Bet-hedging and the orientation of juvenile passerines in fall migration

Summary

1.  Bet-hedging of innate migratory orientation of juvenile passerines may be a fitness-enhancing strategy for fall migration. Experimental studies support the view that juvenile passerines on their first migration to unknown winter grounds orient on a predetermined vector programme and make little or no adjustment for wind displacement. This trait, coupled with the unpredictable profile of wind speed and direction that the juvenile will encounter during migration, suggests that the fitness of a parent’s juvenile offspring will be highly variable from year to year. Under these circumstances, within-clutch phenotypic variation in migratory orientation may be evolutionarily favoured.

2.  To explore this hypothesis, a migration model is developed for a small passerine with breeding grounds in New England and winter grounds in the Caribbean. Parameterization is based on life history data of the neotropical migrant Dendroica caerulescens, the black-throated blue warbler. The model is simulated for the offspring of 20 000 adult females under each of a wide range of potential orientation programmes, incorporating stochastic wind profiles along potential migratory routes, based on 7 years of wind data for eastern North America.

3.  Under these simulations, bet-hedging in the form of within-clutch variation of migratory orientation strongly dominates within-clutch homogeneity, yielding higher geometric mean fitness in all vector programmes considered.

4.  The simulation results provide a potential explanation for the variation observed in the tracks of juvenile passerines. Bet-hedging also explains the extensively-documented ‘coastal effect’ in which fall banding stations along the Atlantic coast of the United States consistently capture a much higher percentage of juvenile birds than do more inland stations.

5.  Bet-hedging is consistent with the published finding that slower flying birds exhibit greater variation in their migratory orientation than faster flying birds.

6.  The bet-hedging model of migratory orientation presented in this paper provides a theoretical structure capable of organizing a diverse collection of field and laboratory observations as predictable consequences of an evolutionarily favoured strategy. This theory may constitute a major advance in our understanding of bird migration and thus justifies the design and execution of new laboratory and field experiments to assess its power and predictive reach.

Introduction

When environmental conditions are stable, organisms should be under selection to produce offspring that all have a phenotype optimized for that stable environment. By contrast, when environmental conditions vary unpredictably, a genotype that hedges its bets by producing a variety of offspring phenotypes may have the advantage in the long run. Compared with a single-phenotype strategy, such a bet-hedging strategy involves trading a reduction in expected offspring fitness within a single year for an increase in offspring fitness (and an associated reduction in extinction probability) across future years (e.g. Seger & Brockman 1987; Frank & Slatkin 1990; Saski & Ellner 1995).

While bet-hedging in birds has been considered in connection with clutch size determination (Sæther & Bakke 2000), hatching asynchrony (Amundsen & Slagsvold 1998), the selection of mates (Handel & Gill 2000; Jennions & Petrie 2000), and the rearing of unrelated offspring (Connor & Curry 1995), its potential role in the selection of an optimal migratory strategy for juvenile passerines has not been explored. It is well documented that adult passerines are able to navigate to their winter grounds with precision. First-year fall migrants, never having been to the winter grounds, do not possess this ability. Instead, the migratory orientation of first year birds appears to be genetically determined, or established during brooding (Able 1991; Wiltschko & Wiltschko 1991; Helbig 1996). In an early study highlighting the difference between juvenile and adult navigational ability, a large sample of adult and juvenile starlings (Sturnus vulgaris) were captured in the Netherlands during fall migration (Perdeck 1958, 1967). Some of these birds were transported south to release locations in Switzerland, while the remaining birds were released where captured. Subsequent same-season recaptures demonstrated that displaced adults changed direction from a south-westerly to a new north-westerly orientation to stay on course for their intended winter grounds. By contrast, the displaced juveniles continued on their original, but now improper, south-westerly orientation, taking them away from their intended winter grounds. The non-displaced juveniles, following this same south-westerly orientation, were recovered on their intended winter grounds. A recent test of these findings for a native North American migrant, Zonotrichia leucophrys gambelii, Gambel’s white-crowned sparrow, was conducted by Thorup et al. (2007a). In their study, 15 adults and 15 juveniles were captured in Washington State during their south-southeast fall migration to winter grounds in the south-western United States and north-western Mexico. The birds were transported by airplane to Princeton, New Jersey, and fitted with radio transmitters. The birds were then released and tracked by airplane for c. 125 km. The adults quickly adjusted their headings to a new west-south-west track that would take them toward their winter grounds. The juveniles, however, continued on the now inappropriate south-southeast heading that they were on prior to the displacement. The work of Thorup et al., together with other investigations conducted subsequent to the original study by Perdeck (Wiltschko & Wiltschko 1988; Berthold 1990; Helbig 1996; Mouritsen 1998), implies that first-year birds are not goal directed, but rather migrate using vector-navigation based on predetermined directions (tied to celestial and magnetic cues), coupled with a time or distance programme.

It has been shown that first-year birds exhibit considerably more variation in their preferred migratory direction than adults, and that the distribution of this juvenile variation appears to be genetically transmitted and differs across populations (Able 1977; Moore 1984; Helbig 1996; Woodrey 2000). For example, when individual blackcaps (Sylvia atricapilla) from populations differing in orientation distribution were crossed, the F1 offspring displayed an intermediate orientation distribution (Helbig 1996). Orientation cage experiments suggest that variation in initial migratory orientation can be present within a clutch (Helbig 1992), and that such variation might be 10° or more about the mean heading. Statistically, this evidence is inconclusive, however, due to the large variation in within-individual observations commonly found in orientation cage experiments. See Thorup, Rabol & Erni (2007b) for a review.

Small migrating passerines are strongly affected by prevailing winds and show limited ability to adjust their heading in real time to compensate (Able 1974; Richardson 1991; Backman & Alerstam 2003). However, an adult bird blown off course can adjust its bearing on subsequent flights to navigate to its intended winter destination (Bingman, Able & Kerlinger 1982; Able & Bingman 1987; Backman & Alerstam 2003). The well-established result that adults are able to return with precision to their winter grounds is evidence that they make these adjustments (Holmes & Sherry 1992; Koronkiewicz et al. 2006). Field studies of the flight of free flying migrants are constrained by the inability to distinguish juveniles from adults, one species from another or, in many cases, relative size of the birds. Tracking individual birds at night for significant distances requires airplane monitoring such as that employed by Thorup et al. (2007a). It is not possible, therefore, to draw clear conclusions on the navigational ability of juvenile passerines in fall migration from the available studies, excepting those of Perdeck and Thorup et al. Orientation experiments have identified a range and potential hierarchy of the compass cues employed by migrating passerines (Muheim, Moore & Phillips 2006), but none of these experiments provide direct evidence on compensation for wind drift by juveniles. Evans (1968) conducted a set of orientation cage experiments on a small sample of juvenile European passerines, presumed to be displaced westerly by wind, and suggested that their post-displacement S or SE orientation was compensation that would return them to their preferred SSW route. That analysis, by assuming that the juveniles in the sample possessed a SSW preferred heading, missed the possibility that, given an east-to-west wind pattern and the inability to employ goal-directed navigation, a S or SE orientation could be an appropriate strategy. In summary, there is no evidence to suggest that juvenile passerine migrants are capable of making significant heading adjustments in response to wind drift during nocturnal flight.

For this study, we assume that a young passerine displaced by winds during fall migration may be expected to continue migrating on its initial, although displaced, vector. Unpredictable winds, coupled with the genetically-predetermined vector-orientations of these young birds, suggest that bet-hedging may be a selective strategy. Genetic transmission of orientational variation to the offspring in a given clutch does not require the specification of a particular probability distribution. Rather it requires only that the orientation mechanism be somewhat imprecise. Imparting an exact heading to all offspring is likely more difficult to achieve than is imparting a ‘general’ direction which manifests itself through orientational variation across the clutch. The design of a guidance system that will reliably hit a small target is more challenging than designing one that will hit somewhere in a neighbourhood of that target. From an evolutionary perspective, the bet-hedging hypothesis would imply that beyond some point, a more precise mechanism for imparting migratory orientation to one’s offspring will have a negative fitness consequence and thus may not be selective. This paper develops a simulation model of annual passerine fitness which emphasizes the interaction between juvenile migrants and a stochastic wind profile. The model is parameterized for the neotropical migrant Dendroica caerulescens Gmelin (black-throated blue warbler) whose life history is comparatively well known (e.g. Holmes, Rodenhouse & Sillett 2005). By employing geometric mean fitness as the measurement standard, we show that bet-hedging strategies strictly outperform non-bet-hedging strategies across a wide range of plausible parameter values.

Model and methods

The purpose of this simulation model is not to provide an accurate description of the migratory ecology of a particular species, but rather to suggest, by employing plausible parameterizations for one well-known neotropical migrant (D. caerulescens), that bet-hedging deserves careful consideration as a potential explanation of the persistence of variation in the migratory orientation of juvenile passerines. Our approach can be divided into two main parts: (A) simulating migration from a New England breeding ground to Caribbean winter ground and (B) calculating and comparing fitness values associated with bet-hedging and non-bet-hedging strategies. All simulations were written and performed in the open-source programming language r. Model source code is available by request.

Simulating migration

The simulation is based on a simplified map of the eastern US and Caribbean (Fig. 1). Migrants are assumed to follow two consecutive vectors with a predetermined number of migration days on each. The first vector is south-westerly and the second more southerly. Each night’s flight consists of a certain number of hours at a given speed and heading. Estimates of flight speed and maximum duration are calculated with the program flight 1.17 (Pennycuick 2006). If a flight ends over water, the bird is assumed to continue flying for up to its maximum duration in an effort to reach land. Heading is calculated by adding a random wind vector to the bird’s innate flight vector. Wind speed and direction are drawn from a set of distributions calculated from National Oceanic and Atmospheric Administration (NOAA) data that depend on the bird’s geographic position.

Figure 1.  Geography of the model. All simulations begin with a bird at the natal site.

Calculating fitness

To calculate survival during fall migration, we simulated the migratory tracks of a large sample of ‘clutches.’ Bet-hedging of migratory orientation was introduced by simulating clutches with variation in orientation among siblings. Survival estimates from these fall migration simulations were combined with literature estimates of survival for the rest of the year. The bet-hedging and non-bet-hedging strategies were compared by calculating the geometric mean fitness of females producing clutches of each type of offspring. Sensitivity to parameter values was explored for flight duration, fat score (in relation to flight speed and maximum flight duration), total nights of migration and clutch size.

Model details and justifications

Simulating migration

Geography

Thorup & Rahbek (2004) and Erni, Liechti & Bruderer (2003, 2005) stress the importance of incorporating key geographical features into migratory simulation studies. For simulation tractability, the U.S. East and Gulf coasts are represented by joining a set of line segments that approximate the principal angles and distances that characterize the region (see Fig. 1). Consistent with a Caribbean land pattern, the winter grounds of D. caerulescens are represented by a region whose northernmost point is Havana, Cuba, and which slopes away to the south-east and south-west. The region has an eastern boundary at the same approximate longitude as that which passes through north-eastern Maine. This representation accommodates the model’s focus on an established breeding area of D. caerulescens in New Hampshire and its preferred winter grounds in the West Indies (Sillett & Holmes 2002).

Migration path

Given that our preliminary simulation results showed that a two-vector path is the simplest configuration capable of producing successful migration between the designated summer and winter grounds of D. caerulescens, we focused on sequential, two-vector migration strategies. This is consistent with experimental evidence (e.g. Perdeck 1958; Helbig 1996; see also Erni et al. 2003). Before adjusting for the influence of prevailing wind, each juvenile passerine is assumed to fly at its no-wind flight speed in the direction (migratory orientation) governed by its innate sequence of two fixed vectors with a fixed number of nocturnal flights on each vector. Simulations were used to select the best division of total nocturnal flights between the first and second vectors.

Number of nocturnal flights

The average number of non-consecutive nights per month with favourable winds (winds with some north to south component) in September and October 1998–2004 for six representative NOAA reporting stations along the eastern seaboard ranged from a low of 8·0 for September at Buffalo, NY, to a high of 13·4 for October at Jacksonville, FL. The average was 10·9. Thus, a juvenile could perhaps expect to complete 10 migratory flights in a month during the September–October period. To provide some variation about this figure, as well as to accommodate both shorter and longer times of passage, we considered migration strategies encompassing 8, 10 and 12 nights of active flight, interspersed with associated stopover periods.

The role of fat reserves: flight speed and maximum duration

Flight speed and flight range of migratory passerines are strongly dependent on avian flight morphology and on the accumulation of energy stores prior to migration and during periods of migratory stopover (Blem 1980; Weber & Houston 1997). flight 1.17, a program developed by Pennycuick (2006), calculates potential flight range and speed for varying assumptions regarding fat reserves, morphological data and altitude. Flight range and flight speed calculations made for our simulations are based on an altitude of 1000 m, using Pennycuick’s program. We used mass in excess of lean body mass as our measure of fat reserves (Woodrey 2000), and the absence of visible subcutaneous fat deposits as a proxy for lean body condition. Based on a sample of 1500 black-throated blue warblers captured in the 2001–2006 period at the Coastal Virginia Wildlife Observatory’s Fall Migration Banding Station at Kiptopeke State Park on Virginia’s Eastern Shore, we regressed body mass on visible fat score and wing length (mm), for a 50–50 mix of males and females, n = 1500. Fat was scaled 0–5, with zero denoting the absence of visible fat deposits in the furcular hollow and 5 denoting a bird with visible fat deposits more than filling the hollow, producing a mounded appearance. The resulting estimating equation has an R² = 0·78 with all coefficients significant (P < 0·0001). This equation, using the sample mean wing length of 61·8 (50–50 male–female mix) and an assumed fat score of 0, yields a lean body mass of 8·86 g. This is the figure we input to Pennycuik’s program.

A subsample of 104 juveniles captured at Kiptopeke in 2001–2006 in the 06:00–07:00 h interval are considered as caught near the end of the night’s migration and prior to significant foraging. These birds had a mean fat score of 1·68. Using this fat score in the above estimating equation implies an average juvenile body mass on arrival of 9·53 g. This estimate matches the directly computed 9·53 g mean for the subsample of 104 arriving juveniles in the 06.00–07.00 h period. A study reporting mass change during stopover for 11 species of neotropical warblers during fall migration through Appledore Island, Maine, found significantly positive mean mass gains (eight species) of 2·1–10·0% of initial body mass (Morris, Holmes & Richmond 1996). Applying this range to the mean arrival mass of 9·53 g for HY black-throated blue warblers at Kiptopeke yields a potential range for mean post-stopover mass of 9·73–10·48 g. Using the above regression, fat scores of 2 and 3 correspond to 9·68 and 10·52 g respectively. Accordingly, our simulations were run for departing juvenile fat scores of 2·0, 2·5 and 3·0.

Duration of a single flight

Summarizing studies based on radar, moonwatch and ceilometer observations of nocturnal passerine migrants over land, Kerlinger & Moore (1989) report that nocturnal passerine migrants commence flight at c. 0·5 h after sunset, that peak migratory volume occurs near 22.00 h, and that migration activity essentially ceases sometime between 24.00 and 02.00 h. As Kerlinger and Moore note, some studies report peak times between 23.00 and 01.00 h, and that some migrants continue to fly as late as 04.30 h (e.g. Lowery 1951). Given this variation, we used alternative ‘primary’ flight durations of 5, 6 and 7 h in our simulations. If a bird was over water at the end of its one-night primary flight (a time when darkness prevails, excepting the level of moonlight), it is assumed to make it to the nearest land provided that land is within 5 km. If not, then the bird enters a secondary flight stage, continuing on its original heading until first light. At that point, if the nearest land is within 113 km (the distance to the horizon for a bird flying at a height of 1000 m), the bird is assumed to make it to the nearest land. If the nearest land is not visible, the bird enters a third and final flight stage, either reaching landfall or exhausting its fuel supply and perishing over water. For each departing fat score in our simulations, Table 2 gives the bird’s average no-wind flight speed and maximum final flight duration as calculated from Pennycuick’s (2006) program. Before adjusting for wind, the figures in Table 2 imply that during the primary migration hours, juvenile migrants within our simulations can fly between 166 km (with fat score = 2 and flight duration = 5 h) and 240 km (with fat score = 3 and flight duration = 7 h).

Table 2.   No-wind flight speeds and maximum flight durations associated with each of the three departing fat score values used in the simulations

Departing fat score	First flight stage no-wind flight speed (kph)	Second flight stage no-wind flight speed (kph)	Third (final) flight stage no-wind flight speed (kph)	Maximum total flight duration (h)
2	33·3	36·9	39·1	15·2
2·5	33·4	37	40·7	21·6
3	34·3	38	42·5	29·9

The role of wind

Timing of migration.  The fall migration of D. caerulescens proceeds southward in a corridor from the Atlantic coast to the Appalachian Mountains (Ralph 1981; Holmes et al. 2005). As is apparent from banding data along the eastern seaboard for the period 1980–2000 (Source: Bird Banding Laboratory, U.S. Geological Survey, Department of the Interior), the passage of D. caerulescens through the eastern U.S. takes place primarily from late August through the months of September and October (see Fig. 2).

Figure 2.  Timing of fall migration for Dendroica caerulescens: per cent of total banding records from New York, Virginia, Georgia and Florida is plotted by 5-day period from August 1 to November 1.

Wind speed and direction.  To provide representative wind conditions that might be encountered by juveniles during migration along this route, 7 years of daily wind data for mid-August through September and October (1998–2004) were obtained from archived NOAA files for each of 15 coastal and inland weather stations arrayed down the eastern seaboard and along the Gulf coast. Figure 3 illustrates the relative location of those stations.

Figure 3.  Mean wind directions and speeds at the National Oceanic and Atmospheric Administration reporting stations (of days when wind direction had some north to south component). Standard deviations in wind directions range from . Standard deviations in wind speeds range from 12·4 to 16·1 kph.

Neotropical migrant passerines such as D. caerulescens are primarily nocturnal migrants (Able & Bingman 1987; Gwinner 1996), and the daily wind observations used in our simulations were those recorded at midnight at altitudes of 100–1600 m, a range that likely would encompass most migratory flight by small passerines. For example, the radar sample of 880 autumn passerine migrants observed by Backman & Alerstam (2003) had a mean altitude of 1233 m with a standard deviation of 662 m. Migration is most intensive on nights when birds have some tailwind, and migration is largely avoided in the face of a headwind (Alerstam 1990). Along the U.S. eastern seaboard favourable tailwinds correspond to north-west, north or north-east originating winds. Hence only nights where the wind was blowing in a destination direction with at least some southerly component (i.e. >90 and <270°) were included in our analysis. The resulting data set was used to obtain an empirical multivariate normal distribution of wind direction and correlated wind speeds covering each of the 15 stations. The mean wind directions and mean wind speeds at each station are reported in Table 1. The wind configuration faced by a particular bird in the simulation on any given night of its migration was determined by a random wind direction and random correlated wind speed drawn from our multivariate normal distribution for the wind station closest to the bird’s position. Mean wind directions and relative mean wind speeds at each of the 15 stations are depicted by arrow directions and lengths in Fig. 3.

Table 1.   1998–2004 August 15 through October 31 destination wind bearing and wind speed parameters for the 15 National Oceanic and Atmospheric Administration reporting stations used in the simulations

n	Station	Lattitude (°N)	Longitude (°W)	Mean wind bearing (°)	Wind bearing SD (°)	Mean wind speed (kph)	Wind speed SD (kph)	Covariance of wind bearing and speed
The sample size on which each station’s wind profile is based is given by n, the number of nights.
655	Gray ME	43·89	70·25	152·3	40·3	29·2	16·1	−61·9
494	Buffalo NY	42·93	78·73	154·9	48·3	27·0	14·2	−235·1
906	Brookhaven NY	40·87	72·87	159·7	45·1	27·0	14·4	−36·1
457	Pittsburgh PA	40·53	80·23	154·6	46·1	25·0	14·5	−135·1
472	Wilmington OH	39·42	83·82	165·5	51·7	24·8	14·5	−202·4
850	Wallops Island VA	37·93	75·48	168·6	49·3	26·5	15·4	53·5
556	Nashville TN	36·25	86·57	179·6	44·1	19·7	11·5	−90·1
848	Morehead City NC	34·7	76·8	186·6	47·5	23·5	14·8	27·3
558	Birmingham AL	33·1	86·7	183·0	45·6	20·4	12·4	−22·4
758	Charleston SC	32·9	80·03	191·9	48·9	23·3	15·4	9·8
808	Jacksonville FL	30·43	81·7	197·5	51·0	23·2	14·0	133·6
748	Slidell LA	30·33	89·82	194·4	44·1	20·3	13·3	49·3
565	Corpus Christi TX	27·77	97·5	212·6	39·5	23·9	12·9	49·3
896	Tampa Bay FL	27·7	82·4	198·8	53·2	20·1	12·7	160·3
769	Key West FL	24·55	81·75	221·2	59·2	22·3	12·8	63·1

Putting it all together

During migratory flight, a bird’s position is updated each hour by adding the hourly random wind vector from the departing location to the bird’s hourly, no-wind, orientation vector. The bird’s no-wind orientation vector is composed of a genetically-programmed heading and a no-wind flight speed. For each combination of flight duration and beginning fat stores considered, no-wind flight speeds were determined by averaging the beginning and ending flight speeds for D. carulesens from Pennycuick’s program. Multiplying the hourly change in the bird’s position by the applicable flight duration gives the total change in the bird’s position during the night.

By way of example, consider the simulation of one night’s flight for a juvenile departing from a point 50 km due west of the Wallops Island reporting station. Suppose the bird is undertaking the third nocturnal flight along its first orientation vector, which has a no-wind bearing of 257°. Assume as well that its departing fat score is 2·5, that its primary flight duration for the night is 6 h, and that the randomly selected wind vector that it encounters has a bearing of 155° and wind speed of 27 kph. (Note from Table 1 that the mean wind bearing and wind speed for Wallops Island on nights where the wind has some north-to-south flow are 168·6° and 26·5 kph respectively.) Based on its departing fat reserves, Table 2 gives its average no-wind flight speed as 33·4 kph during the night’s primary flight period. During the 6 h, the bird’s change in position is the sum of its no-wind effort vector, i.e. (33·4)(6) = 200·4 km, along a 257° heading, plus the displacement due to wind, i.e. (27)(6) = 162 km, along a 155° heading. Vector addition yields a net change in position of 230 km along an actual heading of 213·5°. This movement places the bird over land, and it ends its flight 230 km south-west of its departing location.

Calculating fitness

To generate reasonable values for fitness, we need estimates of survival throughout the whole year, as well as an estimate of female fecundity.

Survival estimates

An adult breeder is assumed to survive to breed again the following year with probability 0·45. This figure encompasses adult mortality across all four seasons. Sillett & Holmes (2002) report mean annual survival rates of males and females as 0·51 and 0·40, respectively, in their multi-year study of the annual cycle of D. caerulescens on their New Hampshire breeding grounds and their Jamaica winter grounds, and as our simulation does not separate males and females, we use 0·45, the midpoint of these two rates. Regarding juvenile survival, Sillett & Holmes (2002) find that once a juvenile reaches the winter grounds successfully, its probability of survival is not significantly different from that of an adult. They place the mean survival probability for the wintering period at 0·93, for the spring migration period at 0·70, and for the summer period at 0·99. Using these values we may estimate the probability that a juvenile, having successfully reached the winter grounds, will survive through the summer breeding season as (0·93)(0·70)(0·99) = 0·64. Sillett and Holmes provide no estimate of the probability that a juvenile successfully completes its first fall migration. We apply their estimate of adult survival during spring migration of 0·70 (equal to the adult mean survival rate for fall migration) as plausibly encompassing juvenile mortality during fall migration from causes other than those associated with the limited capability of the juvenile orientation program. Orientation-related mortality of juvenile passerines is the primary output of our simulation model, and creates the fitness difference between bet-hedging and non-bet-hedging strategies.

Fecundity (estimating ‘clutch’ size)

Based on an extensive study of D. caerulescens on the New Hampshire breeding grounds, this species is frequently doubled-brooded and fledges an average of 4·3 young (males and females) per female each season (Holmes et al. 2005). No data is available on survival of this species in either the post-fledging period or during the fall migration of juveniles. Indeed data on passerine survival during either of these periods is absent for most species. Anders et al. (1997) found a 42·3% post-fledging survival rate in a sample of 45 wood thrushes (Hylocichla mustelina) which they radio-tracked in their study area in Missouri. A current large sample study involving 502 prothonotary warbler (Protonotaria citrea) fledglings places an estimated survival range between 45% and 68% during the post-fledging period (R. J. Reilly, unpublished data). Applying this range to the average 4·3 fledglings per adult female cited by Holmes et al. implies that between 2 and 3 (1·95 and 3·01) juveniles (male and female) survive the post-fledging period, per adult female. Accordingly, our simulations consider 2, 2·5 (half at 2 and half at 3) and 3 juveniles surviving to initiate their first fall migration, per adult female.

Geometric mean fitness

Geometric mean fitness measures the success of a single organism’s offspring as the generations extend into the future. Arithmetic mean fitness measures the average success of the offspring of many organisms across any specified number of generations. The geometric mean is thus the one that matters for the individual (Lewontin & Cohen 1969). If the same probability distribution of random components (such as wind conditions) are assumed to govern the fitness outcome for a single individual in any generation, then that same probability distribution is sufficient to calculate the anticipated outcomes and thus the geometric and arithmetic mean fitness across all generations. To calculate geometric mean fitness of a particular strategy (as well as arithmetic mean fitness) it is necessary to know or to determine the single generation probability distribution of fitness values associated with that strategy. In the model examined here, a strategy consists of the rules governing selection of the two-leg heading vectors for the offspring of one adult. For a particular parameter combination, the single generation probability distribution of outcomes associated with each potential strategy was determined by simulating 20 000 fall migrations of one season’s offspring through the complex, stochastic environment described above. The sample size of 20 000 for each simulation was sufficiently large to yield outcome probabilities that are stable to two decimal places while preserving the computational tractability necessary to complete our sensitivity analysis. These empirically-determined probability distributions were then used to compute geometric mean fitness values for each potential strategy, enabling us to identify the strategy that maximized geometric mean fitness under the specified parameter combination.

Comparing the bet-hedging and non-bet-hedging strategies

Without bet-hedging.  In the absence of bet-hedging, the offspring of an adult female are assumed to employ the common strategy that maximizes geometric mean fitness. For example, if

1. three offspring survive to initiate fall migration,

2. they use a common strategy of 6 flights on a before-wind heading of O₁° followed by 4 flights on a before-wind heading of O₂°,

3. they encounter the location-specific random wind profile where denotes the wind heading and wind speed encountered on the jth flight under heading O_i.

Then the realized fitness value λ for that single simulation is given by

• 

where, as discussed in the Survival estimates section above, 0·45 is the annual survival probability for the adult female and 0·64 is the survival probability (for the balance of the year) for a juvenile who successfully makes it to the winter grounds. Based on 20 000 such simulations, the probabilities that 0,1,2 or 3 offspring make it to the winter grounds are determined as p(0), p(1), p(2) and p(3). The geometric mean fitness for the orientation strategy (O₁,O₂) is then calculated as 0·45^p(0)1·09^p(1)1·73^p(2)2·37^p(3). The optimal orientation strategy in the absence of bet-hedging, , is the one with the highest geometric mean fitness among all orientation strategies.

With bet-hedging.  The bet-hedging case differs from the non-bet-hedging case in the following way: the orientation strategies of individual offspring are now permitted to differ through the introduction of a random component. This is implemented by specifying a pair of headings (O₁,O₂) as above, but individually modifying those headings with a pair of random components (x₁,x₂) drawn for that particular bird from two normal distributions with respective means (O₁,O₂) and standard deviations (σ₁,σ₂). Thus the orientation strategy of that individual offspring is (O₁+x₁, O₂+x₂). In contrast to the non-bet-hedging case, the adult female’s strategy governing the migratory orientation of her offspring has four rather than two components, i.e. . The geometric mean fitness of one such strategy is then determined by simulating the migration of 20 000 sets of offspring to determine p(0), p(1), p(2) and p(3), as discussed above. For tractability, bet-hedging simulations were restricted to combinations of (σ₁,σ₂) about the pair of headings that was determined to be optimal in the absence of bet-hedging. It is plausible, of course, that higher geometric means under bet-hedging might be achieved by letting O₁ and O₂ vary in the bet-hedging optimization in addition to σ₁and σ₂. Practical limitations precluded this. As will be seen, however, this extension was not needed to establish the superiority of bet-hedging strategies.

The bet-hedging hypothesis.  For each of the parameter sets considered, bet-hedging is a superior strategy if .

Results

Establishing the baseline result

To provide a baseline from which sensitivity to parameter changes could be assessed, we begin by estimating the middle case for each of the parameter ranges discussed in the Model and methods section. In particular, the non-bet-hedging baseline simulation was conducted under the following parameterization:

• 

Within the simulation it was determined that a division of the 10 migration flights into six on the first heading and four on the second heading yielded the highest geometric mean fitness. The optimal heading sequence was with an associated maximum geometric mean fitness value of 1·07.

Using the non-bet-hedging optimal heading sequence (257, 157°), we conducted a series of 20 000-run simulations for whole-degree combinations of (σ₁, σ₂), i.e. for levels of parentally-determined variation in the first and second headings of the juvenile offspring. Geometric mean fitness was maximized at . Thus the optimal level of variation about the first heading of 257° is described by a standard deviation of 4°, and the optimal level of variation about the second heading of 157° is 0°. The high optimal variation in the first heading is likely due to the greater danger for unpredictable winds to have fatal consequences during this first segment of migration (e.g. a death in the Atlantic Ocean). The maximized geometric mean fitness value at and is 1·11 compared to 1·07 in the absence of bet-hedging. Thus for the baseline parameterization, bet-hedging provides a fitness advantage over non-bet-hedging. The results for the baseline case are summarized in rows 1 and 2 of Table 3. Columns 3 and 4 of those first two rows illustrate a characteristic property of successful bet-hedging strategies: a reduction in the variation of fitness outcomes (here from 0·44 to 0·33 in column 3) at the expense of some reduction in arithmetic mean fitness (here from 1·27 to 1·26 in column 4).

Table 3.   Sensitivity of fitness results to parameter values

	Offspring surviving to migrate	Departing fat score	Hours of migration per night	Flights along first vector (days)	Flights along second vector (days)	Geometric mean fitness	Variance in fitness outcomes	Arithmetic mean fitness	Mean of first innate heading	Variation about mean of first heading (SD)	Mean of second innate heading	Variation about mean of second heading (SD)
Baseline non-bet-hedging	2·5	2·5	6	6	4	1·07	0·44	1·27	257	0	157	0
Baseline bet-hedging	2·5	2·5	6	6	4	1·11	0·33	1·26	257	4	157	0
Variant 1 non-bet-hedging	2	2·5	6	6	4	0·96	0·28	1·11	257	0	157	0
Variant 1 bet-hedging	2	2·5	6	6	4	0·98	0·23	1·10	257	2	157	0
Variant 2 non-bet-hedging	3	2·5	6	6	4	1·20	0·54	1·44	257	0	157	0
Variant 2 bet-hedging	3	2·5	6	6	4	1·25	0·39	1·42	257	5	157	2
Variant 3 non-bet-hedging	2·5	2	6	6	4	0·75	0·42	0·94	257	0	150	0
Variant 3 bet-hedging	2·5	2	6	6	4	0·79	0·29	0·92	257	5	150	3
Variant 4 non-bet-hedging	2·5	3	6	6	4	1·33	0·34	1·48	261	0	161	0
Variant 4 bet-hedging	2·5	3	6	6	4	1·35	0·29	1·47	261	6	161	3
Variant 5 non-bet-hedging	2·5	2·5	5	6	4	1·04	0·45	1·25	251	0	170	0
Variant 5 bet-hedging	2·5	2·5	5	6	4	1·08	0·35	1·24	251	2	170	0
Variant 6 non-bet-hedging	2·5	2·5	7	6	4	1·00	0·45	1·21	266	0	150	0
Variant 6 bet-hedging	2·5	2·5	7	6	4	1·04	0·33	1·20	266	5	150	2
Variant 7 non-bet-hedging	2·5	2·5	6	5	3	0·94	0·46	1·15	249	0	170	0
Variant 7 bet-hedging	2·5	2·5	6	5	3	0·98	0·34	1·14	249	5	170	1
Variant 8 non-bet-hedging	2·5	2·5	6	6	6	1·10	0·43	1·30	263	0	158	0
Variant 8 bet-hedging	2·5	2·5	6	6	6	1·14	0·32	1·28	263	7	158	3

Sensitivity analysis

To examine the sensitivity of our results to plausible variations in the underlying parameters, we conducted simulations for eight variants from the baseline, both with and without bet-hedging. Rows 3–18 of Table 3 report the sensitivity results by variant. The choice of these variants was motivated in Model details and justifications.

Variants 1 and 2 (rows 3–6) in Table 3, along with the baseline results in rows 1–2, illustrate the unsurprising result that geometric mean fitness is directly related to the number of juvenile offspring that survive to initiate migration. As discussed in the Calculating fitness section, the average number of offspring (both sexes) per adult female that begin migration likely falls between 2 and 3. The baseline and variants 1 and 2 show that between 2 and 3 juvenile offspring, geometric mean fitness varies form 0·96 to 1·20 in the absence of bet-hedging, and from 0·98 to 1·25 with bet-hedging. The optimal levels of heading variation for variants 1 and 2 are standard deviation pairs of respectively. Thus bet-hedging outperforms non-bet-hedging in the baseline and under both variants. Figure 4 illustrates the fitness benefit that bet-hedging can achieve. Under variant 2, in each of three sample years, at least one offspring survives migration under very diverse environmental conditions. The average number of migrating offspring must exceed one for bet-hedging to confer a fitness advantage.

Figure 4.  Representative migratory tracks of three offspring in three different years under variant 2 of Table 3. The adult female employs a bet-hedging strategy with . As the panels illustrate, the interaction of her offsprings’ headings and wind variation between years can produce very different migratory paths. Note, however, that the bet-hedging strategy employed delivers at least one offspring safely to the winter grounds in all 3 years.

Variants 3 and 4 (rows 9–10) in Table 3 demonstrate that fitness outcomes are very strongly dependent on the energetic condition of the migrating juveniles. As Table 2 indicated, overall flight range diminishes rapidly as fat reserves at time of departure decline. This is especially critical when a juvenile finds itself over water at the time it would normally land. Variants 3 and 4 show the effects of varying the departing fat score by 0·5 units off the baseline. Geometric mean fitness in the absence of bet-hedging varies between 0·75 and 1·33 over this interval. Geometric mean fitness under bet-hedging varies from 0·81 to 1·35 over the same interval with associated standard deviation pairs of . Note that geometric mean fitness with bet-hedging exceeds the corresponding value in the absence of bet-hedging, regardless of the level of departing fat reserves. Juveniles that are unable to locate good stopover habitat and/or are inept at exploiting it, will suffer heavy losses according to these simulation results.

Variants 5 and 6 in Table 3, together with the baseline, show the interesting impact of varying the hours of one night’s primary flight between 5 and 7. Highest geometric mean fitness is achieved with 6-h primary flights. Shortening or extending this period reduces fitness with a somewhat greater reduction occurring when flight time is extended. These results are consistent with empirical studies of typical flight duration as cited in the duration of a single flight section. Once again, regarding this paper’s central hypothesis, geometric mean fitness with bet-hedging is higher than without bet-hedging under both variants. Geometric mean fitness in the absence of bet-hedging is 1·04 and 1·00 under variants 5 and 6 respectively. With bet-hedging, the corresponding geometric means are 1·08 and 1·04, with associated standard deviation pairs of .

Finally, variants 7 and 8 in Table 3 examine the impact of changing the total number of nocturnal flights from 10 in the baseline to either eight flights under variant 7 or 12 flights under variant 8. If the number of fights is reduced to eight, the highest geometric mean fitness is achieved by allocating five flights to the first heading and three to the second heading. In comparison with the baseline (in which 10 flights are optimally allocated with six to the first heading and four to the second) geometric fitness is reduced from 1·07 to 0·94 in the absence of bet-hedging and from 1·11 to 0·98 with bet-hedging, with associated standard deviation pairs of . As discussed the number of nocturnal flights section, weather patterns, in combination with stopover requirements and other biological constraints, may preclude extension of the migratory period to accommodate 12 or more flights. These considerations notwithstanding, the extension from 10 to 12 flights (with an accompanying optimal allocation of six nights to each heading) would achieve an increase in fitness from 1·07 to 1·10 in the absence of bet-hedging and from 1·11 to 1·14 with bet-hedging, using standard deviation pairs respectively. Under both variants 7 and 8, bet-hedging yields higher geometric mean fitness than non-bet-hedging.

As an interesting by-product of the model, simulation of 20 000 single juveniles under the baseline parameters in the absence of bet-hedging yields a 0·49 probability that a given juvenile reaches the winter grounds. Recall from the Survival estimates section that we use a 0·64 probability that a juvenile who has successfully reached the winter grounds will survive to breed. Recall as well that the baseline’s average of 2·5 juveniles surviving to initiate fall migration implies a survival probability for the post-fledging period of 0·58 , where 4·3 is the average number of fledglings per adult female. These probabilities, taken together, imply that the first-year survival probability for a given fledgling may be estimated as (0·58)(0·49)(0·64)=0·18. Dropping the post-fledging-period survival rate to the lower bound of 0·47 results in a first-year survival rate of 0·15. Thomson, Baillie & Peach (1999) estimated the survival rate of juvenile song thrushes (Turdus philomelos) at 17%.

Simulation of 20 000 single juveniles under the baseline parameters with bet-hedging yields a 0·48 probability that a given juvenile reaches the winter grounds. Note that this probability is lower than the 0·49 without bet-hedging. This is as expected. Under bet-hedging, the likelihood that any particular juvenile reaches the winter ground is lower than without bet-hedging. However, the likelihood that at least one juvenile does so is higher under bet-hedging than without (0·79 under bet-hedging vs. 0·69 without bet-hedging). This latter difference is a principal source of the fitness advantage from bet-hedging, as captured by its higher geometric mean.

Lastly, we note that bet-hedging’s dominance in geometric mean fitness over non-bet-hedging, in the baseline and in all eight variants, implies that a population of adults who adopt the optimal, bet-hedging strategy for the migratory orientation of its juveniles will not be subject to invasion and displacement by a population of otherwise similar adults who do not.

Discussion

The use of simple vector navigation and the interaction with winds that are unpredictable in speed and direction make the first fall migration of juvenile passerines an especially risky undertaking. Based on a long-term distribution of wind patterns during the fall migration period, there is a single combination of two sequential headings that maximizes the average number of offspring that reach the winter grounds in a given year. Conceivably, an adult female’s strategy could call for sending all offspring along this same vector combination. Given the high variability in wind conditions that may be encountered by an adult’s migrating offspring in successive years, however, this strategy is dangerous. In particular, a string of unfavourable wind events in successive years could leave that adult without breeding progeny and thus mark the end of its genetic contribution. These are classically the type of circumstances in which bet-hedging may be adopted as a mechanism for reducing the variation in fitness outcomes at the expense of some reduction in mean fitness. The advantage gained is the reduced vulnerability of the genetic line to annual environmental stochasticity. Simulations of the model reported here show that within-clutch bet-hedging in migratory orientation of juveniles maximizes geometric mean fitness of the adult and thus improves the chances for long-term survival of an adult’s progeny. Sensitivity analysis demonstrates that this finding is robust to plausible variation in the underlying parameters. The strict dominance of bet-hedging over all common-heading strategies under each parameterization strongly suggests that bet-hedging of juvenile migratory orientation may be an evolutionarily stable strategy. In particular, it suggests that a population employing a bet-hedging strategy of the type considered here could not be successfully invaded and displaced by an otherwise similar population that relies on common sequential headings for its offspring. It is certainly possible that other forms of within-clutch variation could be sources of fitness-enhancing bet-hedging. For example, within-clutch variation in the timing of migration could provide bet-hedging benefits, but only if no common weather pattern (e.g. similar wind profile) is likely to predominate during the cold fronts that fall within the window during which successful migration is feasible. The necessity of migrating in association with such cold fronts implies that the delay of a few days or a week will frequently fail to push the migratory initiation of any two siblings into different cold fronts, thus frustrating the effort to bet-hedge by that method.

The bet-hedging hypothesis clearly provides a potential explanation for the well-documented persistence of variation in the migratory orientation of juvenile passerines. It predicts as well a difference in the optimal level of bet-hedging, depending on a species’ vulnerability to stochastic wind events. For example, faster-flying birds are less affected by stochastic wind profiles, and thus their gain in geometric mean fitness from a given unit of heading variation should be smaller. Hence optimal bet-hedging for faster-flying birds should involve less within-clutch variation in juvenile migratory orientation. Consistent with this prediction, Backman & Alerstam (2003) found that faster-flying migrants passing through their radar beam in Lund, Sweden, exhibited less variation in their flight tracks (inclusive of wind influence) than did slower-flying migrants. This difference in scatter was especially pronounced in their fall migration sample. Orientation cage experiments designed to provide controlled comparisons of the within-clutch scatter in faster vs. slower flying species could provide additional evidence on the prevalence of bet-hedging. Species, or potentially different populations of the same species, whose winter grounds are significantly closer to their breeding grounds might be expected to derive less benefit from bet-hedging. Similarly, species whose suitable winter grounds encompass a relatively wider geographical area (and thus present a broader migratory target) would receive less benefit from bet-hedging. Evidence on these predictions could be derived as well from a set of well designed orientation cage experiments. Once the technology develops to the point that low cost satellite transmitters are available in bulk, such orientation cage experiments could be replaced or supplemented by tracking studies, controlled for wind conditions.

Finally, we might expect bet-hedging to produce a relatively sharp gradient in the age composition of migrants in the vicinity of a well-defined geographical hazard. Bet-hedging thus provides a new explanation for the ‘coastal effect’ in which juveniles constitute a much higher percentage of migrating passerines along the Atlantic coast of the United States than at inland locations. This phenomenon has been widely documented for most small passerines (Ralph 1981; Able & Bingman 1987) and applies to D. caerulescens as well. For example, during September and October for the years 1980–2000, 68% of the 13 621 known-age black-throated blue warblers banded at the Allegheny Front Banding Station in north-eastern West Virginia (∼355 km from the Atlantic coast) were juveniles. By contrast, 95% of 4328 birds of that species banded at Kiptopeke Banding Station on Virginia’s eastern shore for the same 21-year time period were juveniles (Bird Banding Laboratory Database, U.S. Geological Survey, Department of the Interior). Ralph (1981) proposed that many of these juveniles arrive at the coast as a result of following suboptimal headings. Under the bet-hedging hypothesis, the key point to recognize is that the orientation of juveniles observed at the coast is not inherently suboptimal a priori. Rather such headings have proven to be suboptimal only in combination with the stochastic wind profile that was actually encountered that year. In other years, facing different stochastic wind profiles, those same headings might have carried those juvenile migrants safely to favourable winter habitat. Adult birds, by contrast, are able to make daily heading corrections as they navigate toward known winter sites, and would be expected to avoid the coast when possible, as unpredictable winds and close proximity to the ocean could easily constitute a fatal combination for any small passerine migrant.

Acknowledgements

We thank, without implicating, Steve Ellner and the Ecological Theory Lab Group at Cornell University, the United States Bird Banding Laboratory, the Coastal Virginia Wildlife Observatory, and the banders, past and present, at the Kiptopeke and Allegheny Front Banding Stations for the banding data cited in the paper. This paper benefited substantially from the suggestions of four referees and an associate editor. Our research was supported in part by a National Science Foundation Graduate Research Fellowship to James Reilly.