6.4.1 Construction of a migration network
The breeding, wintering and resting areas of the respective bird species, i.e. the network nodes, were determined based on the velocity with which the birds were moving. For each position we calculated the approaching and departing velocities vi,1 = (x(t) −x(t−∆t))/∆t and vi,2 = (x(t+ ∆t)−x(t))/∆t if previous and subsequent positions were taken less then one week before or after the present position. Locations were selected as resting localisations (see
6.4 Methods Fig. 6.2) when the approaching and departing velocities were less than 50 km/day (Gerkmann
& Riede, 2005) for the white stork and 20 km/day for the greater white-fronted geese. The so specified localisations were then clustered hierarchically by a combination of their pairwise Loxodrome distances (Wiltschko & Wiltschko, 1995) and circular time differences (Batschelet, 1981). Loxodromes were selected, because birds have been proposed to fly along these routes of constant bearing rather than along the orthodromes of shortest distance (Berthold, 2001a).
The cut off values for clusters were selected according to maximum daily average flight dis-tances and dendrogram patterns, 600 km/day for the white stork and 350 km/day for the white-fronted goose. All clusters that contained more than 3 resting localisations were finally selected to represent the set of typical breeding, resting and wintering regions of the considered population of each species.
6.4.2 Calculation of the links and their transition rates
The proposed migration model assumes that the continuous time transition ratesrij(τ)are of a von Mises functional form (eq. 6.2). To parameterise this function one first has to approximate a discrete set of transition fluxes Jij(τ, τ + ∆τ) and bird densities Nj(τ) from the data for small, discrete time steps∆t. We selected ∆t to be around 1 week ( ∆t= 0.13 if 1 year = 2π), thus there are 48 different seasonal time intervals(τ, τ+ ∆t). Jij(τ, τ+ ∆t)between all pairs of nodesi,j were counted for each time interval and converted into transition rates
rij(τ) = Jij(τ, τ+ ∆t)
Nj(τ)∆t . (6.3)
Bird densities as extracted from the data are included here, but are not general solutions of our model. The determination of smooth transition rates allows for the assignment of generalised population densities, and not just assertions about the small set of individual birds the data of which is available.
For each realised transition link the von Mises parametersϕij,σij andωij were estimated.
As this estimation is often based on very few data it is sufficient to calculate parameters with circular statistics (Batschelet, 1981) as follows:
ϕij = arctan
P
τ rij(τ) sin(τ) P
τ rij(τ) cos(τ)
(6.4)
σij = vu ut 1
P
τ
rij(τ) X
τ
rij(τ) (|τ −ϕij| mod (2π)) (6.5) ωij = max
τ (rij(τ)). (6.6)
Here,ϕij is the circular mean of the transition times fromj toiwith respect to the transition rates, σij is the circular standard deviation of the transition times, and ωij the maximum transition intensity (see also Fig. 6.1). Using these parameters one can calculate a smooth, estimated transition raterˆij(τ) for any season τ using eq. (6.2).
6 Paper V. A periodic Markov model of bird migration on a network
6.4.3 Estimation of the transition probabilities for the non-homogeneous Markov chain
In this work, we use the discrete Markov chain that corresponds to the continuous model.
Reasons were the relatively sparse data that are obtainable for parameterisation and the readily available methods for the analysis of Markov chains (Norris, 1997). The density of birdsNi(t) thus develops as
Ni(t+ ∆t) =X
j6=i
pij(τ; ∆t) Nj(t) +pii(τ; ∆t) Ni(t), (6.7) pij(τ; ∆t) being the transition probability at season τ for the time interval length ∆t. The two descriptions of the transition process can be transformed into one another by
P(τ; ∆t) =eR(τ) ∆t, (6.8)
using the Chapman-Kolmogorov differential equations (Logofet & Lesnaya, 2000), P(τ; ∆t) is the matrix (pij(τ; ∆t))ij and R(τ) corresponds to (rij(τ))ij. Because of the monotony of the exponential the functional forms of the transition probabilitiespij(τ; ∆t) also resemble the von Mises function.
To solve the model, probability matrices P(τ; ∆t) were calculated from the transition rate matrices R(τ) (eq. 6.8) for∆t= 0.13 (one week). Then, the stable distributions Ni∗(τ) can be calculated by using the Markov chain equations (eq. 6.7) and calculating the Poincar´e map from one year to the next through each time step τ by matrix multiplication
S(τ) =
TY−∆t
s=0
P(τ +s; ∆t). (6.9)
Each matrix S(τ) is a regular stochastic matrix and has a dominant eigenvalue λ(τ) = 1 (Perron-Frobenius Theorem). The corresponding eigenvector is the stable distributionNi∗(τ), i.e. S(τ) =S(τ)N∗(τ).
6.4.4 Characterisation of the migration network and Markov chain
Season specific migration network topologies were derived such that a pair of nodes is con-nected by a directed link fromj to iwhenever pij(τ; ∆t)>0.05. For all species, additionally, a cumulative network was constructed by merging the time specific networks. First, we want to examine which important network characteristics the cumulative migration networks have in common with other transportation networks (e.g. the internet, metabolic networks and the worldwide airport network). We calculated global and local efficiencies and network costs (La-tora & Marchiori, 2001), as well as average shortest topological path distances and clustering coefficients (Watts & Strogatz, 1998) to characterise if the two migration networks are similar to small worlds (Watts, 1999). This is a sign of networks optimised for quick and efficient transport of entities between the nodes. Furthermore, we analysed network robustness to ran-dom or selective deletion of strongly connected nodes (Albert et al., 2000). High robustness
6.4 Methods can be another indication for small world behaviour, but also suggests how strong flow on networks depends on each single node.
For further specification of migration network properties in the light of optimisation for trans-port we examined motif distributions (Miloet al., 2002), i.e. the “network building blocks”, and the networks’ symmetry. The latter shows which proportion of the migration routes is used in both directions, i.e. during spring as well as autumn migration.
For detailed characterisation of each resting area’s (node’s) importance for migration flow we determined degreesk, betweenness centralities bc(Freeman, 1979), closeness centralitycc (Wassermann & Faust, 1994), clustering coefficients C, cumulative densities, mean staying times following the highest density htsi and mean first passage times htf pi (Norris, 1997).
Betweenness centrality is the number of toplogically shortest paths in the network that pass a node, closeness centrality is a measure of the shortest distance of one to all other nodes.
These mesasures characterise how much each node contributes to network connectivity. The cumulative densities, staying times and mean first passage times indicate which nodes are frequently reached and used often and by a large number of birds. First passage times are studied in more detail, because they seem especially interesting in the light of bird migration.
How long does it take to move from one resting area to the other and when is it likely to return to a node? First passage times vary with the time of the year one starts off a region.
Because we are interested in the behaviour of the majority of individuals we have selected the first passage time starting in the season of highestNi∗(τ) for each transition from ito j and return time toi.
The topologies of the bird migration networks change with season. Therefore, to just examine the cumulative network characteristics is not sufficient. Additionally, we present simple network measures for each of the season-specific networks. The changes of the mean degree as well as the size of the giant component (i.e. largest connected subnetwork) with season are indications of network changes.
6.4.5 Model and parameterisation verification and sensitivity analysis
For the verification of how good node positions coincide with regions that are valid for breeding, resting and wintering of the white stork (Gerkmann & Riede, 2005) or greater white-fronted goose, respectively, we determined a dominant land cover type in the area of a radius of 200 km around each node centre. They were compared with the habitat preferences of the specific bird species. Land cover data were obtained from the USGS Land Cover Project (http://edc2.usgs.gov/glcc/).
To validate the appropriateness of our model for mapping bird migration, we explored the quality of rate fitting and how well the solutions Ni∗(τ) represent counted nonzero numbers of birds in the nodes. Differences and similarities ofrij(τ)andrˆij(τ)were analysed by a two-sample paired Wilcox test. To assess the fit of bird numbers by the stationary densitiesNi∗(τ), G-Test (Sokal & Rohlf, 1995) results were evaluated. Furthermore, we examined the sensitivity of the model outcomeNi∗(τ) to changes of the phasesϕij, widths σij and amplitudesωij of single transition rates, again by G-Tests.
6 Paper V. A periodic Markov model of bird migration on a network
Figure 6.2: Determining resting localisations by flight velocity. (a) for a white stork individual and(c)a greater white-fronted goose approaching (red dots) and descending velocities (blue dots) along a two-year trajectory. Empirically given thresholds for velocities of resting and migration movement are shown as green dotted lines. For the stork(b)and the goose individual(d)selected breeding, resting and wintering localisations are mapped as green dots; red dots depict localisations of fast migration movement.
Resting areas for the white stork are located in northern Germany, the Sudan area and southern Africa.
The goose individual winters in the Netherland area, migrating via a number of stopovers in wetland areas in northern Poland, the Baltic coast and Russia up to the island of Kolguiev.