Zheng 2,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2. 

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024

Received  January 2014 Revised  May 2015 Published  July 2015

This paper studies the dynamic behavior of solutions to a modified Lotka-Volterra reaction-diffusion system with homogeneous Neumann boundary conditions, for which a protection zone should be created to prevent the extinction of the prey only if the prey's growth rate is small. We find a critical size of the protection zone, determined by the ratio of the predation rate and the refuge ability, to ensure the existence, uniqueness and global asymptotic stability of positive steady states for general predator's growth rate $\mu>0$. Bellow the critical size the dynamics of the model would be similar to the case without protection zones. The known uniqueness results for the protection problems with other functional responses, e.g., Holling II model, Leslie model, Beddington-DeAngelis model, were all required that the predator's growth rate $\mu>0$ is large enough. Such a large $\mu$ assumption is not needed for the uniqueness and asymptotic results to the modified Lotka-Volterra reaction-diffusion system considered in this paper.
Keywords: Bifurcation, predator-prey, diffusion, protection..
Mathematics Subject Classification: Primary: 35J47, 35K57, 92D4.
Citation: Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027
References:
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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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