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Markov-Dyck shifts, neutral periodic points and topological conjugacy
Combined effects of the spatial heterogeneity and the functional response
1. | School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China |
2. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
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.
References:
[1] |
J. R. Beddington,
Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
|
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[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. |
[8] |
D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill,
A model for trophic interaction, Ecology, 56 (1975), 881-892.
|
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
C. B. Huffaker,
Experimental studies on predator: Dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383.
|
[23] |
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. |
[24] |
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. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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.
|
[29] |
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.
|
[30] |
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. |
[31] |
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. |
[32] |
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. |
[33] |
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. |
[34] |
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. |
[35] |
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. |
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.
|
[2] |
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. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[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. |
[8] |
D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill,
A model for trophic interaction, Ecology, 56 (1975), 881-892.
|
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
C. B. Huffaker,
Experimental studies on predator: Dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383.
|
[23] |
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. |
[24] |
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. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[28] |
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.
|
[29] |
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.
|
[30] |
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. |
[31] |
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. |
[32] |
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. |
[33] |
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. |
[34] |
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. |
[35] |
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. |
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