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

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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

Abstract
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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.

Keywords: prey, adaptive dispersal, persistence, Predator, bifurcation., population dynamics.

Mathematics Subject Classification: Primary: 92D25; Secondary: 34C2.

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.
[2]	D. M. Buss and H. Greiling, Adaptive individual differences,, Journal of Personality, 67 (1999), 209. doi: 10.1111/1467-6494.00053.
[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.
[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.
[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.
[6]	S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology, and Conservation,, Princeton University Press, (2002).
[7]	R. Cressman and V. Křivan, Migration dynamics for the ideal free distribution,, The American Naturalist, 168 (2006), 384. doi: 10.1086/506970.
[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.
[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.
[10]	S. D. Fretwell and J. S. Calver, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37.
[11]	J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 388. doi: 10.1137/0520025.
[12]	J. Hofbauer and K. Sigmund, Evolutionray Games and Population Dynamics,, Cambridge University Press, (1998). doi: 10.1017/CBO9781139173179.
[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.
[14]	V. Křivan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs,, The American Naturalist, 170 (2007), 771.
[15]	V. Křivan and R. Cressman, On evolutionary stability in predator-prey models with fast behavioral dynamics,, Evolutionary Ecology Research, 11 (2009), 227.
[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.
[17]	S. A. Levin, Dispersion and population interactions,, The American Naturalist, 108 (1974), 207. doi: 10.1086/282900.
[18]	J. D. Murray, Mathematical Biology, I, An Introduction,, Springer, (2002).
[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.
[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.
[21]	J. E. Staddon, Adaptive Behaviour and Learning,, CUP Archive, (1983).
[22]	C. Starr, R. Taggart, C. Evers and L. Starr, Biology: The Unity And Diversity of Life,, Yolanda Cossio, (2009).
[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.
[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.
[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.

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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.
[2]	D. M. Buss and H. Greiling, Adaptive individual differences,, Journal of Personality, 67 (1999), 209. doi: 10.1111/1467-6494.00053.
[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.
[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.
[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.
[6]	S. Creel and N. M. Dreel, The African Wild Dog: Behavior, Ecology, and Conservation,, Princeton University Press, (2002).
[7]	R. Cressman and V. Křivan, Migration dynamics for the ideal free distribution,, The American Naturalist, 168 (2006), 384. doi: 10.1086/506970.
[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.
[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.
[10]	S. D. Fretwell and J. S. Calver, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37.
[11]	J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 388. doi: 10.1137/0520025.
[12]	J. Hofbauer and K. Sigmund, Evolutionray Games and Population Dynamics,, Cambridge University Press, (1998). doi: 10.1017/CBO9781139173179.
[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.
[14]	V. Křivan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs,, The American Naturalist, 170 (2007), 771.
[15]	V. Křivan and R. Cressman, On evolutionary stability in predator-prey models with fast behavioral dynamics,, Evolutionary Ecology Research, 11 (2009), 227.
[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.
[17]	S. A. Levin, Dispersion and population interactions,, The American Naturalist, 108 (1974), 207. doi: 10.1086/282900.
[18]	J. D. Murray, Mathematical Biology, I, An Introduction,, Springer, (2002).
[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.
[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.
[21]	J. E. Staddon, Adaptive Behaviour and Learning,, CUP Archive, (1983).
[22]	C. Starr, R. Taggart, C. Evers and L. Starr, Biology: The Unity And Diversity of Life,, Yolanda Cossio, (2009).
[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.
[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.
[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.

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