# American Institute of Mathematical Sciences

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 and 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), 80-96. doi: 10.1086/284253. [2] P. A. Abrams, The impact of habitat selection on the heterogeneity of resources in varying environments, Ecol, 81 (2000), 2902-2913. doi: 10.2307/177350. [3] P. A. Abrams, Habitat choice in predator-prey systems: Spatial instability due to interacting adaptive movements, Am Nat, 169 (2007), 581-594. doi: 10.1086/512688. [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), 505-518. doi: 10.1086/511963. [5] K. Argasinski, Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept, Math Biosci, 202 (2006), 88-114. doi: 10.1016/j.mbs.2006.04.007. [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), 451-471. doi: 10.1002/bimj.4710210507. [7] J. S. Brown and B. P. Kotler, Hazardous duty pay and the foraging cost of predation, Ecol Lett, 7 (2004), 999-1014. doi: 10.1111/j.1461-0248.2004.00661.x. [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), 385-399. doi: 10.2307/1383287. [9] E. L. Charnov, Optimal foraging: Attack strategy of a mantid, Am Nat, 110 (1976), 141-151. doi: 10.1086/283054. [10] R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press, Cambridge, MA, USA, 2003. [11] R. Cressman and J. Garay, The effects of opportunistic and intentional predators on the herding behavior of prey, Ecol, 92 (2011), 432-440. doi: 10.1890/10-0199.1. [12] R. Cressman and V. Křivan, Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, J Math Biol, 67 (2013), 329-358. doi: 10.1007/s00285-012-0548-3. [13] M. M. Dehn, Vigilance for predators: Detection and dilution effects, Behav. Ecol. Sociobiol., 26 (1990), 337-342. [14] F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes, Princeton University Press, Princeton, USA, 2008. [15] U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612. doi: 10.1007/BF02409751. [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), 466-467. doi: 10.1038/293466a0. [17] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press, Cambridge, MA, USA, 1998. [18] G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934. doi: 10.1097/00010694-193602000-00018. [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), 35-57. [20] J. Hofbauer and E. Hopkins, Learning in perturbed asymmetric games, Games Econ Behav, 52 (2005), 133-152. doi: 10.1016/j.geb.2004.06.006. [21] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988. [22] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179. [23] C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398. doi: 10.4039/Ent91385-7. [24] R. Huey and E. R. Pianka, Ecological consequences of foraging mode, Ecol, 62 (1981), 991-999. doi: 10.2307/1936998. [25] V. Křivan, Optimal foraging and predator-prey dynamics, Theor Popul Biol, 49 (1996), 265-290. [26] V. Křivan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs, Am Nat, 170 (2007), 771-782. [27] V. Křivan and E. Sirot, Habitat selection by two competing species in a two-habitat environment, Am Nat, 160 (2002), 214-234. [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), 1001-1015. [29] R. H. MacArthur and E. R. Pianka, On optimal use of a patchy environment, Am Nat, 100 (1966), 603-609. doi: 10.1086/282454. [30] M. Parker and A. Kamenev, Mean extinction time in predator-prey model, J Stat Phys, 141 (2010), 201-216. doi: 10.1007/s10955-010-0049-y. [31] M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am Nat, 97 (1963), 209-223. doi: 10.1086/282272. [32] L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games, J. Econ. Theory, 57 (1992), 363-391. doi: 10.1016/0022-0531(92)90041-F. [33] I. Scharf, E. Nulman, O. Ovadia and A. Bouskila, Efficiency evaluation of two competing foraging modes under different conditions, Am Nat, 168 (2006), 350-357. doi: 10.1086/506921. [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 ), 256-278, Oxford University Press, 2005. [35] O. J. Schmitz, V. Křivan and O. Ovadia, Trophic cascades: The primacy of trait-mediated indirect interactions, Ecol Lett, 7 (2004), 153-163. doi: 10.1111/j.1461-0248.2003.00560.x. [36] T. W. Schoener, Theory of feeding strategies, Annu Rev Ecol Syst, 2 (1971), 369-404. doi: 10.1146/annurev.es.02.110171.002101. [37] J. G. Skellam, The mathematical foundations underlying the use of line transects in animal ecology, Biometrics, 14 (1958), 385-400. doi: 10.2307/2527881. [38] D. W. Stephens and J. R. Krebs, Foraging Theory, Princeton University Press, Princeton, New Jersey, USA, 1986. [39] T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, Cambridge, UK, 2005. doi: 10.1017/CBO9780511542633. [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), 242-272. doi: 10.1086/285537. [41] W. B. Yapp, The theory of line transects, Bird Study, 3 (1956), 93-104. doi: 10.1080/00063655609475840.

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##### References:
 [1] P. A. Abrams, Foraging time optimization and interactions in food webs, Am Nat, 124 (1984), 80-96. doi: 10.1086/284253. [2] P. A. Abrams, The impact of habitat selection on the heterogeneity of resources in varying environments, Ecol, 81 (2000), 2902-2913. doi: 10.2307/177350. [3] P. A. Abrams, Habitat choice in predator-prey systems: Spatial instability due to interacting adaptive movements, Am Nat, 169 (2007), 581-594. doi: 10.1086/512688. [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), 505-518. doi: 10.1086/511963. [5] K. Argasinski, Dynamic multipopulation and density dependent evolutionary games related to replicator dynamics. A metasimplex concept, Math Biosci, 202 (2006), 88-114. doi: 10.1016/j.mbs.2006.04.007. [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), 451-471. doi: 10.1002/bimj.4710210507. [7] J. S. Brown and B. P. Kotler, Hazardous duty pay and the foraging cost of predation, Ecol Lett, 7 (2004), 999-1014. doi: 10.1111/j.1461-0248.2004.00661.x. [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), 385-399. doi: 10.2307/1383287. [9] E. L. Charnov, Optimal foraging: Attack strategy of a mantid, Am Nat, 110 (1976), 141-151. doi: 10.1086/283054. [10] R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press, Cambridge, MA, USA, 2003. [11] R. Cressman and J. Garay, The effects of opportunistic and intentional predators on the herding behavior of prey, Ecol, 92 (2011), 432-440. doi: 10.1890/10-0199.1. [12] R. Cressman and V. Křivan, Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds, J Math Biol, 67 (2013), 329-358. doi: 10.1007/s00285-012-0548-3. [13] M. M. Dehn, Vigilance for predators: Detection and dilution effects, Behav. Ecol. Sociobiol., 26 (1990), 337-342. [14] F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes, Princeton University Press, Princeton, USA, 2008. [15] U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612. doi: 10.1007/BF02409751. [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), 466-467. doi: 10.1038/293466a0. [17] D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press, Cambridge, MA, USA, 1998. [18] G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, 1934. doi: 10.1097/00010694-193602000-00018. [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), 35-57. [20] J. Hofbauer and E. Hopkins, Learning in perturbed asymmetric games, Games Econ Behav, 52 (2005), 133-152. doi: 10.1016/j.geb.2004.06.006. [21] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988. [22] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179. [23] C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385-398. doi: 10.4039/Ent91385-7. [24] R. Huey and E. R. Pianka, Ecological consequences of foraging mode, Ecol, 62 (1981), 991-999. doi: 10.2307/1936998. [25] V. Křivan, Optimal foraging and predator-prey dynamics, Theor Popul Biol, 49 (1996), 265-290. [26] V. Křivan, The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs, Am Nat, 170 (2007), 771-782. [27] V. Křivan and E. Sirot, Habitat selection by two competing species in a two-habitat environment, Am Nat, 160 (2002), 214-234. [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), 1001-1015. [29] R. H. MacArthur and E. R. Pianka, On optimal use of a patchy environment, Am Nat, 100 (1966), 603-609. doi: 10.1086/282454. [30] M. Parker and A. Kamenev, Mean extinction time in predator-prey model, J Stat Phys, 141 (2010), 201-216. doi: 10.1007/s10955-010-0049-y. [31] M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am Nat, 97 (1963), 209-223. doi: 10.1086/282272. [32] L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games, J. Econ. Theory, 57 (1992), 363-391. doi: 10.1016/0022-0531(92)90041-F. [33] I. Scharf, E. Nulman, O. Ovadia and A. Bouskila, Efficiency evaluation of two competing foraging modes under different conditions, Am Nat, 168 (2006), 350-357. doi: 10.1086/506921. [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 ), 256-278, Oxford University Press, 2005. [35] O. J. Schmitz, V. Křivan and O. Ovadia, Trophic cascades: The primacy of trait-mediated indirect interactions, Ecol Lett, 7 (2004), 153-163. doi: 10.1111/j.1461-0248.2003.00560.x. [36] T. W. Schoener, Theory of feeding strategies, Annu Rev Ecol Syst, 2 (1971), 369-404. doi: 10.1146/annurev.es.02.110171.002101. [37] J. G. Skellam, The mathematical foundations underlying the use of line transects in animal ecology, Biometrics, 14 (1958), 385-400. doi: 10.2307/2527881. [38] D. W. Stephens and J. R. Krebs, Foraging Theory, Princeton University Press, Princeton, New Jersey, USA, 1986. [39] T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, Cambridge, UK, 2005. doi: 10.1017/CBO9780511542633. [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), 242-272. doi: 10.1086/285537. [41] W. B. Yapp, The theory of line transects, Bird Study, 3 (1956), 93-104. doi: 10.1080/00063655609475840.
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