Optimal migration

references, abstracts and comments. Where there is no abstract, an abstract has been written, where abstracts are too long they have been abridged. Abstracts in languages other than English have been translated into English. The comment is personal, it points out errors and possible follow-ups, it is begun: CP:

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Alerstam, T. & Å. Lindström (1990): Optimal bird migration: the relative importance of time, energy and safety. - In: Gwinner, E. (ed.) Bird Migration: The Physiology and Ecophysiology. Springer, Berlin, Heidelberg, pp. 331 - 351.

Charnov, E. L. (1976): Optimal foraging: the marginal value theorem. Theor. Pop. Biol. 9: 129 - 136.

Farmer, A. H. & J. A. Wiens (1998): Optimal migration schedules depend on the landscape and the physical environment: a dynamic modeling view. J. Av. Biol. 29: 405 - 415.

We developed a dynamic state variable model of individual migrating shorebirds for use in testing hypotheses about spring migration strategies of the Pectoral Sandpiper Calidris melanotos. We conducted model sensitivity analyses to determine how predicted migration schedules might vary with respect to the landscape and the physical environment.
In landscapes with closely spaced,high-quality stopovers, female pectoral Sandpipers can vary widely in their migration schedules and still arrive on the breeding grounds early enough and with sufficient energy reserves to achieve maximum reproductive success. Such a population might appear quite variable, and show no stopover patterns, even if all individuals were making optimal decisions. Latitudinal gradients in temperature and photoperiod differentially affect a bird's energy budget as it moves northwards in the spring. Stopovers at more northerly locations are associated with higher metabolic rates, lower food abundance in early spring, and longer days for feeding. The optimal migration schedule in these conditions can be quite different from that in a homogeneous environment, and patterns observed in the field can be misinterpreted if the environmental gradients are not considered.
The landscape and the physical environment shape migration schedules and influence one's ability to interpret patterns observed at stopovers. Modeling these factors may lead to new insights about migration adaptations in heterogeneous environments.

Gudmundsson, G. A., Lindström, Å. & T. Alerstam (1991): Optimal fat loads and long distance flights by migrating Knots, Sanderlings and Turnstones. Ibis 133: 140 - 152.

Hedenström, A. & T. Alerstam (1997): Optimal fuel loads in migratory birds: distinguishing between time and energy minimization. J. theor. Biol. 189: 227 - 234.

By combining the potential flight range of fuel with different migration policies, the optimum departure fuel load for migratory birds can be calculated. We evaluate the optimal departure fuel loads associated with minimization of three different currencies: (1) overall time of migration, (2) energy cost of transport and (3) total energy cost of migration. Predicted departure loads are highest for (1), lowest for (2) and intermediate for (3). Further, currencies (1) and (3) show departure loads dependent on the fuel accumulation rate at stopovers, while (2) is not affected by variation in the rate of fuel accumulation. Furthermore, fuel loads optimized with respect to currency (3) will differ depending on the size (body mass) of the bird and the energy density of the fuel. We review ecological situations in which the various currencies may apply, and suggest how a combination of stopover decisions and observations of flight speed may be used to decide among the three cases of migration policies. Finally, we calculate that the total energy cost of migration is roughly divided between flight and stopover as 1:7, probably with a relatively longer stopover time in larger species. Hence, we may expect strong selection pressures to optimize the fuel accumulation strategies during stopover episodes.

Houston, A. I. (1998): Models of optimal avian migration: state, time and predation. J. Av. Biol. 29: 395 - 404.

Taking as my starting point the paper by Alerstam and Lindström (1990), i review models of optimal foraging and fuel loads during avian migration. As well as summarising previous results, I highlight topics where further work is required. I argue that we now have a good understanding of simple deterministic models based on time and energy. Discrepancies between such models and data have proved instructive. We also have some results about the influence of predation risk. Some work has been done on state-dependent models.This approach needs to be extended to models that can take account of changes in various aspects of internal state, and that put migration into the context of a bird's annual routine. I suggest that a combination of simple analytic models of some aspects of state, together with dynamic programming models of annual migration routines may be a fruitful research plan. We also need to explore the consequences of birds making decisions on the basis of climatic conditions.

Lindström, Å. & T. Alerstam (1992): Optimal fat loads in birds: a test of the time minimisation hypothesis. Am. Nat. 140: 477 - 491.

Weber, T. P., Houston, A. I. & B. J. Ens (1994): Optimal Departure Fat Loads and Stopover Site Use in Avian Migration: An Analytical Model. Proc. Biol. Sc. 258: 29-34.

We develop an analytical model which determines optimal departure fat loads and stopover site use for time-minimizing bird migration with any number of sites, to identify conditions under which birds carry more fat than necessary to reach the next stopover site (overloads) or skip suitable stopover sites. The model is analysed with three sites: the wintering ground, one stopover site (both of which are characterized by their daily fattening rates), and the breeding ground. Overloads and skipping canoccur if the sites decrease in quality in the direction of migration. Departure fat loads are usually either at the level necessary to reach the stopover site or at the level for a non-stop flight to the breeding ground; overloads are only deposited under rare circumstances. To model the effects of stochasticity in fat deposition we include explicit functions for the dependence of fitness on arrival time. If overloads are deposited, they are largest when the fitness function is concave, lowest when convex, and intermediate when linear.

Weber, T. P. & A.I. Houston (1997): Flight costs, flight range and the stopover ecology of migrating birds. J. An. Ecol. 66: 297-306.

1. Flight range equations are a central component in the analysis of avian migration strategies. These equations relate the distance that can be covered to the fuel load that the birds carry. Models of stopover decisions deal with the question of how birds should react to variations in fuel deposition rates. Time-minimization models generally predict an increasing relationship between departure fuel load and fuel deposition rate.
2. We show that quantitative details of predictions derived from optimality models depend critically on the flight range equation that is used. We use two classes of flight range equations; one class is based on theoretical assumptions of aerodynamics, the other is based on empirical measurements of metabolism during flight.

3. Most empirically derived equations can be written as Y(x) = c[1-(1+x)-D], where 0 < D < 1, and c is a constant that includes morphological traits and lean body mass.

4.Patterns of site use and departure loads in environments with discrete stopover sites depend in significant ways on flight costs.

5. Flight range estimates that are based on empirically derived, multivariate equations are sensitive to errors in the estimates of exponents of the equations. Varying some exponents within their confidence limits can alter flight ranges by an order of magnitude.

Weber, T. P. & A.I. Houston (1997): A general model for time-minimising avian migration. J. theor. Biol. 185: 447 - 458.

We develop an optimality model for time-minimising bird migration for any distribution of feeding habitats and qualities along the migratory route. If the fuel deposition rate is constant along the route and feeding is possible everywhere, it is optimal to divide the journey into steps of equal length. The relationship between fuel deposition rate and departure fuel load is not continuous. If the fuel deposition rate is changing monotonically along the route, step lengths are not constant. Step lengths increase if fuel deposition rates increase along the route and decrease if fuel deposition rates decrease along the route. In both cases the relationship between fuel deposition rate and departure fuel load is also not continuous. These qualitative patterns remain unaffected if different flight range equations are used and if the dependence of time spent flying on departure fuel load is included. If ecological barriers are introduced where fuel deposition is impossible it is in most cases optimal to make a stop at an edge of the barrier.

Weber, T. P., Ens, B.J. & A. I. Houston (1998): Optimal avian migration: A dynamic model of fuel stores and site use. Ev. Ecology 12: 377-401.

Birds migrating between widely separated wintering and breeding grounds may choose among a number of potential stopover sites by using different itineraries. Our aim is to predict the optimal migration schedule in terms of (1) rates of fuel deposition; (2) departure fuel loads and (3) stopover site use, when only a handful of such sites are available. We assume that reproductive success depends on the date and fuel load at arrival on the breeding grounds. On migration, the birds face a trade-off between gaining fuel and avoiding predation. To allow the optimal decision at any given moment to depend on the fuel load and the location of the bird, as well as on unpredictability in conditions, we employed stochastic dynamic programming. This technique assumes that the birds know the probability distribution of conditions in all sites, but not the particular realization they will encounter. We examined the consequences of varying aspects of the model, like (1) the shape of the relationship between arrival date and fitness, (2) the presence of stochasticity in fuel disposition rates and wind conditions, and (3) the nature of predation (i.e. whether predation risk depends on the fuel load of the birds or their feeding intensity).
Optimal fuel deposition rates are predicted to be constant if there are either only predation risks of maintaining stores or only risks of acquiring fuel stores. If only fuel acquisition is risky, fuel deposition rates can be below maximum, especially if there also is an intermediate best arrival time at the breeding ground. The food deposition rate at a site then depends not just on the site's quality but on the qualities of all visited sites. In contrast, rates of fuel deposition are not constant if both the acquisition and the maintenance of fuel stores carry risk. Optimal departure fuel loads are just enough to reach the next site if the environment is deterministic and are simply set by the energetic cost of covering the distance. As with time-minimizing models, more fuel than necessary to reach a site is only deposited under very restricted circumstances. Such overloads are more likely to be deposited if either fuel gains or expenditure are stochastic. The size of overloads is then determined by the variance in fuel gain at the target site and the worst possible conditions during flight.

Weber, T. P., Alerstam, T. & A. Hedenström (1998): Stopover decisions under wind influence. J. Av. Biol. 29: 552-560.

Despite evidence that wind conditions are an important factor in determining stopover decisions, models of time-minimizing bird migration have up to now emphasized the optimal response of the migrants to variations in fuel acquisition rates. We present a simple model of a time-minimizing migrant faced with two potential wind conditions on each day, which occur with a fixed probability. Wind assistance is modelled as a multiplicative factor in the flight range equation. We identify conditions under which birds leave the stopover site even with no tailwinds and conditions where the birds leave only with tailwinds in cases with global and local variation of the fuel deposition rate. The optimal policy depends on the probability and amount of wind assistance. In all cases there is an initial period at a stopover site when the bird should stay and build up its initially small fuel reserves irrespective of wind. After this initial time, there is a period when the optimal departure decision is to leave when tailwinds occur but stay and continue fuel deposition in other winds. If the probability of tailwinds is low the bird should at some later time change its policy to leave even in unfavourable winds. However, if a certain threshold value of the probability of favourable winds is reached the birds should never leave without wind assistance. These patterns lead to a complex relationship between departure load and fuel depositiobn rate. We compare our predictions with a null-model where the birds simply leave as soon as favourable winds occur. We further show that the inclusion of wind assistance cannot explain the discrepancy between observed and predicted values of departure loads under local variation in fuelling rates.

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