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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

C. B. Huffaker, Experimental studies on predator: Dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

C. B. Huffaker, Experimental studies on predator: Dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019214

[2]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[3]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[4]

Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831

[5]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026

[6]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189

[7]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173

[8]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701

[9]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005

[10]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

[11]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[12]

Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101

[13]

Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141

[14]

Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185

[15]

Arnaud Ducrot, Vincent Guyonne, Michel Langlais. Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 67-82. doi: 10.3934/dcdss.2011.4.67

[16]

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124

[17]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[18]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[19]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209

[20]

Henri Berestycki, Alessandro Zilio. Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7141-7162. doi: 10.3934/dcds.2019299

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (153)
  • HTML views (137)
  • Cited by (0)

Other articles
by authors

[Back to Top]