- Email: [email protected]
On the evolutionary stability of partial migration
Journal of Theoretical Biology 321 (2013) 36–39
Contents lists available at SciVerse ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
On the evolutionary stability of partial migration Per Lundberg n Department of Biology, Theoretical Population Ecology and Evolutionary Biology Group, Ecology Building, Lund University, SE-223 62 Lund, Sweden
H I G H L I G H T S c c c
The evolutionary solution to the partial migration game is not necessarily an ESS. Individual-based evolutionary solution deviates from the optimal solution for the population. Partial migration is unlikely to evolve without frequency- or density-dependent selection.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2012 Received in revised form 17 December 2012 Accepted 18 December 2012 Available online 7 January 2013
The evolution of partial migration in birds is typically assumed to be the result of an optimization process. The fitness rewards for individuals choosing to migrate are balanced against the rewards of remaining in the breeding area all year around. This balancing is often thought to result in an evolutionarily stable strategy (ESS) such that an optimal fraction of the population becomes migratory through adaptive evolution. Here I show that this solution can indeed be reached through adaptive evolution, but that the equilibrium is a neutral or ‘‘weak’’ ESS. The equilibrium fraction of migrants is more reminiscent of the Fisherian sex ratio. I also show that this individual-based evolutionary solution may deviate significantly from the optimal solution for the population (maximum population size), quite in line with previous findings. Finally, I show that partial migration is very unlikely without density- or frequency-dependent selection. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Partial migration Evolution Frequency-dependence Density-dependence ESS
1. Introduction In strongly seasonal environments, some periods of the year offer ample resources for survival and reproduction, while others put the basics of life to a challenge. In order to cope with adverse times, organisms have evolved three principal strategies: to migrate, to become dormant (hibernate or diapause or any other stage of inactivity and lowered metabolism), or simply to remain in the habitat, often associated with for example, changes in diet, movement patterns or micro-habitat use. Birds show a great variety of wintering and migration strategies— from residency and strong site fidelity in the harshest of climates to long-distance migration at continental scales (Berthold, 1996). One particularly intriguing strategy is partial migration, i.e., that a fraction of a breeding population is sedentary throughout the year, whereas the other part of the population undertakes a bona fide migration to wintering areas some distance away from the breeding ground to return there the following spring (e.g., Lundberg, 1988; Berthold, ¨ 1996; Alerstam and Hedenstrom, 1998; Sekercioglu, 2010). The mechanisms behind partial migration have been much debated and there seems to be an evidence in favor of both genetic control
and largely condition dependent determination of the fraction of migrants in the population each year (e.g., Lundberg, 1987, 1988; Kaitala et al. 1993; Berthold, 1996; Pulido et al., 1996; Alerstam and Hedenstr¨om, 1998; Kokko and Lundberg, 2001; Sekercioglu, 2010). The balance of the fitness rewards for the alternative strategies is, however, not always obvious and the evolutionary game played by migrants and residents remains incompletely understood. The point of departure for previous theoretical analyses of partial migration in birds (Lundberg, 1987, 1988; Kaitala et al., 1993; Kokko and Lundberg, 2001; Taylor and Norris, 2007; Griswold et al,. 2010; Holt and Fryxell, 2011; Kokko, 2011; Shaw and Levin, 2011) has been that negative frequency-dependence (of strategy choice) must be an important mechanism to maintain the balance between fitness costs and gains, regardless of whether it is under direct or indirect genetic control. The intention with this note is to investigate a somewhat neglected aspect of partial migration, namely its evolutionary stability. I use a simple population model with the usual frequency- and density-dependent ingredients and investigate the fitness balance between alternative strategies.
2. Model and results n
Tel.: þ46 46 222 3767; fax: þ 46 70 2522 111. E-mail address: [email protected]
0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.12.017
The strategy game of partial migration amounts to determine the evolutionarily stable probability that an individual will
P. Lundberg / Journal of Theoretical Biology 321 (2013) 36–39
migrate. By ‘‘evolutionarily stable’’, it should be understood that if members of the population should migrate with that probability, no alternative would give individuals higher fitness (it is the optimal probability) and hence not be able to invade the population. We will assume that the population dynamics follows a simple discrete process with two age classes from one year (t) to another (t þ1): Nt þ 1 ¼ sA Nt þbN t sJ
ð1Þ
where sA is adult annual (overwintering) survival probability and sJ is the juvenile annual survival probability, and b is the per capita number of births. I assume that adult survival is both frequency- and density-dependent so that the survival terms in Eq. (1) become sA,M ¼ sA,M,max sM N t p sA,R ¼ sA,R,max sR N t ð1pÞ
ð2Þ
where the subscripts M and R represent migrants and sedentary individuals, respectively. The parameter sA,i,max is the maximal per capita survival rates of adults and si is the strength of (negative) density-dependence. The probability for an individual to migrate is denoted as p. Eq. (2) represents, for simplicity, linear frequencyand density-dependence but these functions can take any suitable form. Juvenile survival is assumed to represent the survival from birth (in spring) to the following fall and equal for juveniles born to resident and migrant parents. It will further be assumed that reproduction is strategy-specific such that bR and bM represent the birth rates of residents and migrants, respectively. If there is no frequency- or density-dependence, the only solution will be complete residency or complete migration (Appendix 1). If we now use Eq. (2) in (1) and the assumption about strategyspecific birth rates then the population dynamics takes the form: Nt þ 1 ¼ sA,M,max sM Nt p pN t þ sA,R,max sR Nt ð1pÞÞð1pÞN t ð3Þ þsJ ðbM p þ bR ð1pÞÞNt The change in population size will depend on the strategyspecific vital rates, but also the proportion of migrants (p) in the population. We are now asking which value of p is evolutionary stable by using standard evolutionary invasion analysis (e.g., Geritz et al., 1998; Kokko, 2007; Vincent and Brown, 2005; Morris and Lundberg, 2011). Let N t þ 1 =Nt ¼ l be mean absolute fitness (per capita population growth rate). At equilibrium l ¼1, i.e., there is no change in population size (density) from one year to another and we let Nn denote the population size at equilibrium. In order to calculate Nn, we use Eq. (3) and knowing that Nt þ 1/Nt ¼1 and after some algebra we have: Nn ¼
sA,M,max p þ sA,R,max ð1pÞ þ sJ ðbM p þbR ð1pÞÞ1
sM p2 þ sR ð1pÞ2
37
value of p—not to be confused with ‘‘resident’’ in the sense of the sedentary strategy). We start with a resident population at equilibrium and consequently l(Nn, p)¼1. That is, the per capita growth rate (fitness) of individuals with migration strategy p at population equilibrium Nn equals unity (by definition). Now, let a rare mutant with strategy value p0 attempt to invade the resident population. This mutant will have a fitness equal to: l p’,Nn ðp Þ ¼ p’ sA,M,max sMM Nn p þ ð1p’ÞsA,R þ ð1p’Þ sA,R,max sR Nn ð1pÞÞ þ sJ ðbM p’ þbR ð1p’ÞÞ
ð5Þ
Note the difference between the p0 indicating an alternative candidate strategy value and p, the value of the fraction of migrants in the current equilibrium population. In order to find the value of p0 that maximizes the invader strategy fitness, we calculate the derivative of l with respect to p0 : @l ¼ sA,M,max sM N n p sA,R,max sR Nn ð1pÞ þ sj ðbM bR Þ @p’
ð6Þ
Eq. (6) describes how invader fitness changes with the alternative (mutant) value of migration probability. Fig. 1 shows how this fitness gradient, which we call g(p), changes for all values of p. As shown, the gradient is positive for smaller values of p, but negative for larger values. At a given intermediate value (between 0.6 and 0.7 in Fig. 1) the gradient is zero which means that there is potentially an evolutionarily stable p-value. Alternative pvalues smaller than the equilibrium value have positive growth and there is selection for higher values, and the reverse for higher p-values than the equilibrium. Can adaptive evolution take the population to that point? The convergence stability criterion (e.g., Eshel, 1983; Geritz et al., 1998; Morris and Lundberg, 2011) is the answer and says that the derivative of the gradient (with respect to p) must be negative. That is, that the equilibrium point is a maximum of the fitness function (Eq. (5)). The derivative of the fitness gradient is rather straightforward to calculate and we get: @g ¼ Nn ðsM þ sR Þ @p
ð7Þ
This expression will always be negative (all the terms in Eq. (7) must be positive) which means that the evolutionary endpoint is convergence stable. The last criterion is that the endpoint should be an ESS (uninvadable) and not something else, e.g., a potential evolutionary branching point from which two different strategies
ð4Þ
In order to calculate the evolutionarily stable fraction of migrants in the population, we now use Eq. (4) as the state of the population (the evolutionary feedback environment (e.g., Heino et al., 1997; Morris and Lundberg, 2011)) in which possible alternative strategies may arise and invade. 2.1. Invasion analysis The invasion analysis is done such that, we assume that the population is at equilibrium (Eq. (4)) and is characterized by a migration strategy p. We then assume that an alternative strategy p0 attempts to invade such a population. Assuming that the alternative strategy initially is rare, it only suffers density- and frequencydependent feedback from individuals in the resident population (‘‘resident’’ here meaning the population characterized by a particular
Fig. 1. The fitness gradient (slope of the fitness function) for an example set of parameter values. The evolutionary equilibrium probability to migrate (pn) is found where the gradient is 0 (here between 0.6 and 0.7). The parameter values are: bM,max ¼3, bR,max ¼ 7, sA,M,max ¼ 0.4, sA,R,max ¼0.7, sM ¼0.01, sR ¼ 0.02, sJ ¼ 0.5.
38
P. Lundberg / Journal of Theoretical Biology 321 (2013) 36–39
can diverge (Abrams and Matsuda, 1993; Geritz et al., 1998; Vincent and Brown, 2005). This is found by examining the second derivative of the fitness gradient (@2 g=@p2 ). When this derivative is negative, we have an ESS and when it is positive, we have a fitness minimum and a branching point. In our case the derivative is exactly zero (p does not occur in Eq. (7)). We call this case a ‘‘weak’’ ESS. It is indeed convergence stable, but it is similar to neutral stability and it is not a branching point. It means that the strategy is the optimal one when invading a resident population. It is also convergence stable (attainable through adaptive evolution), but the resident strategy pn can again be invaded by any mutant. However, this mutant will not be able to grow in numbers because then the resident strategy is once again the optimal strategy. This is analogous to Fisher0 s sex ratio problem (Maynard Smith, 1989). Fig. 2 shows the pair-wise invisability plot (see Kokko, 2007 for a lucid treatment of pair-wise invasibility) of the system illustrating this problem. An additional feature of this partial migration system is the discrepancy between the evolutionarily optimal solution and the solution that maximizes population size. Fig. 3 illustrates the phenomenon. For a wide range of parameter values (here, the per capita birth rate of the migrants is arbitrarily chosen) the p-value that maximizes population size deviates considerably from the evolutionary equilibrium (Fig. 4). This is typically a rather mild problem in this system, but there are circumstances when this can lead as far as to ‘‘evolutionary suicide’’ (cf. Kokko, 2011). This means that adaptive evolution can take the population to dangerously small population sizes.
Fig. 3. The equilibrium population density (N*) as a function of the probability of migrating (p). The maximum N* (here at p E0.53, denoted by an open circle) deviates from the evolutionary solution denoted by a filled circle (at p E0.64).
3. Discussion Partial migration has received considerable interest in later years after quite some time in the shadow (see for example Milner-Gulland et al., 2011 and Chapman et al., 2011 and several
Fig. 4. The deviation between the evolutionary equilibrium p-value and the pvalue maximizing N* is shown as a function of the maximum per capita birth rate of the migrant strategy. The residents have a birth rate¼ 5. Positive (negative) deviations indicate Nn 4pn (Nn o pn).
Fig. 2. The pair-wise invasibility plot (PIP) for the same parameter setting as in Fig. 1. The x-axis is the evolutionary resident strategy value (migration probability) and the y-axis is the strategy value of an alternative (mutant). Fitness is higher for an alternative strategy above and to the left of the one-to-one line and the vertical line at pnresident E0.63, respectively. Fitness is also higher for an alternative below and to the right of the one-to-one line, respectively. Regions with higher mutant fitness are shaded gray with a plus sign. Regions with smaller mutant fitness are white with a minus sign. The equilibrium p is found at the intersection of the one-to-one line and the vertical line in the fitness plane (black dot). The equilibrium is not an ESS (or what could be called a ‘‘weak’’ ESS), but adaptive evolution takes the population there. At equilibrium, any mutant strategy value is equal to the resident value (moving along the vertical isofitness line).
papers in the same issue of Oikos). The first theoretical foundations laid out more than 20 years ago (e.g., Lundberg, 1987, 1988) have been refined and extended quite thoroughly. Territoriality, genetics, energetics and explicit environmental stochasticity have become components of the theoretical framework that now also goes beyond migratory birds. Partial migration is now studied in a wide range of animal taxa (Chapman et al., 2011). Elements of the theoretical framework are empirically testable. Claims that partial migration is, in a formal sense, an ESS solution remains to be shown. Here, I have used a simple and general population model with no assumptions about many of the real life complications in natural populations (e.g., size structure, waiting games, energetics) to show that the putative ESS solutions from many similar models are not strictly speaking ESSs. Whether the ‘‘weak’’ ESS here discovered is a real phenomenon again remains to be shown empirically. The model used here is reasonably generic but the strict linear density-dependence contributes to the stability point being neutral. The properties of the stability point will be difficult to establish empirically but one prediction from this model is that there should be quite some
P. Lundberg / Journal of Theoretical Biology 321 (2013) 36–39
variation of the fraction of migrants over time due, not the least, to environmental stochasticity. In a recent paper, Shaw and Levin (2011) showed that partial migration indeed can be a true ESS, and they also showed how environmental stochasticity can contribute to the evolution of partial migration. The model used by Shaw and Levin (2011) was however, very different from the very simple one used here and they were able to demonstrate how different types of partial migration can evolve by going beyond the classic ‘‘bird type’’ (migration between breeding and wintering grounds, the latter with no reproduction) analyzed here. Lundberg (1987) showed that environmental stochasticity might influence the equilibrium proportion of migrants in the population, but not the existence of an ESS solution. Kaitala et al. (1993) showed that environmental stochasticity is not necessary for partial migration to evolve, at least not to an ESS. In order to attain an ESS fraction of migrants, frequency- or density-dependence must cause evolutionary feedback regardless of environmental variability. As in many fields of biology, there is sometimes a wide abyss between the theoretical development of the field and the data available to test the theoretical assertions. The results presented in this note demonstrate that it is not necessarily an ESS in the strict and formal sense we should be looking for in partially migratory bird populations. There are many models of partial migration subsumed under the notion of optimization and straightforward adaptive evolution of stable fractions of migrants and non-migrants. More empirical data on vital rates and the establishment that putative migrants truly are so would help when deciding how we should view partial migration from a more conceptual point of view.
Acknowledgments I thank Jacob Johansson for illuminating discussions. The Centre for Animal Movement Research at the Biology Department, Lund University, contributed by helping me to bring this problem to my renewed attention. Financial support for this study was received from the Swedish Research Council and the marine ecology group at Gothenburg University provided a stimulating and generous environment when parts of this work were completed. I am grateful for the insightful and important comments of two reviewers for the manuscript.
Appendix 1 We will assume that the population dynamics of the entire population follows the same simple discrete process with two age classes from one year (t) to another (tþ1) as in Eq. (1). With no density-dependence the population model can be written as: Nt þ 1 ¼ Nt p sA,M þbM sJ,M þN t ð1pÞ sA,R þbR sJ,R ðA1Þ The subscripts A and J denote adults and juveniles, respectively and M and R denote migrants and non-migrants, respectively. Again, let N t þ 1 =Nt ¼ l be mean absolute fitness (per capita population
39
growth rate), and after having rewritten Eq. (A1) we have:
l ¼ psA,M þ pbM sJ,M þ ð1pÞsA,R þ ð1pÞbR sJ,R
¼ p sA,M þ bM sJ,M þ ð1pÞ sA,R þ bR sJ,R ¼ p sA,M þ bM sJ,M p sA,R þbR sJ,R þ sA,R þ bR sJ,R
ðA2Þ
To find the optimal strategy (i.e., the optimal probability for an individual to migrate, pn) take the partial derivative of l with respect to p, @l ¼ sA,M þ bM sJ,M sA,R þ bR sJ,R @p
ðA3Þ
The only fitness maxima are pn ¼1 or 0, depending on whether (sA, M þbMsJ,M) or (sA,R þbRsJ, R) is the largest term. That is, all individuals will either migrate or stay as non-migrants. This illustrates that partial migration is unlikely to evolve without frequency- or density-dependent reproduction or survival. Environmental stochasticity alone is also not enough for the evolution of partial migration (Kaitala et al., 1993). References Abrams, P., Matsuda, H., 1993. Effects of adaptive predatory and antipredatory behavior in a 2-prey one-predator system. Evol. Ecol. 7, 312–326. ¨ Alerstam, T., Hedenstrom, A., 1998. The development of bird migration theory. J. Avian Biol. 29, 343–369. Berthold, P., 1996. Control of Bird Migration. Chapman & Hall, London. Chapman, B.B., et al., 2011. The ecology and evolution of partial migration. Oikos 120, 1764–1775. Eshel, I., 1983. Evolutionary and continuous stability. J. Theor. Biol. 103, 99–111. Geritz, S., et al., 1998. Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12, 35–57. Griswold, C.K., et al., 2010. The evolution of migration in a seasonal environment. Proc. R. Soc. London B 277, 2711–2720. Heino, M., et al., 1997. Evolution of mixed maturation strategies in semelparous life histories: the crucial role of dimensionality of feedback environment. Philos. Trans. R. Soc. B 352, 1647–1655. Holt, R.D., Fryxell, J.M., 2011. Theoretical reflections on the evolution of migration. In: Milner-Gulland, E.J., et al. (Eds.), Animal Migration. Oxford University Press, Oxford, pp. 17–31. Kaitala, A., et al., 1993. A theory of partial migration. Am. Nat. 142, 59–81. Kokko, H., 2007. Modelling for Field Biologists. Cambridge University Press, Cambridge. Kokko, H., 2011. Directions in modelling partial migration: how adaptation can cause a population decline and why the rules of territory acquisition matter. Oikos 120, 1826–1837. Kokko, H., Lundberg, P., 2001. Dispersal, migration and offspring retention in saturated habitats. Am. Nat. 157, 188–202. Lundberg, P., 1987. Partial bird migration and evolutionarily stable strategies. J. Theor. Biol. 125 351-340. Lundberg, P., 1988. The evolution of partial migration in birds. Trends Ecol. Evol. 3, 172–175. Milner-Gulland, E.J. (Ed.), Oxford University Press. Morris, D.W., Lundberg, P., 2011. Pillars of Evolution. Oxford University Press, Oxford. Maynard Smith, J., 1989. Evolutionary Genetics. Oxford University Press, Oxford. Pulido, F., et al., 1996. Frequency of migrants and migratory activity are genetically correlated in a bird population. Proc. Natl. Acad. Sci. USA 93, 14642–14647. Sekercioglu, C., 2010. Partial migration in birds: the frontier of movement ecology. J. Anim. Ecol. 79, 933–936. Shaw, A.K., Levin, S.A., 2011. To breed or not to breed: a model of partial migration. Oikos 120, 1871–1879. Taylor, C.M., Norris, D.R., 2007. Predicting conditions for migration: effects of density dependence and habitat quality. Biol. Lett. 3, 280–283. Vincent, T.L., Brown, J.S., 2005. Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics Cambridge University Press, Cambridge.
Contents lists available at SciVerse ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
On the evolutionary stability of partial migration Per Lundberg n Department of Biology, Theoretical Population Ecology and Evolutionary Biology Group, Ecology Building, Lund University, SE-223 62 Lund, Sweden
H I G H L I G H T S c c c
The evolutionary solution to the partial migration game is not necessarily an ESS. Individual-based evolutionary solution deviates from the optimal solution for the population. Partial migration is unlikely to evolve without frequency- or density-dependent selection.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2012 Received in revised form 17 December 2012 Accepted 18 December 2012 Available online 7 January 2013
The evolution of partial migration in birds is typically assumed to be the result of an optimization process. The fitness rewards for individuals choosing to migrate are balanced against the rewards of remaining in the breeding area all year around. This balancing is often thought to result in an evolutionarily stable strategy (ESS) such that an optimal fraction of the population becomes migratory through adaptive evolution. Here I show that this solution can indeed be reached through adaptive evolution, but that the equilibrium is a neutral or ‘‘weak’’ ESS. The equilibrium fraction of migrants is more reminiscent of the Fisherian sex ratio. I also show that this individual-based evolutionary solution may deviate significantly from the optimal solution for the population (maximum population size), quite in line with previous findings. Finally, I show that partial migration is very unlikely without density- or frequency-dependent selection. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Partial migration Evolution Frequency-dependence Density-dependence ESS
1. Introduction In strongly seasonal environments, some periods of the year offer ample resources for survival and reproduction, while others put the basics of life to a challenge. In order to cope with adverse times, organisms have evolved three principal strategies: to migrate, to become dormant (hibernate or diapause or any other stage of inactivity and lowered metabolism), or simply to remain in the habitat, often associated with for example, changes in diet, movement patterns or micro-habitat use. Birds show a great variety of wintering and migration strategies— from residency and strong site fidelity in the harshest of climates to long-distance migration at continental scales (Berthold, 1996). One particularly intriguing strategy is partial migration, i.e., that a fraction of a breeding population is sedentary throughout the year, whereas the other part of the population undertakes a bona fide migration to wintering areas some distance away from the breeding ground to return there the following spring (e.g., Lundberg, 1988; Berthold, ¨ 1996; Alerstam and Hedenstrom, 1998; Sekercioglu, 2010). The mechanisms behind partial migration have been much debated and there seems to be an evidence in favor of both genetic control
and largely condition dependent determination of the fraction of migrants in the population each year (e.g., Lundberg, 1987, 1988; Kaitala et al. 1993; Berthold, 1996; Pulido et al., 1996; Alerstam and Hedenstr¨om, 1998; Kokko and Lundberg, 2001; Sekercioglu, 2010). The balance of the fitness rewards for the alternative strategies is, however, not always obvious and the evolutionary game played by migrants and residents remains incompletely understood. The point of departure for previous theoretical analyses of partial migration in birds (Lundberg, 1987, 1988; Kaitala et al., 1993; Kokko and Lundberg, 2001; Taylor and Norris, 2007; Griswold et al,. 2010; Holt and Fryxell, 2011; Kokko, 2011; Shaw and Levin, 2011) has been that negative frequency-dependence (of strategy choice) must be an important mechanism to maintain the balance between fitness costs and gains, regardless of whether it is under direct or indirect genetic control. The intention with this note is to investigate a somewhat neglected aspect of partial migration, namely its evolutionary stability. I use a simple population model with the usual frequency- and density-dependent ingredients and investigate the fitness balance between alternative strategies.
2. Model and results n
Tel.: þ46 46 222 3767; fax: þ 46 70 2522 111. E-mail address: [email protected]
0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.12.017
The strategy game of partial migration amounts to determine the evolutionarily stable probability that an individual will
P. Lundberg / Journal of Theoretical Biology 321 (2013) 36–39
migrate. By ‘‘evolutionarily stable’’, it should be understood that if members of the population should migrate with that probability, no alternative would give individuals higher fitness (it is the optimal probability) and hence not be able to invade the population. We will assume that the population dynamics follows a simple discrete process with two age classes from one year (t) to another (t þ1): Nt þ 1 ¼ sA Nt þbN t sJ
ð1Þ
where sA is adult annual (overwintering) survival probability and sJ is the juvenile annual survival probability, and b is the per capita number of births. I assume that adult survival is both frequency- and density-dependent so that the survival terms in Eq. (1) become sA,M ¼ sA,M,max sM N t p sA,R ¼ sA,R,max sR N t ð1pÞ
ð2Þ
where the subscripts M and R represent migrants and sedentary individuals, respectively. The parameter sA,i,max is the maximal per capita survival rates of adults and si is the strength of (negative) density-dependence. The probability for an individual to migrate is denoted as p. Eq. (2) represents, for simplicity, linear frequencyand density-dependence but these functions can take any suitable form. Juvenile survival is assumed to represent the survival from birth (in spring) to the following fall and equal for juveniles born to resident and migrant parents. It will further be assumed that reproduction is strategy-specific such that bR and bM represent the birth rates of residents and migrants, respectively. If there is no frequency- or density-dependence, the only solution will be complete residency or complete migration (Appendix 1). If we now use Eq. (2) in (1) and the assumption about strategyspecific birth rates then the population dynamics takes the form: Nt þ 1 ¼ sA,M,max sM Nt p pN t þ sA,R,max sR Nt ð1pÞÞð1pÞN t ð3Þ þsJ ðbM p þ bR ð1pÞÞNt The change in population size will depend on the strategyspecific vital rates, but also the proportion of migrants (p) in the population. We are now asking which value of p is evolutionary stable by using standard evolutionary invasion analysis (e.g., Geritz et al., 1998; Kokko, 2007; Vincent and Brown, 2005; Morris and Lundberg, 2011). Let N t þ 1 =Nt ¼ l be mean absolute fitness (per capita population growth rate). At equilibrium l ¼1, i.e., there is no change in population size (density) from one year to another and we let Nn denote the population size at equilibrium. In order to calculate Nn, we use Eq. (3) and knowing that Nt þ 1/Nt ¼1 and after some algebra we have: Nn ¼
sA,M,max p þ sA,R,max ð1pÞ þ sJ ðbM p þbR ð1pÞÞ1
sM p2 þ sR ð1pÞ2
37
value of p—not to be confused with ‘‘resident’’ in the sense of the sedentary strategy). We start with a resident population at equilibrium and consequently l(Nn, p)¼1. That is, the per capita growth rate (fitness) of individuals with migration strategy p at population equilibrium Nn equals unity (by definition). Now, let a rare mutant with strategy value p0 attempt to invade the resident population. This mutant will have a fitness equal to: l p’,Nn ðp Þ ¼ p’ sA,M,max sMM Nn p þ ð1p’ÞsA,R þ ð1p’Þ sA,R,max sR Nn ð1pÞÞ þ sJ ðbM p’ þbR ð1p’ÞÞ
ð5Þ
Note the difference between the p0 indicating an alternative candidate strategy value and p, the value of the fraction of migrants in the current equilibrium population. In order to find the value of p0 that maximizes the invader strategy fitness, we calculate the derivative of l with respect to p0 : @l ¼ sA,M,max sM N n p sA,R,max sR Nn ð1pÞ þ sj ðbM bR Þ @p’
ð6Þ
Eq. (6) describes how invader fitness changes with the alternative (mutant) value of migration probability. Fig. 1 shows how this fitness gradient, which we call g(p), changes for all values of p. As shown, the gradient is positive for smaller values of p, but negative for larger values. At a given intermediate value (between 0.6 and 0.7 in Fig. 1) the gradient is zero which means that there is potentially an evolutionarily stable p-value. Alternative pvalues smaller than the equilibrium value have positive growth and there is selection for higher values, and the reverse for higher p-values than the equilibrium. Can adaptive evolution take the population to that point? The convergence stability criterion (e.g., Eshel, 1983; Geritz et al., 1998; Morris and Lundberg, 2011) is the answer and says that the derivative of the gradient (with respect to p) must be negative. That is, that the equilibrium point is a maximum of the fitness function (Eq. (5)). The derivative of the fitness gradient is rather straightforward to calculate and we get: @g ¼ Nn ðsM þ sR Þ @p
ð7Þ
This expression will always be negative (all the terms in Eq. (7) must be positive) which means that the evolutionary endpoint is convergence stable. The last criterion is that the endpoint should be an ESS (uninvadable) and not something else, e.g., a potential evolutionary branching point from which two different strategies
ð4Þ
In order to calculate the evolutionarily stable fraction of migrants in the population, we now use Eq. (4) as the state of the population (the evolutionary feedback environment (e.g., Heino et al., 1997; Morris and Lundberg, 2011)) in which possible alternative strategies may arise and invade. 2.1. Invasion analysis The invasion analysis is done such that, we assume that the population is at equilibrium (Eq. (4)) and is characterized by a migration strategy p. We then assume that an alternative strategy p0 attempts to invade such a population. Assuming that the alternative strategy initially is rare, it only suffers density- and frequencydependent feedback from individuals in the resident population (‘‘resident’’ here meaning the population characterized by a particular
Fig. 1. The fitness gradient (slope of the fitness function) for an example set of parameter values. The evolutionary equilibrium probability to migrate (pn) is found where the gradient is 0 (here between 0.6 and 0.7). The parameter values are: bM,max ¼3, bR,max ¼ 7, sA,M,max ¼ 0.4, sA,R,max ¼0.7, sM ¼0.01, sR ¼ 0.02, sJ ¼ 0.5.
38
P. Lundberg / Journal of Theoretical Biology 321 (2013) 36–39
can diverge (Abrams and Matsuda, 1993; Geritz et al., 1998; Vincent and Brown, 2005). This is found by examining the second derivative of the fitness gradient (@2 g=@p2 ). When this derivative is negative, we have an ESS and when it is positive, we have a fitness minimum and a branching point. In our case the derivative is exactly zero (p does not occur in Eq. (7)). We call this case a ‘‘weak’’ ESS. It is indeed convergence stable, but it is similar to neutral stability and it is not a branching point. It means that the strategy is the optimal one when invading a resident population. It is also convergence stable (attainable through adaptive evolution), but the resident strategy pn can again be invaded by any mutant. However, this mutant will not be able to grow in numbers because then the resident strategy is once again the optimal strategy. This is analogous to Fisher0 s sex ratio problem (Maynard Smith, 1989). Fig. 2 shows the pair-wise invisability plot (see Kokko, 2007 for a lucid treatment of pair-wise invasibility) of the system illustrating this problem. An additional feature of this partial migration system is the discrepancy between the evolutionarily optimal solution and the solution that maximizes population size. Fig. 3 illustrates the phenomenon. For a wide range of parameter values (here, the per capita birth rate of the migrants is arbitrarily chosen) the p-value that maximizes population size deviates considerably from the evolutionary equilibrium (Fig. 4). This is typically a rather mild problem in this system, but there are circumstances when this can lead as far as to ‘‘evolutionary suicide’’ (cf. Kokko, 2011). This means that adaptive evolution can take the population to dangerously small population sizes.
Fig. 3. The equilibrium population density (N*) as a function of the probability of migrating (p). The maximum N* (here at p E0.53, denoted by an open circle) deviates from the evolutionary solution denoted by a filled circle (at p E0.64).
3. Discussion Partial migration has received considerable interest in later years after quite some time in the shadow (see for example Milner-Gulland et al., 2011 and Chapman et al., 2011 and several
Fig. 4. The deviation between the evolutionary equilibrium p-value and the pvalue maximizing N* is shown as a function of the maximum per capita birth rate of the migrant strategy. The residents have a birth rate¼ 5. Positive (negative) deviations indicate Nn 4pn (Nn o pn).
Fig. 2. The pair-wise invasibility plot (PIP) for the same parameter setting as in Fig. 1. The x-axis is the evolutionary resident strategy value (migration probability) and the y-axis is the strategy value of an alternative (mutant). Fitness is higher for an alternative strategy above and to the left of the one-to-one line and the vertical line at pnresident E0.63, respectively. Fitness is also higher for an alternative below and to the right of the one-to-one line, respectively. Regions with higher mutant fitness are shaded gray with a plus sign. Regions with smaller mutant fitness are white with a minus sign. The equilibrium p is found at the intersection of the one-to-one line and the vertical line in the fitness plane (black dot). The equilibrium is not an ESS (or what could be called a ‘‘weak’’ ESS), but adaptive evolution takes the population there. At equilibrium, any mutant strategy value is equal to the resident value (moving along the vertical isofitness line).
papers in the same issue of Oikos). The first theoretical foundations laid out more than 20 years ago (e.g., Lundberg, 1987, 1988) have been refined and extended quite thoroughly. Territoriality, genetics, energetics and explicit environmental stochasticity have become components of the theoretical framework that now also goes beyond migratory birds. Partial migration is now studied in a wide range of animal taxa (Chapman et al., 2011). Elements of the theoretical framework are empirically testable. Claims that partial migration is, in a formal sense, an ESS solution remains to be shown. Here, I have used a simple and general population model with no assumptions about many of the real life complications in natural populations (e.g., size structure, waiting games, energetics) to show that the putative ESS solutions from many similar models are not strictly speaking ESSs. Whether the ‘‘weak’’ ESS here discovered is a real phenomenon again remains to be shown empirically. The model used here is reasonably generic but the strict linear density-dependence contributes to the stability point being neutral. The properties of the stability point will be difficult to establish empirically but one prediction from this model is that there should be quite some
P. Lundberg / Journal of Theoretical Biology 321 (2013) 36–39
variation of the fraction of migrants over time due, not the least, to environmental stochasticity. In a recent paper, Shaw and Levin (2011) showed that partial migration indeed can be a true ESS, and they also showed how environmental stochasticity can contribute to the evolution of partial migration. The model used by Shaw and Levin (2011) was however, very different from the very simple one used here and they were able to demonstrate how different types of partial migration can evolve by going beyond the classic ‘‘bird type’’ (migration between breeding and wintering grounds, the latter with no reproduction) analyzed here. Lundberg (1987) showed that environmental stochasticity might influence the equilibrium proportion of migrants in the population, but not the existence of an ESS solution. Kaitala et al. (1993) showed that environmental stochasticity is not necessary for partial migration to evolve, at least not to an ESS. In order to attain an ESS fraction of migrants, frequency- or density-dependence must cause evolutionary feedback regardless of environmental variability. As in many fields of biology, there is sometimes a wide abyss between the theoretical development of the field and the data available to test the theoretical assertions. The results presented in this note demonstrate that it is not necessarily an ESS in the strict and formal sense we should be looking for in partially migratory bird populations. There are many models of partial migration subsumed under the notion of optimization and straightforward adaptive evolution of stable fractions of migrants and non-migrants. More empirical data on vital rates and the establishment that putative migrants truly are so would help when deciding how we should view partial migration from a more conceptual point of view.
Acknowledgments I thank Jacob Johansson for illuminating discussions. The Centre for Animal Movement Research at the Biology Department, Lund University, contributed by helping me to bring this problem to my renewed attention. Financial support for this study was received from the Swedish Research Council and the marine ecology group at Gothenburg University provided a stimulating and generous environment when parts of this work were completed. I am grateful for the insightful and important comments of two reviewers for the manuscript.
Appendix 1 We will assume that the population dynamics of the entire population follows the same simple discrete process with two age classes from one year (t) to another (tþ1) as in Eq. (1). With no density-dependence the population model can be written as: Nt þ 1 ¼ Nt p sA,M þbM sJ,M þN t ð1pÞ sA,R þbR sJ,R ðA1Þ The subscripts A and J denote adults and juveniles, respectively and M and R denote migrants and non-migrants, respectively. Again, let N t þ 1 =Nt ¼ l be mean absolute fitness (per capita population
39
growth rate), and after having rewritten Eq. (A1) we have:
l ¼ psA,M þ pbM sJ,M þ ð1pÞsA,R þ ð1pÞbR sJ,R
¼ p sA,M þ bM sJ,M þ ð1pÞ sA,R þ bR sJ,R ¼ p sA,M þ bM sJ,M p sA,R þbR sJ,R þ sA,R þ bR sJ,R
ðA2Þ
To find the optimal strategy (i.e., the optimal probability for an individual to migrate, pn) take the partial derivative of l with respect to p, @l ¼ sA,M þ bM sJ,M sA,R þ bR sJ,R @p
ðA3Þ
The only fitness maxima are pn ¼1 or 0, depending on whether (sA, M þbMsJ,M) or (sA,R þbRsJ, R) is the largest term. That is, all individuals will either migrate or stay as non-migrants. This illustrates that partial migration is unlikely to evolve without frequency- or density-dependent reproduction or survival. Environmental stochasticity alone is also not enough for the evolution of partial migration (Kaitala et al., 1993). References Abrams, P., Matsuda, H., 1993. Effects of adaptive predatory and antipredatory behavior in a 2-prey one-predator system. Evol. Ecol. 7, 312–326. ¨ Alerstam, T., Hedenstrom, A., 1998. The development of bird migration theory. J. Avian Biol. 29, 343–369. Berthold, P., 1996. Control of Bird Migration. Chapman & Hall, London. Chapman, B.B., et al., 2011. The ecology and evolution of partial migration. Oikos 120, 1764–1775. Eshel, I., 1983. Evolutionary and continuous stability. J. Theor. Biol. 103, 99–111. Geritz, S., et al., 1998. Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12, 35–57. Griswold, C.K., et al., 2010. The evolution of migration in a seasonal environment. Proc. R. Soc. London B 277, 2711–2720. Heino, M., et al., 1997. Evolution of mixed maturation strategies in semelparous life histories: the crucial role of dimensionality of feedback environment. Philos. Trans. R. Soc. B 352, 1647–1655. Holt, R.D., Fryxell, J.M., 2011. Theoretical reflections on the evolution of migration. In: Milner-Gulland, E.J., et al. (Eds.), Animal Migration. Oxford University Press, Oxford, pp. 17–31. Kaitala, A., et al., 1993. A theory of partial migration. Am. Nat. 142, 59–81. Kokko, H., 2007. Modelling for Field Biologists. Cambridge University Press, Cambridge. Kokko, H., 2011. Directions in modelling partial migration: how adaptation can cause a population decline and why the rules of territory acquisition matter. Oikos 120, 1826–1837. Kokko, H., Lundberg, P., 2001. Dispersal, migration and offspring retention in saturated habitats. Am. Nat. 157, 188–202. Lundberg, P., 1987. Partial bird migration and evolutionarily stable strategies. J. Theor. Biol. 125 351-340. Lundberg, P., 1988. The evolution of partial migration in birds. Trends Ecol. Evol. 3, 172–175. Milner-Gulland, E.J. (Ed.), Oxford University Press. Morris, D.W., Lundberg, P., 2011. Pillars of Evolution. Oxford University Press, Oxford. Maynard Smith, J., 1989. Evolutionary Genetics. Oxford University Press, Oxford. Pulido, F., et al., 1996. Frequency of migrants and migratory activity are genetically correlated in a bird population. Proc. Natl. Acad. Sci. USA 93, 14642–14647. Sekercioglu, C., 2010. Partial migration in birds: the frontier of movement ecology. J. Anim. Ecol. 79, 933–936. Shaw, A.K., Levin, S.A., 2011. To breed or not to breed: a model of partial migration. Oikos 120, 1871–1879. Taylor, C.M., Norris, D.R., 2007. Predicting conditions for migration: effects of density dependence and habitat quality. Biol. Lett. 3, 280–283. Vincent, T.L., Brown, J.S., 2005. Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics Cambridge University Press, Cambridge.