We consider a population of long-distance migratory birds that compete for territories at their breeding grounds. Arrival date at the breeding grounds of an individual is denoted x. The beginning and the end of the season are denoted x_b and x_e, respectively. We assume birds raise their young on a temporally restricted resource, such as insect larvae, that is available for only a short period, e.g., [23]. Following Johansson and Jonzén [21], the matching between timing of reproduction and the temporal food abundance distribution is not modelled explicitly. Instead we assume that early arriving individuals can access and use these food resources for a longer period of time compared to later arriving individuals, and thereby obtain a higher reproductive output. With a sufficiently early arrival date, we assume individuals have full access to the food resources and with a sufficiently late arrival date we assume reproductive failure due to lack of food resources. In line with these assumptions we model the expected reproductive success R of an individual as a decreasing, logistic function of arrival date (Figure 1A, [21]): (1)

The parameter R₀ represents the maximal reproductive output. Shifts in the underlying food resource distribution affect the timing of the half saturation point. In this way r can be seen as a proxy for the food peak date [21], and to facilitate reading we will refer to r as such from this point forward. Finally, small (large) values of σ_R result in steep (shallow) slopes around the inflection point, which we interpret as a narrow (broad) resource distribution (Figure 1A, [21]).

We assume that early migration is associated with mortality costs e.g., due to harsh conditions or low resource levels en route or on the breeding grounds. We let survival S be an increasing function of timing of arrival date (Figure 1B, [21]): (2)

With this sigmoidal shape we assume that the prospects of survival improve fastest with arrival date around the half saturation point s. We assume s is associated with (a)biotic conditions such as snow melt or the appearance of food resources and henceforth refer to it as the onset of spring. Note that the survival function derived from first principles by Jonzén et al. [20] also has a sigmoidal shape. The parameters S₀ represents the maximal survival rate and the slope factor σ_S controls the rate at which survial increases with arrival date (cf. σ_R). We treat r and s as separate variables since they represent environmental effects on different parts of the life cycle, which are expected to differ between species [20]. Due to their dependence on shared underlying environmental factors, however, r and s likely are correlated and we consider both joint and separate changes of these parameters below [21].

The population density in a given year t is denoted n_t, and is, more specifically, assumed to be measured in terms of migrants returning to the breeding grounds. For simplicity we assume an asexual model. We let μ denote the mean arrival date in the population and assume that arrival dates follow the distribution g_μ. We assume that μ is under selective control whereas g_μ(x) represents spread of arrival dates as a result of environmental variability [24,25]. Following Johansson and Jonzén [21] we study a case where the arrival dates are uniformly distributed. Thus, we assume that g_μ = 1/v within the interval [μ − v/2, μ + v/2], where v is the length of the arrival interval, and 0 otherwise. The main reason to use a uniform distribution is to facilitate the analysis and thereby maximise the transparency of the model (see also [21]). Other distributions of arrival dates, including a Gaussian distribution, have been used to characterize phenology in bird migration data [26]. However, Johansson and Jonzén explored alternatives to a uniform arrival distribution in a similar model (Online Appendix C in [21]) and found that they had only minor effects on shifts of the ESS timing on arrival due to environmental changes.

Within any given year, the density of individuals that arrives at the breeding grounds increases with arrival date x. Density dependence occurs via territory competition among individuals that survive the pre-breeding state. The daily stream of arriving individuals that survive the pre-breeding period equals g_μ(x)S(x)n_t. The accumulated density of surviving individuals, n_S, on a given day is then: (3) and the total density of surviving individuals after arrival of all individuals is defined as N_S,t (note that N_S,t < n_t) and is given by: (4)

The majority of the population is assumed to be monomorphic with respect to the mean arrival date. Following the adaptive dynamics framework for evolution of continuous traits [27,28], it is assumed that the prevailing trait is subject to invasions of variant traits due to mutation or potentially migration. It is assumed that invasions are rare, and that the environment is otherwise stable such that population dynamics equilibrize between invasion attempts. The prevailing mean arrival date in the population is denoted μ and the variant trait is denoted μ'.

The expected number of offspring w(x) of a bird with arrival date x is assumed to depend on territory quality Q(x) such that the per capita reproductive output is R(x)Q(x). We interpret R(x) as the reproductive output in the best available territory and interpret Q as a discount factor with values between 0 and 1 [21]. We assume that territory quality decreases as individuals occupy the best available territories according to a principle of prior residency [19]. For simplicity we assume that territory quality declines linearly, until 0, with the density of arriving (and surviving) individuals: (5) where the parameter q_c controls how fast the territory quality declines with density. Individuals of the variant population that arrive outside the interval of arrival for the prevailing strategy are assumed to be few and to have a marginal ecological effect, in line with a typical assumptions of the adaptive dynamics approach [22,27,28]. Following Johansson and Jonzén [21] we thus assume Q(x) = Q(μ − v/2) for x < μ − v/2 and Q(x) = Q(μ + v/2) for x > μ + v/2.

The reproductive output in a population with distribution g_μ' is obtained by integrating the product of surviving adults and their reproductive success over the distribution of arrival dates in the population: (6)

Population density is reduced by mortality on wintering grounds and during migration. In the model this is captured by the following dynamics between years: (7) where the constants s_A and s_J denote winter survival of adults and juveniles respectively. We assume however that changes in s_A and s_J due to environmental changes are strongly correlated. For simplicity we thus let s_J = ks_A with k < 1 and consider changes in s_A only. The combined within-year and between-year population dynamic can finally be written as: (8)

The age structure in the population is studied by considering the fraction of young in the population, based on equation 7. The first term in equation 7 represents density of adults that have survived migration to the breeding grounds, and the second term represent the density of individuals born in year t, i.e., first year migrants that survived the migration to the breeding grounds. Thus we define the fraction of young, f, as the fraction of first year migrants within the population of migrating individuals as: (9)

We analyze the model following the adaptive dynamics framework [27,28]. Thus we derive invasion fitness W(μ',μ), understood as the per capita growth rate of the rare invading strategy μ' into a population employing strategy μ, from equation 8 as: (10) where n^* denotes the population density at equilibrium. This equilibrium was obtained by numerical iterations of equation 8 (with 1,000 time steps). With a uniform distribution W can be written: (11)

Following Geritz et al. [25] we then calculate the selection gradient h as: (12)

Assuming that rate of evolutionary change is proportional to h, an end point for evolution is then found where the selection gradient vanishes (D[μ] = 0), provided that this point is also convergence stable and resistant to invasions. Following McGill & Brown [28] we refer to the mean arrival date fulfilling these criteria as an ESS. Note that apart from the assumption of a monomorphic resident population, our analysis is equivalent to analyses based on the quantitative genetic approach to studying gradual evolution under frequency dependent selection [28,29,30].

The right hand derivative (indicated by ∂⁺ and defined for μ' > μ) and the left hand derivative (indicated by ∂⁻ and defined for μ' < μ) of equation 11 with respect to the variant trait are: (13)

Since these derivatives coincide when μ' = μ the selection gradient can be written as [cf. 21]: (14)

We found the ESS values by using numerical calculations to solve h(μ) = 0 and to establish that these points were resistant to invasions and convergence stable [27]. The corresponding equilibrium densities and equilibrium population structures were also calculated numerically.

We assume that the population is in a state of ecological and evolutionary (ESS) equilibrium before the environmental change takes place. This equilibrium can thus be interpreted as an historical baseline [22]. The purpose of this section is to characterise how this historical baseline depends on model parameters, and thereby to provide the necessary background for understanding the effects of environmental changes described in the result section. The ESS mean arrival time predicted by the model depends on different environmental parameters in broad agreement with previous models [20,21,22]. Timing of arrival is early when spring starts early since that allows the territory game to take place early in the season (Figure 2A). Winter survival and productivity on the breeding grounds increase competition which in turn favours early arrival (Figure 2B,C). Broadening the resource peak (increasing σ_R) causes the ESS timing of arrival to be slightly later (Figure 2D). This pattern can be due to better reproductive conditions for late arriving individuals, as explained by Jonzén et al. [20]. With the parameter settings here, however, individuals arrive before r (mean timing of arrival is around 0 and r = 1 in Figure 2D), and thus a broader resource peak means fewer resources for reproduction and a lower population size (Figure 1A, Figure 2I). Hence, here the later ESS timing of arrival is due to reduced competition. With a slower improvement of pre-breeding survival (increased σ_S) we also get a later ESS (Figure 2D). This is also an effect of lower competition, due to reduced survival and thereby lower population densities (Figure 2J), since mean timing of arrival occurs after the inflection point s (note that s = −1 in Figure 2E, cf. Figure 1B). It should be noted that the shape of the pre-breeding survival curve can affect the relationship between the ESS arrival time and the width of the food distribution, which may have a maximum or a minimum (see [20] for details).

Equilibrium population densities and equilibrium age structure in ESS situations follow in a straightforward manner from the underlying biology. Population density increases with earlier spring (Figure 2F), since survival S increases, at least slightly, for any given arrival date. Population density also increases with winter survival (Figure 2G) and with productivity (Figure 2H). Population density decreases with width of the food distribution (Figure 2I) and with slower rate of improvement of pre-breeding survival (Figure 2J), since individuals arrive before r and after s, as mentioned above. Age structure at ESS is only marginally affected by earlier onset of spring (Figure 2K). This may seem a bit surprising since, at least in the short run, earlier spring favours adult survival, and therefore can be expected to change the age structure. The ESS timing of arrival occurs earlier in the year, however (Figure 1A), which in turn counteracts that effect. Age structure is affected by winter survival and productivity on the other hand. When winter survival increases (Figure 2L) the proportion of adults increases which causes the fraction of young to decrease. In contrast, when productivity increases, the fraction of young increases (Figure 2M) due to an increase in per capita reproduction so that more young are recruited into the population. Broadening the resource peak causes a (weak) decline in fraction of young (Figure 2N), due to the reduction in reproductive success as described above. With a slower rate of improvement of pre-breeding survival the fraction of young decreases however (Figure 2O), due to the reduced survival of adults as described above.

We study the effects of change of the environmental variables s and r. In general, environmental variables may undergo sudden shifts, gradual changes and stochastic fluctuations [31,32]. For simplicity we here consider sudden shifts only, but by envisioning gradual or stochastic changes as a series of smaller shifts their qualitative effects can still be discerned from the results.

When the environment changes, the population will no longer be at ecological and evolutionary equilibrium. In the short term, the population densities may increase or decrease, as may the fraction of young in the population. We record these two properties, along with the new selection gradient h(μ) after the population density has settled on the new equilibrium. We assume however that evolutionary change, i.e., change in mean arrival time μ, is slow compared to demographic responses, and therefore assume that it is not affected.

An advanced onset of spring has a strong effect on selection compared to an advanced food peak when spring is already early (Figure 3A), when winter survival is high (Figure 3B), when productivity is high (Figure 3C), when the resource peak is narrow (Figure 3D) and with slow improvement of pre-breeding conditions (Figure 3E). In these cases, the selection after only r has changed is very small (solid lines are close to zero), and the effect of changing s and r jointly (green lines) is very similar to changing s only (blue lines).

Under most settings, advanced food peak leads to selection for earlier arrival (Figure 3A–C,E). When the resource distribution is broad, however, advanced food peak can cause selection for a delayed arrival date (Figure 3D). To understand this phenomenon, note that a changed resource peak affects the adaptive landscape not only by affecting the reproductive success of individuals for different arrival dates, but also by changing the reproductive success as such (Figure 1A), which, via effects on population densities, can affect the competitive pressure. At evolutionary equilibrium there is a balance between selection for earlier arrival to increase reproduction and to avoid competition, and selection for later arrival dates to increase survival (cf. Equation 14). When the food peak advances, selection for earlier reproduction increases but population density decreases (Figure 3I). Thereby also the competitive pressure decreases, which in turn weakens selection towards earlier arrival dates. The effect of a food peak shift on selection is relatively weak when the food distribution is wide, but the effect of population densities (Figure 3I) and thereby on competition may still be substantial, which explains why later arrival can be favoured under these circumstances.

Finally we note that s has the stronger influence on selection when the ESS is early (compare Figure 3A–E with Figure 2A–E). An early ESS can in turn occur either when spring is early or when competition is strong. An early spring leads to a large population, whereas strong competition occurs when the population density is large e.g., due to high winter survival, high productivity, narrow resource distribution or slow rate of improving pre-breeding conditions. Thus, strong influence of s on selection is also predicted by large population size (Figure 2F–J).

The population grows when spring advances and declines when the food resource peak advances in all scenarios (Figure 3F–J). The effect of changed resource peak is very small for many parameter settings investigated (purple lines are close to 1 in large parts of Figure 3F–J), however. In these cases, the model predicts a net increase when both factors advance, indicating that the positive effects on population density from the advanced spring overrides the small negative effects of an advanced food peak. More precisely, this occurs when the baseline ESS mean arrival time is early, which, as discussed above, is in turn predicted by large population densities. In cases where changes in r have the larger effect, namely at low population densities in the baseline (cf. Figure 2F–J), the changes in r can be very strong however, and lead to strong reductions or even extinctions of the population (Figure 3F–J).

The fraction of young in the population decreases both when the onset of spring advances and when the food peak advances. In many cases, the effect of changed resource peak is again very small, however (purple lines are close to 1 in large parts of Figure 3K–O). Effects of s and r on population structure follow the same pattern as the effects of s and r on selection and population density: high population densities (Figure 2F–J) predict more effect of s than of r.

Our eco-evolutionary approach to studying phenological adaptation in migratory birds contributes with several new insights into how demographic changes relate to phenological shifts in migratory birds. To start with, our analysis allows for a comparison of the effects of advanced onset of spring (decreased s) and advanced food peak (decreased r) on total population density. Since advanced onset of spring alone increases pre-breeding survival and advanced food peak reduces reproductive success, the two types of environmental change have opposing effects on population density. When both changes occur in parallel, e.g., reflecting even temperature changes over the season, the net population trend can thus be hard to predict. The model shows, however, that the historical baseline influences which of the two factors has the largest influence. Thus, when the ESS timing of arrival time is early, e.g., due to strong territory competition, the positive effects of improved pre-breeding conditions are strong and we get a net population increase. When the ESS arrival time is late, the negative effects of reduced reproduction are stronger and we get a net population decline.

In many cases, our model predicts weak effects of a changed resource peak on population trends (Figure 3F–J). This observation seems to go counter the emphasis on changed resource peaks to explain population trends in migratory birds [6]. On the other hand, we can also see from our results that, whereas the positive effects of an advanced onset of spring in our model only increases population densities a little, a changed resource peak can cause strong decreases in population size and even extinctions. In this way, our model support concerns that mismatched reproduction can indeed have serious consequences for populations of bird species. Our model moreover suggests indicators for situations where shifts in the food distribution are expected to have strong negative effects on populations, e.g., low winter survival (Figure 3G), low productivity (Figure 3H), and slow improvement of pre-breeding conditions (Figure 3J).

Another interesting result is that the fraction of young decreases both when the resource peak advances and when the onset of spring advances. This result is not surprising when one considers the fact that this measure does not inform on the absolute numbers of young: increased reproductive mismatch (decreased r) reduces the number of young recruited into the population and improved survival due to earlier onset of spring (decreased s) increases the proportion of adults. On the other hand, this result shows that relating phenological shifts solely to food resource peaks can be misleading when interpreting associated demographic responses [9]. Under the assumption that phenological shifts mostly affect demography via reproduction, a reduced fraction of young in a population would be interpreted as a sign of trophic mismatch, whereas in the light of our study, it could well be due to improved pre-breeding conditions. In many cases, our model also predicts a relatively small effect of an advancing food peak on population structure (Figure 3K–O), which reduces the scope for explaining changed population structure with shifts in food resource peaks even further.

When it comes to evolutionary responses to climate change, our analysis highlights the role of pre-breeding survival for phenological adaptation in migratory birds. As it turns out, the survival costs not only modulate responses to changed food peaks, as shown by Jonzén et al. [20], but also act as a key factor for selection on arrival dates on its own. In many cases there is almost no selective response to changes in the resource peak (r), but there are always selective responses when the onset of spring (s) changes (Figure 3A–E). According to our model, adaptive changes in arrival times are more likely to be associated with changes in pre-breeding conditions rather than with changes in food peaks.

In the predecessors [20,21,22] of the model developed here, competitive pressure was treated as a parameter. In the present model, competitive pressure instead depends on the underlying ecology in the model and the strategies employed by the population. We have thereby closed the eco-evolutionary feedback loop. A first insight from this more complete treatment is that parameters that affect the equilibrium population density, such as winter survival and productivity, by adjusting the competitive pressure, also affect the evolutionarily stable timing of arrival (Figure 3B,C). We have also shown that advancing the resource peak can favour later arrival dates under the influence of eco-evolutionary feedbacks (Figure 3D). This example of selection counter the direction of environmental change underscores the fact that competitive interaction can affect phenological adaptation in non-intuitive ways and contributes a new adaptive mechanism to explain variation in phenological trends.

The enhanced realism in the model analysed here also allows a closer connection to empirical observations, compared to previous models of this kind. Below we compare our model predictions with phenological responses in the Pied flycatcher. Thereafter we discuss different possibilities to further explore this eco-evolutionary approach to studying phenological adaptation in a changing climate.

A general result from our study is that ecological differences can cause variation in adaptive phenological responses to climate change. Naturally, species-specific properties will give rise to a certain amount of variation in phenology. Therefore, to reduce confounding factors, our hypotheses should be compared to phenological responses within a single species. Interestingly, breeding populations of pied flycatchers, Ficedula hypoleuca, have shown markedly different responses to environmental change across Western Europe [33]. The pied flycatcher is a long-distance migrant that breeds in Eurasia in summer and overwinters in Africa, south of the Sahara desert [34,35]. In The Netherlands, population numbers have declined in areas where a large mismatch exists between the timing of egg-laying and peak in food availability [36]. The timing of arrival did not change but the laying date has advanced. However, this change has been insufficient to avoid demographic consequences, as shown by declined population numbers [5]. Pied flycatchers in Britain also show large declines in population numbers. A study by Goodenough et al. [12] suggested, however, that only part of this decline can be explained by reduced breeding performance.

In a study on Pied flycatchers in Finland it was found that laying date was delayed in comparison to measures of spring phenology [37]. Judging from these trends, and the fact that clutch sizes also decreased, Laaksonen et al. [37] speculate that some trophic mismatch has occurred. The density of breeding pairs showed a quadratic trend over the study period, however, with a minimum for intermediate years [38]. Thus, the increased mismatch has been accompanied by an increase of the population towards the end of the study period.

A first observation we can do from these studies is that the mismatch between laying date and resource food peak alone does not predict trends in population change. This is in line with a general result of our study that the effect of changed food peak date on population densities may vary significantly depending on the historical baseline and on changes in other environmental variables (Figure 2).

Some of the differences between these responses can also be related to factors that influence demographic responses in our model. For the British population, for example, there is a relatively long time interval between arrival and breeding (around 30 days [39]). In relation to our model, this could be interpreted as an historical adaptation to a relative large difference between s and r. Under these circumstances, our model predicts a relatively small effect of a changed resource peak on population density (low value of s in Figure 3F). In the Dutch population in contrast, there is a short time interval between arrival and breeding (around 10 days [5]). This is a situation where the model predicts a relatively strong effect (high value of s in Figure 3F). In this way, the model predictions are in agreement with population decline, depending little on resource peak changes in Britain and depending more on resource peak changes in The Netherlands, as suggested by these studies.

It is also interesting to note that one factor that explained the decline in the British population [12] was Winter NAO index, which, as suggested by the authors, is likely to influence food availability on the wintering grounds and during migration. This could be interpreted as reduced winter survival or reduced productivity, which in our model also can explain reduced population density (Figure 2G,H).

For early arriving individuals in the Finnish population, reproductive mismatch and increased population density seems to have occurred simultaneously. This may seem strange, but interestingly, temperatures during the early part of their migration have increased as well [39], which could be interpreted as reduced survival cost for early arrival, i.e., earlier s in our model. If the majority of this population arrives early, then this change can indeed explain increased population density in our model (Figure 3F–J), even when trophic mismatch via advanced food peak occurs simultaneously (advanced r). In this population, there was also a change in the arrival date for early migrants [39], which is adaptive according to our model when s decreases (Figure 3A) and is in line with long term increase of population density as well (Figure 2F).

In sum, we find that a number of observations from these systems are in line with predictions from the model. These comparisons also serve as an illustration for the motivation of our study; namely the need to take the whole life history and a broad set of factors into account when predicting demographic changes involved in phenological shifts in migratory birds.