March  2016, 21(2): 677-697. doi: 10.3934/dcdsb.2016.21.677

On a two-patch predator-prey model with adaptive habitancy of predators

1. 

Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada

2. 

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  July 2014 Revised  January 2015 Published  November 2015

A two-patch predator-prey model with the Holling type II functional response is studied, in which predators are assumed to adopt adaptive dispersal to inhabit the better patch in order to gain more fitness. Analytical conditions for the persistence and extinction of predators are obtained under different scenarios of the model. Numerical simulations are conducted which show that adaptive dispersal can stabilize the system with either weak or strong adaptation, when prey and predators tend to a globally stable equilibrium in one isolated patch and tend to limit cycles in the other. Furthermore, it is observed that the adaptive dispersal may cause torus bifurcation for the model when the prey and predators population tend to limit cycles in each isolated patch.
Citation: Xiaoying Wang, Xingfu Zou. On a two-patch predator-prey model with adaptive habitancy of predators. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 677-697. doi: 10.3934/dcdsb.2016.21.677
References:
[1]

P. A. Abrams, R. Cressman and V. Křivan, The role of behavioral dynamics in determining the patch distributions of interacting species,, The American Naturalist, 169 (2007), 505.  doi: 10.1086/511963.  Google Scholar

[2]

D. M. Buss and H. Greiling, Adaptive individual differences,, Journal of Personality, 67 (1999), 209.  doi: 10.1111/1467-6494.00053.  Google Scholar

[3]

R. S. Cantrell, C. Cosner, D. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy,, Journal of Biological Dynamics, 1 (2007), 249.  doi: 10.1080/17513750701450227.  Google Scholar

[4]

R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments,, Journaal of Mathematical Biology, 65 (2012), 943.  doi: 10.1007/s00285-011-0486-5.  Google Scholar

[5]

M. E. Clark, T. G. Wolcott, D. L. Wolcott and A. H. Hines, Foraging behavior of an estuarine predator, the blue crab Callinectes sapidus in a patchy environment,, Ecography, 23 (2000), 21.   Google Scholar

[6]

S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology, and Conservation,, Princeton University Press, (2002).   Google Scholar

[7]

R. Cressman and V. Křivan, Migration dynamics for the ideal free distribution,, The American Naturalist, 168 (2006), 384.  doi: 10.1086/506970.  Google Scholar

[8]

R. Cressman and V. Křivan, Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds,, Journal of Mathematical Biology, 67 (2013), 329.  doi: 10.1007/s00285-012-0548-3.  Google Scholar

[9]

R. Cressman, V. Křivan and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments,, The American Naturalist, 164 (2004), 473.  doi: 10.1086/423827.  Google Scholar

[10]

S. D. Fretwell and J. S. Calver, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37.   Google Scholar

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 388.  doi: 10.1137/0520025.  Google Scholar

[12]

J. Hofbauer and K. Sigmund, Evolutionray Games and Population Dynamics,, Cambridge University Press, (1998).  doi: 10.1017/CBO9781139173179.  Google Scholar

[13]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation,, Memoirs of the Entomological Society of Canada, 97 (1965), 5.  doi: 10.4039/entm9745fv.  Google Scholar

[14]

V. Křivan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs,, The American Naturalist, 170 (2007), 771.   Google Scholar

[15]

V. Křivan and R. Cressman, On evolutionary stability in predator-prey models with fast behavioral dynamics,, Evolutionary Ecology Research, 11 (2009), 227.   Google Scholar

[16]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments,, Mathematical Biosciences, 120 (1994), 77.  doi: 10.1016/0025-5564(94)90038-8.  Google Scholar

[17]

S. A. Levin, Dispersion and population interactions,, The American Naturalist, 108 (1974), 207.  doi: 10.1086/282900.  Google Scholar

[18]

J. D. Murray, Mathematical Biology, I, An Introduction,, Springer, (2002).   Google Scholar

[19]

I. Scharf, E. Nulman, O. Ovadia and A. Bouskila, Efficiency evaluation of two competing foraging modes under different conditions,, The American Naturalist, 168 (2006), 350.  doi: 10.1086/506921.  Google Scholar

[20]

R. D. Seitz, R. N. Lipcius, A. H. Hines and D. B. Eggleston, Density-dependent predation, habitat variation, and the persistence of marine bivalve prey,, Ecology, 82 (2001), 2435.   Google Scholar

[21]

J. E. Staddon, Adaptive Behaviour and Learning,, CUP Archive, (1983).   Google Scholar

[22]

C. Starr, R. Taggart, C. Evers and L. Starr, Biology: The Unity And Diversity of Life,, Yolanda Cossio, (2009).   Google Scholar

[23]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM Journal on Mathematical Analysis, 24 (1993), 407.  doi: 10.1137/0524026.  Google Scholar

[24]

W. Wang and Y. Takeuchi, Adaptation of prey and predators between patches,, Journal of Theoretical Biology, 258 (2009), 603.  doi: 10.1016/j.jtbi.2009.02.014.  Google Scholar

[25]

D. K. Wasko and M. Sasa, Food resources influence spatial ecology, habitat selection, and foraging behavior in an ambush-hunting snake (Viperidae: Bothrops asper): an experimental study,, Zoology, 115 (2012), 179.  doi: 10.1016/j.zool.2011.10.001.  Google Scholar

show all references

References:
[1]

P. A. Abrams, R. Cressman and V. Křivan, The role of behavioral dynamics in determining the patch distributions of interacting species,, The American Naturalist, 169 (2007), 505.  doi: 10.1086/511963.  Google Scholar

[2]

D. M. Buss and H. Greiling, Adaptive individual differences,, Journal of Personality, 67 (1999), 209.  doi: 10.1111/1467-6494.00053.  Google Scholar

[3]

R. S. Cantrell, C. Cosner, D. L. Deangelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy,, Journal of Biological Dynamics, 1 (2007), 249.  doi: 10.1080/17513750701450227.  Google Scholar

[4]

R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments,, Journaal of Mathematical Biology, 65 (2012), 943.  doi: 10.1007/s00285-011-0486-5.  Google Scholar

[5]

M. E. Clark, T. G. Wolcott, D. L. Wolcott and A. H. Hines, Foraging behavior of an estuarine predator, the blue crab Callinectes sapidus in a patchy environment,, Ecography, 23 (2000), 21.   Google Scholar

[6]

S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology, and Conservation,, Princeton University Press, (2002).   Google Scholar

[7]

R. Cressman and V. Křivan, Migration dynamics for the ideal free distribution,, The American Naturalist, 168 (2006), 384.  doi: 10.1086/506970.  Google Scholar

[8]

R. Cressman and V. Křivan, Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds,, Journal of Mathematical Biology, 67 (2013), 329.  doi: 10.1007/s00285-012-0548-3.  Google Scholar

[9]

R. Cressman, V. Křivan and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments,, The American Naturalist, 164 (2004), 473.  doi: 10.1086/423827.  Google Scholar

[10]

S. D. Fretwell and J. S. Calver, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37.   Google Scholar

[11]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 388.  doi: 10.1137/0520025.  Google Scholar

[12]

J. Hofbauer and K. Sigmund, Evolutionray Games and Population Dynamics,, Cambridge University Press, (1998).  doi: 10.1017/CBO9781139173179.  Google Scholar

[13]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation,, Memoirs of the Entomological Society of Canada, 97 (1965), 5.  doi: 10.4039/entm9745fv.  Google Scholar

[14]

V. Křivan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs,, The American Naturalist, 170 (2007), 771.   Google Scholar

[15]

V. Křivan and R. Cressman, On evolutionary stability in predator-prey models with fast behavioral dynamics,, Evolutionary Ecology Research, 11 (2009), 227.   Google Scholar

[16]

Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two-patch environments,, Mathematical Biosciences, 120 (1994), 77.  doi: 10.1016/0025-5564(94)90038-8.  Google Scholar

[17]

S. A. Levin, Dispersion and population interactions,, The American Naturalist, 108 (1974), 207.  doi: 10.1086/282900.  Google Scholar

[18]

J. D. Murray, Mathematical Biology, I, An Introduction,, Springer, (2002).   Google Scholar

[19]

I. Scharf, E. Nulman, O. Ovadia and A. Bouskila, Efficiency evaluation of two competing foraging modes under different conditions,, The American Naturalist, 168 (2006), 350.  doi: 10.1086/506921.  Google Scholar

[20]

R. D. Seitz, R. N. Lipcius, A. H. Hines and D. B. Eggleston, Density-dependent predation, habitat variation, and the persistence of marine bivalve prey,, Ecology, 82 (2001), 2435.   Google Scholar

[21]

J. E. Staddon, Adaptive Behaviour and Learning,, CUP Archive, (1983).   Google Scholar

[22]

C. Starr, R. Taggart, C. Evers and L. Starr, Biology: The Unity And Diversity of Life,, Yolanda Cossio, (2009).   Google Scholar

[23]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM Journal on Mathematical Analysis, 24 (1993), 407.  doi: 10.1137/0524026.  Google Scholar

[24]

W. Wang and Y. Takeuchi, Adaptation of prey and predators between patches,, Journal of Theoretical Biology, 258 (2009), 603.  doi: 10.1016/j.jtbi.2009.02.014.  Google Scholar

[25]

D. K. Wasko and M. Sasa, Food resources influence spatial ecology, habitat selection, and foraging behavior in an ambush-hunting snake (Viperidae: Bothrops asper): an experimental study,, Zoology, 115 (2012), 179.  doi: 10.1016/j.zool.2011.10.001.  Google Scholar

[1]

Wei Feng, Jody Hinson. Stability and pattern in two-patch predator-prey population dynamics. Conference Publications, 2005, 2005 (Special) : 268-279. doi: 10.3934/proc.2005.2005.268

[2]

Peter A. Braza. Predator-Prey Dynamics with Disease in the Prey. Mathematical Biosciences & Engineering, 2005, 2 (4) : 703-717. doi: 10.3934/mbe.2005.2.703

[3]

Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114

[4]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[5]

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

[6]

Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046

[7]

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

[8]

Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020132

[9]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877

[10]

Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020180

[11]

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

[12]

Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020130

[13]

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

[14]

Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Wendi Wang. Population dispersal and disease spread. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 797-804. doi: 10.3934/dcdsb.2004.4.797

[20]

Peter A. Braza. A dominant predator, a predator, and a prey. Mathematical Biosciences & Engineering, 2008, 5 (1) : 61-73. doi: 10.3934/mbe.2008.5.61

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]