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1 2 Supplementary Material for: Social networks in changing environments 3 Behavioral Ecology and Sociobiology 4 5 6 A.D.M. Wilson*, S. Krause, I.W. Ramnarine, K.K. Borner, R.J.G. Clement, R.H.J.M. Kurvers & J. Krause 7 8 *Centre for Integrative Ecology, Deakin University, 75 Pigdons Road, Waurn Ponds, Victoria 9 3216 Australia 10 11 *Corresponding Author:[email protected] 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Markov chain modelling approach and parameter estimation Our observations consist of sequences of ‘behavioural states’. In the presence of k potential neighbours at each time point a focal fish can either be with a nearest neighbour g, 1 ≤ g ≤ k, denoted by sg or alone (no conspecific within 4 body lengths) denoted by a. A focal fish is regarded as being social, if it is in state sg for some neighbour g. ‘Being social’ is not an explicit state in the model, but is implicitly defined by the set of states s1 ... sk. Following Wilson & Krause et al. (2014) we used the three probabilities pleave_nn = Prob(staten+1 ≠ sg | staten =sg). psa = Prob(staten+1 = a | staten {s1, ..., sk}), and pas = Prob(staten+1 {s1, ..., sk} | staten = a) to construct a model that describes the transitions between these states. Here, pleave_nn denotes the probability of leaving the current nearest neighbour, and psa (pas) the probability that the focal fish will be alone (social) in the next state when it is currently social (alone). These probabilities can be estimated based on relative frequencies (Fink 2008). Following Wilson & Krause et al. 2014 we did not take the specific individual identities into account when estimating the model probabilities, i.e. we counted the state changes regardless of the individual identities. Therefore, our model describes the general dynamics common to all observed individuals. Figure 1 in the main text shows a graphical representation of the resulting model. Its probabilities define the transition probabilities of a (first-order) Markov chain without having to introduce any further assumptions or parameters. From these probabilities the probability pswitch_nn (of switching the nearest neighbours while staying social) and pretain_nn (of retaining the nearest neighbour) can be derived as follows. pswitch_nn = pleave_nn psa. pretain_nn = 1 pswitch_nn psa. 43 44 45 46 47 A Markov model can be used to produce state sequences by simply ‘running’ the model. In our case, this allows predictions regarding the frequency distributions of the lengths of contact with a specific neighbour, of phases of being social, and of phases of being alone. These predictions can be used to analyse the goodness of fit of our model, which is described in more detail in the next section. 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Goodness of fit of the Markov chain model The time spent in a state of a Markov chain follows a geometric distribution. In our study system this means, the frequencies of phase lengths of, e.g., being social should decrease exponentially with increasing phase length. To compare the model predictions with the observed data we simulated observations of the model’s behaviour where we took into account the 2 min observation time per focal individual. This is necessary because incompletely observed phases (that started or ended outside the observation period) will lead to higher numbers of short phases than theoretically expected. We repeated the simulation 10 4 times and computed the mean frequencies and the 2.5% and 97.5% percentiles for each phase length. The simulation was based on the estimated probabilities and did not take into account their confidence intervals. Therefore, the predicted percentile ranges are conservative. Our results show that the observed data are well approximated by the model predictions (Figure S1). 64 65 Movement simulation 66 67 68 69 In order to construct a movement simulation that can be used as a null model for the investigation of density changes we tried to introduce as few parameters as possible. We used a discrete-time simulation where between two successive time points an individual can be moving or resting. At each time point, an individual can retain its state (moving or resting) with fixed probabilities pmoving and presting, respectively, or change it. A moving individual moves by a fixed distance l in a direction given by the individual’s heading h. Additionally, at each time point an individual can decide to change its heading with probability pchange_heading. A new heading is computed by adding a randomly chosen value α to the current heading. We tried out two different probability distributions for α, a uniform distribution on the range (-π/2, +π/2) and a von Mises distribution, which is a circular analogue of the normal distribution, on the range (-π, +π) with concentration 1. This choice had little influence on our results and the trends were exactly the same. Therefore, we decided to only use the von Mises distribution as it seemed more natural. We also tried out two different ways to define the simulation ‘world’ in which the individuals move, a circle with fixed radius r and a torus constructed from a square by pasting the opposite edges together. Our results did not seem to depend very much on these variants and we decided to use a circular world with a fixed radius as this seemed more natural to us. However, in this case we need to prevent individuals from ‘leaving’ the simulation world. We did this by letting individuals that are about to leave choose a new heading (using the von Mises distribution) until a movement in this direction of length l is within the circle. 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 Summarized, our movement simulation consists of the 3 probabilities pmoving, presting, and pchange_heading, the step length l, and the choices regarding the probability distribution of heading changes and the shape and size of the simulation world. To be able to draw comparisons with our real observations we additionally introduced a distance d that defines the distance within which two individuals are regarded as ‘neighbours’ having social contact. We examined a wide range of parameter settings for the 3 probabilities, the step length l, and the distance d and found that regardless of the settings a) the lengths of contact with a particular neighbour, of being social and of being alone could be described by a Markov chain model with parameters estimated from the simulated movements and b) the estimated model probabilities pas (of ending being alone) and pswitch_nn (of switching the current neighbour) increased with increasing density always following the same pattern (a linear function, see Figure S2 for an illustration of this for specific parameter settings). This means, although we do not know whether our movement simulation exactly describes the real movements of our fish it makes sense to use the movement model to assess the influence of density changes on the estimated model probabilities pas and pswitch_nn. The same holds for the probability psa (of ending being social) which equals pleave_nn - pswitch_nn and therefore decreases with increasing density. The probability pleave_nn (of leaving the current neighbour) does not depend on the density (however, see the remarks in the next paragraph). By changing the number of individuals or the radius r of the simulation world while keeping constant all other parameters of the movement simulation we can investigate the relative magnitude of changes to the model probabilities caused by increased or decreased density. 110 111 112 113 114 115 We found that the probability pleave_nn slightly increased with increasing density although these changes were very minor (Figure S2). One reason for this might be that with higher density it happens more often that the nearest neighbour of the focal individual changes because a third individual approaches the focal individual. This leads to shorter contact phases with the same individual and thus increases pleave_nn. 116 117 118 119 References 120 Fink GA (2008) Markov Models for Pattern Recognition. Springer-Verlag 121 122 Figures 123 124 Figure S1. Frequency distributions of the lengths of contact with a particular nearest neighbour, the lengths of social contact, and the lengths of being alone in the observed data (circles) before manipulations for a) pool 1a, b) pool 1b, c) pool 2, and d) pool 3. Also shown are the means (x’s) and the 2.5% and 97.5% percentiles as predicted by our Markov chain models. (Note that 0 values cannot be displayed in a logarithmic plot and are omitted.) 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 Figure S2. Markov chain model probabilities pleave_nn (open circles), psa (filled circles), pas (squares), and pswitch_nn (triangles) as a function of the density. The R-squared values of all fitted lines were > 0.98. The size of the simulated world was constant and the number of individuals was increased to increase the density. Note that the density values do not have an absolute meaning because the size of the world does not have any unit. The probabilities were estimated from ‘observations’ of simulated movements. The simulation parameters were set to the values pmoving = 0.05, presting = 0.2, pchange_heading = 0.05, l = 0.025, and d = 0.1. 142 Figure S1 143 a) 50 20 5 10 Frequency 20 10 Frequency 5 2 1 1 2 2 1 152 153 154 2 4 6 8 10 12 2 4 Number of consecutive time points b) 6 8 10 2 12 4 6 Social contact 10 12 Being alone 5 Frequency 5 10 Frequency 10 20 20 50 50 100 50 Contact with a particular neighbour 8 Number of consecutive time points Number of consecutive time points 20 Frequency 157 158 159 160 200 50 151 155 156 100 200 50 20 Frequency 10 5 149 150 Being alone Social contact 100 146 147 148 Contact with a particular neighbour 10 144 145 2 4 6 8 10 1 1 1 2 2 2 5 161 162 163 164 2 12 4 c) 10 12 2 4 Contact with a particular neighbour 6 8 10 12 Number of consecutive time points Being alone 100 50 Social contact 50 100 20 10 5 Frequency 175 1 1 174 2 4 6 8 10 12 1 2 2 2 5 5 10 Frequency 10 Frequency 20 20 169 170 171 172 173 8 50 165 166 167 168 6 Number of consecutive time points Number of consecutive time points 2 4 Number of consecutive time points 6 8 10 12 2 4 Number of consecutive time points 6 8 10 12 Number of consecutive time points 176 d) 50 5 Frequency 5 10 Frequency 10 20 20 50 20 2 1 2 Frequency 10 5 2 1 185 50 100 179 180 181 182 183 184 Being alone Social contact Contact with a particular neighbour 1 177 178 2 4 6 8 10 Number of consecutive time points 12 2 4 6 8 10 Number of consecutive time points 12 2 4 6 8 10 Number of consecutive time points 12 186 Figure S2 187 1.0 188 189 196 197 198 199 200 201 202 203 0.4 0.2 195 0.0 193 194 Probability 191 192 0.6 0.8 190 6 8 10 Density 12 14