This paper deals with a predator-prey model with Beddington-DeAngelis functional response, in which a protection zone is created for the prey species. Whether the combination of the protection zone and the Beddington-DeAngelis functional response can yield new results or not is of interest. The result reveals that they jointly produce a new critical value, which is smaller than that determined by either the protection zone or the functional response singly. As a result, rather different stationary patterns can be found and the combined effects are very prominent. Then the effect of the parameter $k$ in the Beddington-DeAngelis functional response is studied. The result deduces that as $k$ is large enough, there exists a unique positive stationary solution and it is linearly stable except a special case. Actually, we can obtain that the positive stationary solution is globally asymptotically stable.
Citation: |
J. R. Beddington
, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975)
, 331-340.
![]() |
|
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.![]() ![]() ![]() |
|
W. Chen
and M. Wang
, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005)
, 31-44.
doi: 10.1016/j.mcm.2005.05.013.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
R. Cui
, J. Shi
and B. Wu
, Strong allee effect in a diffusive predator-prey system with a
protection zone, J. Differential Equations, 256 (2014)
, 108-129.
doi: 10.1016/j.jde.2013.08.015.![]() ![]() ![]() |
|
E. N. Dancer
, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983)
, 131-151.
doi: 10.1016/0022-247X(83)90098-7.![]() ![]() ![]() |
|
E. N. Dancer
and Y. Du
, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002)
, 292-314.
doi: 10.1137/S0036141001387598.![]() ![]() ![]() |
|
D. L. DeAngelis
, R. A. Goldstein
and R. V. O'Neill
, A model for trophic interaction, Ecology, 56 (1975)
, 881-892.
![]() |
|
D. T. Dimitrov
and H. V. Kojouharov
, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Appl. Math. Comput., 162 (2005)
, 523-538.
doi: 10.1016/j.amc.2003.12.106.![]() ![]() ![]() |
|
J. Dockery
, V. Hutson
, K. Mischaikow
and M. Pernarowski
, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998)
, 61-83.
doi: 10.1007/s002850050120.![]() ![]() ![]() |
|
Y. Du
, Effects of a degeneracy in the competition model, part Ⅰ. classical and generalized steady-state solutions, J. Differential Equations, 181 (2002)
, 92-132.
doi: 10.1006/jdeq.2001.4074.![]() ![]() ![]() |
|
Y. Du
, Effects of a degeneracy in the competition model, part Ⅱ. perturbation and dynamical behaviour, J. Differential Equations, 181 (2002)
, 133-164.
doi: 10.1006/jdeq.2001.4075.![]() ![]() ![]() |
|
Y. Du
and S. B. Hsu
, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004)
, 331-364.
doi: 10.1016/j.jde.2004.05.010.![]() ![]() ![]() |
|
Y. 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.![]() ![]() ![]() |
|
Y. Du
, R. Peng
and M. 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.![]() ![]() ![]() |
|
Y. Du
and J. 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.![]() ![]() ![]() |
|
Y. Du
and J. Shi
, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007)
, 4557-4593.
doi: 10.1090/S0002-9947-07-04262-6.![]() ![]() ![]() |
|
Y. Du
and M. Wang
, Asymptotic behaviour of positive steady states to a predator-prey model, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006)
, 759-778.
doi: 10.1017/S0308210500004704.![]() ![]() ![]() |
|
G. Guo
and J. 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.![]() ![]() ![]() |
|
X. He
and S. Zheng
, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017)
, 239-257.
doi: 10.1007/s00285-016-1082-5.![]() ![]() ![]() |
|
X. He
and S. Zheng
, Protection zone in a modified Lotka-Volterra model, Discrete Contin. Dyn. Syst.B, 20 (2015)
, 2027-2038.
doi: 10.3934/dcdsb.2015.20.2027.![]() ![]() ![]() |
|
C. B. Huffaker
, Experimental studies on predator: Dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958)
, 343-383.
![]() |
|
V. Hutson
, Y. Lou
, K. Mischaikow
and P. Poláčik
, Competing species near a degenerate limit, SIAM. J. Math. Anal., 35 (2003)
, 453-491.
doi: 10.1137/S0036141002402189.![]() ![]() ![]() |
|
K. Kuto
, Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment, Nonlinear Anal. RWA, 10 (2009)
, 943-965.
doi: 10.1016/j.nonrwa.2007.11.015.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
Y. Lou
, S. Martínez
and P. Poláčik
, Loops and branches of coexistence states in a LotkaVolterra competition model, J. Differential Equations, 230 (2006)
, 720-742.
doi: 10.1016/j.jde.2006.04.005.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
W. Ruan
and W. Feng
, On the fixed point index and multiple steady states of reaction-diffusion systems, Differential Integral Equations, 8 (1995)
, 371-391.
![]() ![]() |
|
G. T. Skalski
and J. F. Gilliam
, Functional responses with predator interference: viable alternatives to the Holling type Ⅱ model, Ecology, 82 (2001)
, 3083-3092.
![]() |
|
M. Wang
and P. Y. H. Pang
, Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model, Appl. Math. Lett., 21 (2008)
, 1215-1220.
doi: 10.1016/j.aml.2007.10.026.![]() ![]() ![]() |
|
Y. X. Wang
and W. T. Li
, Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system, Nonlinear Anal. RWA, 14 (2013)
, 1235-1246.
doi: 10.1016/j.nonrwa.2012.09.015.![]() ![]() ![]() |
|
Y. X. Wang
and W. T. Li
, Fish-hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross-diffusion, J. Differential Equations, 251 (2011)
, 1670-1695.
doi: 10.1016/j.jde.2011.03.009.![]() ![]() ![]() |
|
Y. X. Wang
and W. T. Li
, Spatial degeneracy vs functional response, Discrete Contin. Dyn. Syst. B, 21 (2016)
, 2811-2837.
doi: 10.3934/dcdsb.2016074.![]() ![]() ![]() |
|
Y. X. Wang
and W. T. Li
, Uniqueness and global stability of positive stationary solution for a predator-prey system, J. Math. Anal. Appl., 462 (2018)
, 577-589.
doi: 10.1016/j.jmaa.2018.02.032.![]() ![]() ![]() |
|
X. Zeng
, J. Zhang
and Y. Gu
, Uniqueness and stability of positive steady state solutions for a ratio-dependent predator-prey system with a crowding term in the prey equation, Nonlinear Anal. RWA, 24 (2015)
, 163-174.
doi: 10.1016/j.nonrwa.2015.02.005.![]() ![]() ![]() |