January  2019, 39(1): 19-39. doi: 10.3934/dcds.2019002

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

E-mail address: wangyux10@163.com (Y.-X. Wang, Corresponding author)

Received  July 2017 Revised  May 2018 Published  October 2018

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: Yu-Xia Wang, Wan-Tong Li. Combined effects of the spatial heterogeneity and the functional response. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 19-39. doi: 10.3934/dcds.2019002
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. CuiJ. 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. DeAngelisR. 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. DockeryV. HutsonK. 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. DuR. 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. HutsonY. LouK. 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. LouS. 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. ZengJ. 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. CuiJ. 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. DeAngelisR. 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. DockeryV. HutsonK. 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. DuR. 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. HutsonY. LouK. 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. LouS. 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. ZengJ. 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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