Abstract
The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, “triple-threshold” strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.
Similar content being viewed by others
References
Baguette, M., & Van Dyck, H. (2007). Landscape connectivity and animal behavior: functional grain as a key determinant for dispersal. Landsc. Ecol., 22, 1117–1129.
Balkau, B. J., & Feldman, M. W. (1973). Selection for migration modification. Genetics, 74, 171–174.
Bull, J. J., Thompson, C., Ng, D., & Moore, R. (1987). A model for natural selection of genetic migration. Am. Nat., 129, 143–157.
Cadet, C., Ferrière, R., Metz, J. A. J., & van Baalen, M. (2003). The evolution of dispersal under demographic stochasticity. Am. Nat., 162, 427–441.
Chitty, D. (1967). The natural selection of self-regulatory behavior in animal populations. Proc. Ecol. Soc. Aust., 2, 51–78.
Comins, H. N. (1985). Evolutionarily stable dispersal strategies for localized dispersal in two dimensions. J. Theor. Biol., 94, 579–606.
Comins, H. N., Hamilton, W. D., & May, R. M. (1980). Evolutionarily stable dispersal strategies. J. Theor. Biol., 82, 205–230.
Dieckmann, U., & Law, R. (1996). The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol., 34, 579–612.
Dieckmann, U., Heino, M., & Parvinen, K. (2006). The adaptive dynamics of function-valued traits. J. Theor. Biol., 241, 370–389.
Dobson, F. S., & Jones, W. T. (1985). Multiple causes of dispersal. Am. Nat., 126, 855–858.
Doebeli, M., & Ruxton, G. D. (1997). Evolution of dispersal rates in metapopulation models: branching and cyclic dynamics in phenotype space. Evolution, 51, 1730–1741.
Durinx, M., Metz, J. A. J., & Meszéna, G. (2008). Adaptive dynamics for physiologically structured population models. J. Math. Biol., 56, 673–742.
Errington, P. L. (1946). Predation and vertebrate populations. Q. Rev. Biol., 21, 144–177.
Gandon, S., & Michalakis, Y. (2001). Multiple causes of the evolution of dispersal. In J. Clobert, E. Danchin, A. A. Dhondt, & J. D. Nichols (Eds.), Dispersal (pp. 155–167). London: Oxford University Press.
Geritz, S. A. H., Metz, J. A. J., Kisdi, É., & Meszéna, G. (1997). Dynamics of adaptation and evolutionary branching. Phys. Rev. Lett., 78, 2024–2027.
Geritz, S. A. H., Kisdi, É., Meszéna, G., & Metz, J. A. J. (1998). Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol., 12, 35–57.
Geritz, S., Gyllenberg, M., & Ondráček, P. (2009). Evolution of density-dependent dispersal in a structured metapopulation. Math. Biosci., 219, 142–148.
Greenwood, P. J., Harvey, P. H., & Perrins, C. M. (1978). Inbreeding and dispersal in great tit. Nature, 271, 52–54.
Gyllenberg, M., & Metz, J. A. J. (2001). On fitness in structured metapopulations. J. Math. Biol., 43, 545–560.
Gyllenberg, M., Parvinen, K., & Dieckmann, U. (2002). Evolutionary suicide and evolution of dispersal in structured metapopulations. J. Math. Biol., 45, 79–105.
Gyllenberg, M., Kisdi, E., & Utz, M. (2008). Evolution of condition-dependent dispersal under kin competition. J. Math. Biol., 57(2), 285–307.
Gyllenberg, M., Kisdi, É., & Utz, M. (2011a). Body condition dependent dispersal in a heterogeneous environment. Theor. Popul. Biol., 79, 139–154.
Gyllenberg, M., Kisdi, É., & Utz, M. (2011b). Variability within families and the evolution of body condition dependent dispersal. J. Biol. Dyn., 5, 191–211.
Hamilton, W. D. (1964a). The genetical evolution of social behaviour. I. J. Theor. Biol., 7, 1–16.
Hamilton, W. D. (1964b). The genetical evolution of social behaviour. II. J. Theor. Biol., 7, 17–52.
Hamilton, W. D., & May, R. M. (1977). Dispersal in stable habitats. Nature, 269, 578–581.
Hanski, I. (2005). The shrinking world: ecological consequences of habitat loss. Oldendorf/Luhe: International Ecology Institute.
Hanski, I., & Mononen, T. (2011). Eco-evolutionary dynamics of dispersal in spatially heterogeneous environments. Ecol. Lett., 14, 1025–1034.
Hastings, A. (1983). Can spatial variation alone lead to selection for dispersal? Theor. Popul. Biol., 24, 244–251.
Hepburn, H. R. (2006). Absconding, migration and swarming in honeybees: an ecological and evolutionary perspective. In V. E. Kipyatkov (Ed.), Life cycles in social insects: behaviour, ecology and evolution (pp. 121–135). St. Petersburg: St. Petersburg University Press.
Holt, R. D. (1985). Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution. Theor. Popul. Biol., 28, 181–208.
Holt, R. D., & McPeek, M. (1996). Chaotic population dynamics favors the evolution of dispersal. Am. Nat., 148, 709–718.
Kisdi, É. (2002). Dispersal: risk spreading versus local adaptation. Am. Nat., 159, 579–596.
Kisdi, É. (2004). Conditional dispersal under kin competition: extension of the Hamilton–May model brood size-dependent dispersal. Theor. Popul. Biol., 66, 369–380.
Korona, R. (1991). Genetic-basis of behavioral strategies—dispersal of female flour beetles, Tribolium confusum, in a laboratory system. Oikos, 62(3), 265–270.
Krebs, C. J., Wingate, I., Leduc, J., Redfield, J., Taitt, M., & Hilborn, R. (1976). Microtus population biology—dispersal in fluctuating populations of Microtus townsendii. Can. J. Zool., 54, 79–95.
Kuno, E. (1981). Dispersal and the persistence of populations in unstable habitats: a theoretical note. Oecologia, 49, 123–126.
Lack, D. (1966). Population studies of birds. Oxford: Oxford University Press.
Lambin, X., Aars, J., & Piertney, S. B. (2001). Dispersal, intraspecific competition, kin competition and kin facilitation: a review of the empirical evidence. In J. Clobert, E. Danchin, A. A. Dhondt, & J. D. Nichols (Eds.), Dispersal (pp. 110–122). London: Oxford University Press.
Lawson-Handley, L. J., & Perrin, N. (2007). Advances in our understanding of mammalian sex-biased dispersal. Mol. Ecol., 16, 1559–1578.
Le Galliard, J.-F., Ferriére, R., & Dieckmann, U. (2005). Adaptive evolution of social traits: origin, trajectories and correlations of altruism and mobility. Am. Nat., 165, 206–224.
Matthysen, E. (2005). Density-dependent dispersal in birds and mammals. Ecography, 28, 403–416.
Mayr, E. (1963). Animal species and evolution. London: Oxford University Press.
Metz, J. A. J., & Gyllenberg, M. (2001). How should we define fitness in structured metapopulation models? Including an application to the calculation of ES dispersal strategies. Proc. R. Soc. Lond. B, Biol. Sci., 268, 499–508.
Metz, J. A. J., Jong, T. J., & Klinkhamer, P. G. L. (1983). What are the advantages of dispersing: a paper by Kuno explained and extended. Oecologia, 57, 166–169.
Metz, J. A. J., Nisbet, R. M., & Geritz, S. A. H. (1992). How should we define “fitness” for general ecological scenarios? Trends Ecol. Evol., 7, 198–202.
Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. J. A., & van Heerwaarden, J. S. (1996). Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In S. J. van Strien & S. M. Verduyn Lunel (Eds.), Stochastic and spatial structures of dynamical systems (pp. 183–231). Amsterdam: North-Holland.
Moore, J., & Ali, R. (1982). Are dispersal and inbreeding avoidance related? Anim. Behav., 32, 94–112.
Motro, U. (1982a). Optimal rates of dispersal I. Haploid populations. Theor. Popul. Biol., 21, 394–411.
Motro, U. (1982b). Optimal rates of dispersal II. Diploid populations. Theor. Popul. Biol., 21, 412–429.
Motro, U. (1983). Optimal rates of dispersal III. Parent offspring conflict. Theor. Popul. Biol., 23, 159–168.
Nagy, J. D. (1996). Evolutionarily attracting dispersal strategies in vertebrate metapopulations. Ph.D. thesis, Arizona State University, Tempe, AZ, USA.
Nurmi, T., & Parvinen, K. (2011). Joint evolution of specialization and dispersal in structured metapopulations. J. Theor. Biol., 275, 78–92.
Ogden, J. C. (1970a). Artificial selection for dispersal in flour beetles (tenebrionidae: Tribolium). Ecology, 51, 130–133.
Ogden, J. C. (1970b). Aspects of dispersal in tribolium flour beetles. Physiol. Zool., 42, 124–131.
Parvinen, K. (1999). Evolution of migration in a metapopulation. Bull. Math. Biol., 61, 531–550.
Parvinen, K. (2002). Evolutionary branching of dispersal strategies in structured metapopulations. J. Math. Biol., 45, 106–124.
Parvinen, K. (2006). Evolution of dispersal in a structured metapopulation model in discrete time. Bull. Math. Biol., 68, 655–678.
Parvinen, K. (2011). Adaptive dynamics of altruistic cooperation in a metapopulation: evolutionary emergence of cooperators and defectors or evolutionary suicide? Bull. Math. Biol., 73, 2605–2626.
Parvinen, K., & Metz, J. A. J. (2008). A novel fitness proxy in structured locally finite metapopulations with diploid genetics, with an application to dispersal evolution. Theor. Popul. Biol., 73, 517–528.
Parvinen, K., Dieckmann, U., Gyllenberg, M., & Metz, J. A. J. (2003). Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity. J. Evol. Biol., 16, 143–153.
Parvinen, K., Dieckmann, U., & Heino, M. (2006). Function-valued adaptive dynamics and the calculus of variations. J. Math. Biol., 52, 1–26.
Parvinen, K., Heino, M., & Dieckmann, U. (2012). Function-valued adaptive dynamics and optimal control theory. J. Math. Biol. doi:10.1007/s00285-012-0549-2.
Peacock, M. M., & Smith, A. T. (1997). The effect of habitat fragmentation on dispersal patterns, mating behavior, and genetic variation in a pika (Ochotona princeps) metapopulation. Oecologia, 112, 524–533.
Perrin, N., & Goudet, J. (2001). Inbreeding, kinship, and the evolution of natal dispersal. In J. Clobert, E. Danchin, A. A. Dhondt, & J. D. Nichols (Eds.), Dispersal (pp. 123–142). London: Oxford University Press.
Perrins, C. (2008). Survival of young swifts in relation to brood size. Nature, 201, 1147–1148.
Roff, D. (1977). Dispersal in dipterans—its costs and consequences. J. Anim. Ecol., 46, 443–456.
Roff, D. A., & Fairbairn, D. J. (2001). The genetic basis of dispersal and migration, and its consequences for the evolution of correlated traits. In J. Clobert, E. Danchin, A. A. Dhondt, & J. D. Nichols (Eds.), Dispersal (pp. 191–202). London: Oxford University Press.
Ronce, O., & Olivieri, I. (2004). Life history evolution in metapopulations. In I. Hanski & O. E. Gaggiotti (Eds.), Ecology, genetics, and evolution of metapopulations (pp. 227–257). Amsterdam: Elsevier.
Ronce, O., Perret, F., & Olivieri, I. (2000a). Evolutionarily stable dispersal rates do not always increase with local extinction rates. Am. Nat., 155, 485–496.
Ronce, O., Perret, F., & Olivieri, I. (2000b). Landscape dynamics and evolution of colonizer syndromes: interactions between reproductive effort and dispersal in a metapopulation. Evol. Ecol., 14, 233–260.
Smith, A. T. (1974a). The distribution and dispersal of pikas: consequences of insular population structure. Ecology, 55, 1112–1119.
Smith, A. T. (1974b). The distribution and dispersal of pikas: influences of behaviour and climate. Ecology, 55, 1368–1376.
Smith, A. T. (1980). Temporal changes in insular populations of the pika (ochotona princeps). Ecology, 61, 8–13.
Teague, R. (1977). A model of migration modification. Theor. Popul. Biol., 12, 86–94.
Van Valen, L. (1971). Group selection and the evolution of dispersal. Evolution, 25, 591–598.
Williams, G. C. (1966). Adaptation and natural selection. Princeton: Princeton University Press.
Acknowledgements
Authors wish to thank two anonymous reviewers for valuable remarks to improve this manuscript. This study was funded by the Academy of Finland, project number 128323 to K.P.
Author information
Authors and Affiliations
Corresponding author
Appendix: Relatedness and the Benefit of Dispersal
Appendix: Relatedness and the Benefit of Dispersal
According to Hamilton’s rule (Hamilton 1964a, 1964b), a costly altruistic behavior can evolve if the cost to the actor, C, is less than R g ⋅B, where R g is the genetic relatedness between the actor and the recipient, and B is the fitness benefit to the recipient. To study how kin-cooperation and kin-competition act as evolutionary forces driving dispersal rates in the metapopulation, we therefore need to define a relatedness measure. Costs and benefits can be measured in terms of the fitness proxy (Metz and Gyllenberg 2001), which is the expected amount of successful mutant dispersers emigrated from a clan initiated by a single mutant immigrant over the life span of the colony:
where w n,n′ is the expected amount of time that an average clan will spend in a state with n residents and n′ mutants. For a detailed explanation and a method to calculate w n,n′, see Metz and Gyllenberg (2001) and Parvinen and Metz (2008). Relatedness must therefore also be defined in terms of the clan in the special case that the mutant and resident strategies are the same (note that therefore we are only considering an approximation where we assume that the difference of the mutant and resident is so slight that we can neglect the alteration of the population dynamics brought about by the presence of mutants). Our density-dependent relatedness measure is the probability throughout the life-time of the colony that a randomly chosen individual is a mutant, with the condition that the local population size in the patch at the sampling moment is equal to k:
We can decompose the invasion fitness proxy into parts
where p k is the probability that in a metapopulation-dynamical resident equilibrium, a randomly chosen patch has k residents. Furthermore, the conditional production E k is the expected amount of successful mutant dispersers emigrated from a clan which initially consists of k−1 residents and a single mutant.
Consider a local population with exactly k individuals, of which at least one is a mutant. Suppose that a specific mutant in this local population is given the option to emigrate immediately. The direct difference in benefits and costs of a positive emigration decision is 1−E k : 1 for the emigrating mutant, and −E k for the loss of expected average disperser production of a clan founded by this individual at the decision moment. The indirect effect comes through possible relatives. The expected amount of relatives (i.e., mutants) present in the patch at the decision moment is ρ k k, of which 1 is the specific individual itself. The fitness difference for each relative is E k−1−E k , because the population size in the patch after the emigration decision will be k−1 if the specific mutant emigrates and k if it does not. To summarize, the additional benefits of adult dispersal in a local population with size k compared with the nondispersing case are
Benefits in natal dispersal goes slightly differently. Consider again a mutant in a local population of k individuals, but now at the moment of birth of one offspring to the mutant. Now the parent (or the offspring) is given the choice of making the offspring to disperse or not. If the offspring disperses, the direct fitness benefits of the parent and offspring together are 1+E k , and the costs are 2E k+1. The indirect benefits to the possible relatives are E k −E k+1 for each. In total, the additional benefits for a parent that sends its offspring to disperse, compared to parent that does not, are
Because we assume clonal reproduction, there is no parent-offspring conflict here.
In principle, if these additional benefits are positive in a local population of size k, the fitness gradient for the emigration strategy component e k should be positive, and vice versa. However, the quantities (9) and (10) do not take into account how w n,n′, the expected time that a mutant clan spends in a state with n residents and n′ mutants, will change when the strategy of the mutant changes. Thus, the quantities (9) and (10) are only indicative, but can to some extent explain observed dispersal behavior in the model. The complete calculation is taken into account in the actual calculation of the fitness gradient, derived by Parvinen (2011).
Rights and permissions
About this article
Cite this article
Parvinen, K., Seppänen, A. & Nagy, J.D. Evolution of Complex Density-Dependent Dispersal Strategies. Bull Math Biol 74, 2622–2649 (2012). https://doi.org/10.1007/s11538-012-9770-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-012-9770-9