December  2014, 19(10): 3397-3432. doi: 10.3934/dcdsb.2014.19.3397

Evolution of mobility in predator-prey systems

1. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada, Canada

2. 

Biology Centre ASCR, Institute of Entomology and Department of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia, Branišovská 31, 370 05 České Budějovice, Czech Republic

Received  June 2013 Revised  October 2013 Published  October 2014

We investigate the dynamics of a predator-prey system with the assumption that both prey and predators use game theory-based strategies to maximize their per capita population growth rates. The predators adjust their strategies in order to catch more prey per unit time, while the prey, on the other hand, adjust their reactions to minimize the chances of being caught. We assume each individual is either mobile or sessile and investigate the evolution of mobility for each species in the predator-prey system. When the underlying population dynamics is of the Lotka-Volterra type, we show that strategies evolve to the equilibrium predicted by evolutionary game theory and that population sizes approach their corresponding stable equilibrium (i.e. strategy and population effects can be analyzed separately). This is no longer the case when population dynamics is based on the Holling II functional response, although the strategic analysis still provides a valuable intuition into the long term outcome. Numerical simulation results indicate that, for some parameter values, the system has chaotic behavior. Our investigation reveals the relationship between the game theory-based reactions of prey and predators, and their population changes.
Citation: Fei Xu, Ross Cressman, Vlastimil Křivan. Evolution of mobility in predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3397-3432. doi: 10.3934/dcdsb.2014.19.3397
References:
[1]

P. A. Abrams, Foraging time optimization and interactions in food webs,, Am Nat, 124 (1984), 80. doi: 10.1086/284253. Google Scholar

[2]

P. A. Abrams, The impact of habitat selection on the heterogeneity of resources in varying environments,, Ecol, 81 (2000), 2902. doi: 10.2307/177350. Google Scholar

[3]

P. A. Abrams, Habitat choice in predator-prey systems: Spatial instability due to interacting adaptive movements,, Am Nat, 169 (2007), 581. doi: 10.1086/512688. Google Scholar

[4]

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

[5]

K. Argasinski, Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept,, Math Biosci, 202 (2006), 88. doi: 10.1016/j.mbs.2006.04.007. Google Scholar

[6]

L. Arnold, W. Horsthemke and J. W. Stucki, The influence of external real and white noise on the Lotka-Volterra model,, Biom. J., 21 (1979), 451. doi: 10.1002/bimj.4710210507. Google Scholar

[7]

J. S. Brown and B. P. Kotler, Hazardous duty pay and the foraging cost of predation,, Ecol Lett, 7 (2004), 999. doi: 10.1111/j.1461-0248.2004.00661.x. Google Scholar

[8]

J. S. Brown, J. W. Laundré and M. Gurung, The ecology of fear: Optimal foraging, game theory, and trophic interactions,, J Mammal, 80 (1999), 385. doi: 10.2307/1383287. Google Scholar

[9]

E. L. Charnov, Optimal foraging: Attack strategy of a mantid,, Am Nat, 110 (1976), 141. doi: 10.1086/283054. Google Scholar

[10]

R. Cressman, Evolutionary Dynamics and Extensive Form Games,, MIT Press, (2003). Google Scholar

[11]

R. Cressman and J. Garay, The effects of opportunistic and intentional predators on the herding behavior of prey,, Ecol, 92 (2011), 432. doi: 10.1890/10-0199.1. Google Scholar

[12]

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

[13]

M. M. Dehn, Vigilance for predators: Detection and dilution effects,, Behav. Ecol. Sociobiol., 26 (1990), 337. Google Scholar

[14]

F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes,, Princeton University Press, (2008). Google Scholar

[15]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes,, J. Math. Biol., 34 (1996), 579. doi: 10.1007/BF02409751. Google Scholar

[16]

W. A. Foster and J. E. Treherne, Evidence for the dilution effect in the selfish herd from fish predation on a marine insect,, Nature, 293 (1981), 466. doi: 10.1038/293466a0. Google Scholar

[17]

D. Fudenberg and D. K. Levine, The Theory of Learning in Games,, MIT Press, (1998). Google Scholar

[18]

G. F. Gause, The Struggle for Existence,, Williams and Wilkins, (1934). doi: 10.1097/00010694-193602000-00018. Google Scholar

[19]

S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol., 12 (1998), 35. Google Scholar

[20]

J. Hofbauer and E. Hopkins, Learning in perturbed asymmetric games,, Games Econ Behav, 52 (2005), 133. doi: 10.1016/j.geb.2004.06.006. Google Scholar

[21]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems,, Cambridge University Press, (1988). Google Scholar

[22]

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

[23]

C. S. Holling, Some characteristics of simple types of predation and parasitism,, Can. Entomol., 91 (1959), 385. doi: 10.4039/Ent91385-7. Google Scholar

[24]

R. Huey and E. R. Pianka, Ecological consequences of foraging mode,, Ecol, 62 (1981), 991. doi: 10.2307/1936998. Google Scholar

[25]

V. Křivan, Optimal foraging and predator-prey dynamics,, Theor Popul Biol, 49 (1996), 265. Google Scholar

[26]

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

[27]

V. Křivan and E. Sirot, Habitat selection by two competing species in a two-habitat environment,, Am Nat, 160 (2002), 214. Google Scholar

[28]

J. H. Lü, G. R. Chen and S. C. Zhang, Dynamical analysis of a new chaotic attractor,, Int. J. Bifur. Chaos Appl. Sci. Eng., 12 (2002), 1001. Google Scholar

[29]

R. H. MacArthur and E. R. Pianka, On optimal use of a patchy environment,, Am Nat, 100 (1966), 603. doi: 10.1086/282454. Google Scholar

[30]

M. Parker and A. Kamenev, Mean extinction time in predator-prey model,, J Stat Phys, 141 (2010), 201. doi: 10.1007/s10955-010-0049-y. Google Scholar

[31]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions,, Am Nat, 97 (1963), 209. doi: 10.1086/282272. Google Scholar

[32]

L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games,, J. Econ. Theory, 57 (1992), 363. doi: 10.1016/0022-0531(92)90041-F. Google Scholar

[33]

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

[34]

O. J. Schmitz, Behavior of predators and prey and links with population level processes., In Ecology of Predator-Prey Interactions (eds. P. Barbosa and I. Castellanos ), (2005), 256. Google Scholar

[35]

O. J. Schmitz, V. Křivan and O. Ovadia, Trophic cascades: The primacy of trait-mediated indirect interactions,, Ecol Lett, 7 (2004), 153. doi: 10.1111/j.1461-0248.2003.00560.x. Google Scholar

[36]

T. W. Schoener, Theory of feeding strategies,, Annu Rev Ecol Syst, 2 (1971), 369. doi: 10.1146/annurev.es.02.110171.002101. Google Scholar

[37]

J. G. Skellam, The mathematical foundations underlying the use of line transects in animal ecology,, Biometrics, 14 (1958), 385. doi: 10.2307/2527881. Google Scholar

[38]

D. W. Stephens and J. R. Krebs, Foraging Theory,, Princeton University Press, (1986). Google Scholar

[39]

T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511542633. Google Scholar

[40]

E. E. Werner and B. R. Anholt, Ecological consequences of the trade-off between growth and mortality rates mediated by foraging activity,, Am Nat, 142 (1993), 242. doi: 10.1086/285537. Google Scholar

[41]

W. B. Yapp, The theory of line transects,, Bird Study, 3 (1956), 93. doi: 10.1080/00063655609475840. Google Scholar

show all references

References:
[1]

P. A. Abrams, Foraging time optimization and interactions in food webs,, Am Nat, 124 (1984), 80. doi: 10.1086/284253. Google Scholar

[2]

P. A. Abrams, The impact of habitat selection on the heterogeneity of resources in varying environments,, Ecol, 81 (2000), 2902. doi: 10.2307/177350. Google Scholar

[3]

P. A. Abrams, Habitat choice in predator-prey systems: Spatial instability due to interacting adaptive movements,, Am Nat, 169 (2007), 581. doi: 10.1086/512688. Google Scholar

[4]

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

[5]

K. Argasinski, Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept,, Math Biosci, 202 (2006), 88. doi: 10.1016/j.mbs.2006.04.007. Google Scholar

[6]

L. Arnold, W. Horsthemke and J. W. Stucki, The influence of external real and white noise on the Lotka-Volterra model,, Biom. J., 21 (1979), 451. doi: 10.1002/bimj.4710210507. Google Scholar

[7]

J. S. Brown and B. P. Kotler, Hazardous duty pay and the foraging cost of predation,, Ecol Lett, 7 (2004), 999. doi: 10.1111/j.1461-0248.2004.00661.x. Google Scholar

[8]

J. S. Brown, J. W. Laundré and M. Gurung, The ecology of fear: Optimal foraging, game theory, and trophic interactions,, J Mammal, 80 (1999), 385. doi: 10.2307/1383287. Google Scholar

[9]

E. L. Charnov, Optimal foraging: Attack strategy of a mantid,, Am Nat, 110 (1976), 141. doi: 10.1086/283054. Google Scholar

[10]

R. Cressman, Evolutionary Dynamics and Extensive Form Games,, MIT Press, (2003). Google Scholar

[11]

R. Cressman and J. Garay, The effects of opportunistic and intentional predators on the herding behavior of prey,, Ecol, 92 (2011), 432. doi: 10.1890/10-0199.1. Google Scholar

[12]

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

[13]

M. M. Dehn, Vigilance for predators: Detection and dilution effects,, Behav. Ecol. Sociobiol., 26 (1990), 337. Google Scholar

[14]

F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes,, Princeton University Press, (2008). Google Scholar

[15]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes,, J. Math. Biol., 34 (1996), 579. doi: 10.1007/BF02409751. Google Scholar

[16]

W. A. Foster and J. E. Treherne, Evidence for the dilution effect in the selfish herd from fish predation on a marine insect,, Nature, 293 (1981), 466. doi: 10.1038/293466a0. Google Scholar

[17]

D. Fudenberg and D. K. Levine, The Theory of Learning in Games,, MIT Press, (1998). Google Scholar

[18]

G. F. Gause, The Struggle for Existence,, Williams and Wilkins, (1934). doi: 10.1097/00010694-193602000-00018. Google Scholar

[19]

S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evol. Ecol., 12 (1998), 35. Google Scholar

[20]

J. Hofbauer and E. Hopkins, Learning in perturbed asymmetric games,, Games Econ Behav, 52 (2005), 133. doi: 10.1016/j.geb.2004.06.006. Google Scholar

[21]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems,, Cambridge University Press, (1988). Google Scholar

[22]

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

[23]

C. S. Holling, Some characteristics of simple types of predation and parasitism,, Can. Entomol., 91 (1959), 385. doi: 10.4039/Ent91385-7. Google Scholar

[24]

R. Huey and E. R. Pianka, Ecological consequences of foraging mode,, Ecol, 62 (1981), 991. doi: 10.2307/1936998. Google Scholar

[25]

V. Křivan, Optimal foraging and predator-prey dynamics,, Theor Popul Biol, 49 (1996), 265. Google Scholar

[26]

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

[27]

V. Křivan and E. Sirot, Habitat selection by two competing species in a two-habitat environment,, Am Nat, 160 (2002), 214. Google Scholar

[28]

J. H. Lü, G. R. Chen and S. C. Zhang, Dynamical analysis of a new chaotic attractor,, Int. J. Bifur. Chaos Appl. Sci. Eng., 12 (2002), 1001. Google Scholar

[29]

R. H. MacArthur and E. R. Pianka, On optimal use of a patchy environment,, Am Nat, 100 (1966), 603. doi: 10.1086/282454. Google Scholar

[30]

M. Parker and A. Kamenev, Mean extinction time in predator-prey model,, J Stat Phys, 141 (2010), 201. doi: 10.1007/s10955-010-0049-y. Google Scholar

[31]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions,, Am Nat, 97 (1963), 209. doi: 10.1086/282272. Google Scholar

[32]

L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games,, J. Econ. Theory, 57 (1992), 363. doi: 10.1016/0022-0531(92)90041-F. Google Scholar

[33]

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

[34]

O. J. Schmitz, Behavior of predators and prey and links with population level processes., In Ecology of Predator-Prey Interactions (eds. P. Barbosa and I. Castellanos ), (2005), 256. Google Scholar

[35]

O. J. Schmitz, V. Křivan and O. Ovadia, Trophic cascades: The primacy of trait-mediated indirect interactions,, Ecol Lett, 7 (2004), 153. doi: 10.1111/j.1461-0248.2003.00560.x. Google Scholar

[36]

T. W. Schoener, Theory of feeding strategies,, Annu Rev Ecol Syst, 2 (1971), 369. doi: 10.1146/annurev.es.02.110171.002101. Google Scholar

[37]

J. G. Skellam, The mathematical foundations underlying the use of line transects in animal ecology,, Biometrics, 14 (1958), 385. doi: 10.2307/2527881. Google Scholar

[38]

D. W. Stephens and J. R. Krebs, Foraging Theory,, Princeton University Press, (1986). Google Scholar

[39]

T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511542633. Google Scholar

[40]

E. E. Werner and B. R. Anholt, Ecological consequences of the trade-off between growth and mortality rates mediated by foraging activity,, Am Nat, 142 (1993), 242. doi: 10.1086/285537. Google Scholar

[41]

W. B. Yapp, The theory of line transects,, Bird Study, 3 (1956), 93. doi: 10.1080/00063655609475840. Google Scholar

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