Model setup

The model was created using the open-source software NetLogo ver. 5.0.4 [28]. All empirical values for model parameterization were obtained from the existing literature and our unpublished results obtained from Wood Warbler studies in the Wielkopolska National Park (western Poland). Unpublished data that were used are included as Supplementary Material (S1 File). Below, we briefly describe the basic model features: the simulation space, agents and model implementation. The model and full ODD protocol [29], [30] are provided as Supplementary Material (S2 File, S3 File).

The simulation space was designed to represent one of the study plots (Wiry Forest) in our study area—the Wielkopolska National Park. It was a discrete 2D square grid of 200 × 200 cells. Each cell represented an empirical distance of 10 m. Hence, the modeled space covered an area of 4 km², which is approximately equal to the area of the Wiry Forest (4.65 km²). The simulation space did not contain obstacles or artificial boundaries and was modeled as a torus.

Each run of this model consisted of two stages: the simulation of birds’ settling and the foraging behavior of predators. We simulated two types of predators i.e., Wild Boar (Sus scrofa) and Red Fox (Vulpes vulpes), which differed in their foraging behavior. During the first stage, time was not specified (see S3 File for details). The simulation time of the second stage was scaled based on data regarding Red Fox movement during foraging bouts [20], [31]. Time moved forward in discrete steps of fixed size, i.e., 1 tick per step in all simulations. We considered the following: (1) predators moved forward 1 grid cell per time step; (2) in nature, foxes often move ~2 km per foraging bout [31]; and (3) a single cell represented an empirical distance of 10 m; we defined one day as 200 time steps. Because the nest cycle of Wood Warblers lasts 32 days, all simulations were run for 6400 time steps. We assumed constant predation risk in each model run.

Bird and nest agents

The model included three sub-types of bird agents that exhibited a different settling strategy: cue-providers, cue-users and random-settlers.

The cue-providers represented birds that made settlement decisions based on personal information. During each model initialization, the cue-providers were distributed randomly within a simulation space at least 10 cells away from the nearest neighbor because the mean nearest-neighbor distance between territories of Wood Warblers in the Wiry Forest (Wielkopolska National Park) equals 102.21 m (n = 92 pairs) (Szymkowiak and Kuczyński, S1 File).

The cue-users represented birds that relied on social information, i.e., birds that made settlement decisions based on the location of other individuals who had already occupied territories. During each model initialization, the cue-users were distributed randomly within a simulation space but were at least 10 cells away from the nearest cue-provider. After all cue-users were placed in the modeled space, each of them started prospecting behavior, during which they moved in a correlated random walk (independent from each other) and became territorial after acquiring social cues. The correlated random walk is a practicable approach to describing animal movement patterns in which an individual’s trajectory through space is regarded as a sequence of distinct and randomly oriented steps, with the direction of a single step related to the direction of a previous step [32].

The random-settlers represented birds that settled at random, i.e., birds that did not rely on any specific type of information when making settlement decisions. After both other types of bird agents became territorial, the random-settlers were placed within a simulation space. Although they represent randomly settled individuals, random-settlers were distributed under the following two restrictions: individuals were least 10 cells away from other random-settlers and at least 20 cells away from the nearest cluster of cue-providers and cue-users. The first restriction resulted from the average nearest-neighbor distance between Wood Warbler territories in the Wiry Forest (Wielkopolska National Park). The second limitation had to be introduced because simulations included all bird agent types. Hence, there was a risk that random-settlers in a purely random distribution would be located in the middle of a group of clustered birds (i.e., cue-providers and cue-users), which might influence the results to a great extent and lead to misleading conclusions. It is also important to note that from a modeling perspective, both cue-providers and random-settlers were initially distributed similarly within a simulation space. However, in a biological sense, we implicitly assumed that cue-providers were birds that relied on personal information, such as their own breeding experience from the previous year, whereas random settlers made uninformed decisions. Moreover, both agent types could be distinguished because cue-providers were involved in social interactions with cue-users, which was not the case for random-settlers.

The nests were stationary agents and were distributed randomly within the bird agent’s territories. Although each bird agent occupied a territory with a radius of 5 cells (representing an empirical distance of 50 m), nests were sprouted randomly within a radius of 3 cells measured from the territory center. This approach was intended to account for the mean empirical distance between the territory center and nest location, which equals 32.45 m (n = 47) in our study plot (Szymkowiak and Kuczyński, S1 File).

Conspecific attraction behavior

Two types of conspecific attraction strategies were implemented, and all cue-users followed one of the strategies in a particular simulation run. Therefore, separate sets of simulations were performed for exploring the costs and benefits of each strategy.

Basic conspecific attraction

In the basic conspecific attraction strategy, the cue-users moved in a correlated random walk, one cell per time step. The movement heading was drawn at each time step from a normal distribution with a mean of 0° and a standard deviation defined by the observer (see S3 File, A.3.3.1. for details about choosing SD values for cue-user prospecting behavior). At each time step, the non-territorial cue-users were asked whether there was another bird agent in a radius of 10 cells that had already occupied the territory. If another agent was present, the cue-user perceived this information as a social cue, stopped and became territorial (note that after this it could be perceived as a social cue for others); if there was no such individual, the cue-user continued prospecting behavior. This procedure was completed after all territories had been occupied by cue-users.

Conspecific attraction with Bayesian updating

In this strategy, the cue-users performed a more sophisticated conspecific attraction strategy that included a form of Bayesian updating of prior settlement decisions. First, each cue-user followed a basic conspecific attraction strategy. However, at each time step, all cue-providers and those cue-users that acquired social cues (and were settled) were asked how many other individuals were in a radius of 15 cells. If the number exceeded a threshold value defined by the observer (k parameter), bird agents moved (one cell per time step) according to a correlated random walk with a heading drawn from a normal distribution (mean = 0°, SD = 360°). Because the turning angle was high, this process resulted in a very tight, convoluted movement that was continued until the number of individuals in a radius of 15 cells was equal to or lower than the threshold value. If this assumption was fulfilled, the bird agent stopped immediately. Because the birds were asked to assess the number of other territorial individuals in a 15-cell buffer at each time step, this “updating of territory borders” could be induced repeatedly if there were newly arriving birds at a particular location. In some scenarios, however, newly arriving birds could potentially displace individuals that have already occupied territories, which is unlikely from an empirical perspective but was allowed for coding simplicity because it did not affect the results of our analysis. The “updating territory borders” procedure was completed after all cue-providers and cue-users occupied stable territories. In a biological sense, we modeled settlement behavior in which the cue-users followed a conspecific attraction strategy; however, together with the cue-providers, they also perceived the risk associated with clustering and performed Bayesian updating of prior settlement decisions. This strategy is based on the implicit assumption that songbirds are able to adjust site selection decisions when information allowing them to assess habitat quality more reliably becomes available [21]. Such behavior is supported by abundant empirical data demonstrating habitat selection as a dynamic decision-making process during which initial settlement locations could be shifted when more timely information emerges or ecological conditions change (e.g., [33], [34])

Predator agents

Two types of predator agents were simulated in this model, Wild Boar and Red Fox, which differed in their foraging behavior. Both species are common in our study area and are well known predators of Wood Warbler nests [35], [36]. All predator agents were randomly distributed within a simulation space at the beginning of each model run and were able to eat nests if they occupied the same cell. In each model run, predators could consume an infinite number of nests at any time because we assumed that it would be unlikely for a boar or fox to be satiated after consuming a typical Wood Warbler clutch containing 6 small eggs [35].

The Wild Boar represented an incidental predator of Wood Warbler nests. The boars moved forward in a correlated random walk, one grid cell per time step. The direction of movement was drawn at each time step from a normal distribution with a mean of 0° and standard deviation defined by the observer (turning.boar parameter). All standard deviations for predator movement parameters used in the final experiments were chosen based on sensitivity analyses and made constant across the simulations (see Efficient predator setup for details). In nature, Wild Boars and Red Foxes move different distances during foraging bouts; whereas foxes move ~2 km daily [20], [31], boars are expected to move ~1 km daily [37]. The time of the entire simulation was scaled based on data for the average distance moved by Red Foxes during foraging bouts. Hence, to account for differences in the activity patterns of both predators, boar activity was constrained to 100 time steps per day (see A.3.3.4. in S3 File for details).

The Red Fox represented a predator with a capacity to perform more intensive searching of nearby areas after encountering a nest. Normally, foxes moved in a correlated random walk with a movement heading that was drawn, at each time step, from a normal distribution with a mean of 0° and a standard deviation defined by the observer (turning.fox parameter). However, foxes changed their movement strategy and performed area-restricted searches (ARSs) after encountering a nest. During the ARS movements, the foxes’ turning angle increased (strength.ARS parameter), and they explored nearby areas instead of moving further away. The duration of ARS movements was controlled by the observer-defined parameter (duration.ARS parameter). If the fox agent did not find any other nests during this time, it resumed movement in a standard correlated random walk. The ARS foraging strategy is based on the implicit assumption made by a predator about the higher probability of encountering other food resources near a previously located nest. As noted by Ringelman [20], this movement strategy is qualitatively similar to Lévy flight behavior, i.e., random walks in which the short movement bouts are interspersed with long movements [38]; foxes have been shown to follow this strategy when searching for prey [20], [31], [39].

Model initialization

The model was initialized with 80 total bird agents (and their nests), which is equivalent to the density of breeding Wood Warbler pairs in the Wiry Forest of the Wielkopolska National Park. Wood Warblers exhibit little site tenacity, with a median of 11% of males returning to previous breeding areas (n = 4 publications, based on Table 1 in [40]). We assumed that these males rely on personal information when making settlement decisions; thus, 8 cue-providers were initially placed in the simulation space. Among the remaining 72 bird agents, 36 were modeled as cue-users and 36 as random-settlers.

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Table 1. Parameter values (on the logit scale) of the model relating nest success when foxes forage with the threshold value that initiated Bayesian updating of prior settlement decisions.

https://doi.org/10.1371/journal.pone.0119132.t001

We are not aware of up-to-date estimates of predator densities at our study site; therefore, we preliminarily explored the model setup to select the optimal number of predators for final simulations. We systematically varied the number of boars and foxes (range: 1–8 individuals, 50 simulations for each combination) and calculated the nest survival of clustered and randomly distributed birds. Simulations were performed separately for each predator type. After preliminary model exploration, we decided to simulate 6 predators that caused the mean nest failures within the observed ranges of Wood Warbler nest success (ca. 20–60%) [35], [36], [41].

Simulation runs and statistical analysis

First, we systematically varied the parameter space of boar and fox movement behavior, attempting to find a setup that was most efficient from the predator’s perspective, causing the lowest nest survival of the entire bird population. Each parameter setup was modeled in 50 simulations. The optimal foraging strategy could largely depend on nest distribution [20]; thus, we modeled either 80 random-settlers (random distribution) or 8 cue-providers with 72 cue-users performing a basic conspecific attraction strategy (clustered distribution). Then, the efficiency of the predators was explored separately for the two bird communities.

After creating efficient boars and foxes, we performed a set of experiments to explore the predation-related costs and benefits of various settling strategies in landscapes with different predator communities. In each experiment, a mixed bird community was simulated, and the number of each bird agent sub-type was constant between experiments. Then, we compared the fitness costs and benefits of following a particular settling strategy by comparing the nest success of cue-providers, cue-users and random-settlers.

Data were analyzed in a Bayesian approach using a binomial GLM with a logit link function (which is equivalent to a logistic regression). When testing for differences in nest success between birds applying different settling strategies, a simple model contrasting groups was used:

Equation 1: where p_i is the probability of success for the i-th simulation, s_i is the number of successful nests for the i-th simulation, α_j(i) is the logit-scale parameter for expected probability of success for birds applying strategy j, and N_i is the known overall number of nests involved in each simulation. This approach is equivalent to a means parameterization in binomial GLMs [42].

This model was slightly modified when testing for differences between strategies in a multi-predator environment:

Equation 2: where Nfoxes_i is the number of foxes in a predator community for the i-th simulation, β_j(i) is the logit-scale parameter for slopes for the relationship between number of foxes and nest success for birds applying strategy j, and the other parameters were the same as in the previous model. This parameterization is equivalent to a binomial ANCOVA [42].

For testing whether Bayesian birds achieved any fitness gain and how this was related to the number of individuals in a close neighborhood that initiated the updating of prior settlement decisions, the following model was used:

Equation 3: where α is an overall mean nest success (on logit-scale), β_j(i) is the logit-scale parameter for expected probability of success for birds applying strategy j and γ_k(i) is the logit-scale parameter for the expected probability of success for birds applying Bayesian updating when the local density is higher than k. The value of k = 0 was assigned to the naive strategy, i.e., to birds that did not perform updating of settlement decisions. To make the model identifiable, priors of the first levels of both predictors were set to zero, which is equivalent to the effects parameterization in GLMs [42]. Thus, the estimated parameters provide contrast and express differences between the group means and appropriate reference levels (cue-providers in the case of β and naive birds in the case of γ).

Statistical analyses were performed using JAGS Gibbs-sampling environment [43] and R 3.0.3 [44]. Both environments were integrated via the R2Jags library [45]. Differences in parameters (e.g., group probabilities, slopes) were tested by computing the contrasts (reciprocal cross differences) within JAGS and checking whether the 95% credible intervals (CI thereafter) of those contrasts contained zero. Vague normal density priors were used for all covariate parameters. Gibbs sampling was performed with three independent chains for 10000 iterations each. The first 4000 iterations of each chain were discarded as burn-in, and only samples from every 10th iteration were stored. The JAGS code is provided as Supplementary Material (S4 File).

Efficient predator setup

First, we varied the parameters that determined the straightness of predator movement. The nest survival increased when predator paths became more convoluted and reached a minimum value when the turning parameter equaled 10°, irrespective of the nest distribution. Then, we set turning.fox = 10° and duration.ARS = 125 time steps and systematically varied the strength.ARS parameter. We found that predators caused the lowest nest survival when strength.ARS was between 40 and 70° for both clustered and randomly distributed nests. Next, we set turning.fox = 10° and strength.ARS = 55° and systematically varied the duration.ARS parameter. The results revealed that the survival of randomly distributed nests increased when foxes spent more time performing area-restricted searches, whereas for clustered nests, this relationship was reversed. A reasonable trade-off was chosen, and we set a duration.ARS = 100 time steps for further analyses. Overall, our sensitivity analysis revealed that the parameter values for efficient predator setup were similar to those in Ringelman’s model [20].

Conspecific attraction costs and benefits

In the first experiment (100 simulations), 6 Wild Boars were simulated to test whether the cue-providers and cue-users benefited from clustered distribution in a landscape dominated by incidental predators. We found no differences in nest success between birds involved in a conspecific attraction mechanism (cue-providers: 64.6%, CI: 61.3–67.9; cue-users: 63.1%, CI: 61.5–64.6). Moreover, the nest success of random-settlers (63.9%, CI: 62.3–65.4) was similar and did not differ from any of the “social” groups (Fig. 1).

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Fig 1. Posterior distributions of different of nest success probabilities of birds following particular settling strategies.

The left panel refers to a landscape dominated by boars, and the right panel refers to a fox-only environment. Vertical solid lines indicate zero, and dashed lines indicate 95% credible intervals around the mean difference. In a landscape dominated by incidental predators (boars), there were no significant differences (credible intervals always contain zero), meaning that the three strategies performed equally well. When predators were able to perform area-restricted search (foxes), a random settling strategy significantly outperformed both social strategies, whereas there were no significant differences in nest success between cue-users and cue-providers.

https://doi.org/10.1371/journal.pone.0119132.g001

In the subsequent experiment (100 simulations), 6 Red Foxes were modeled to test whether clustering as a by-product of conspecific attraction resulted in fitness costs in a landscape dominated by predators that were capable of performing area-restricted searches. The birds that settled at random performed better, i.e., had a higher nest success (posterior mean: 43.6%, CI: 42.0–45.2), than those with clustered territories (cue-providers: 39.6%, CI: 36.3–43.1; cue-users: 39.8%, CI: 38.3–41.5). Furthermore, as indicated by the comparison of posterior distributions of cross-differences (Fig. 1), the nests of clustered birds had similar survival rates; however, the nest success for both the cue-providers and cue-users differed from the random-settlers.

In the next experiment, we examined the fitness costs of random-settling and conspecific attraction in a multi-predator landscape. We systematically varied the numbers of boars and foxes (range: 0–6 individuals, 100 simulations for each combination) to simulate a mixed predator community. However, the total number of predators in each set of simulations was held constant at 6. Although we included only the number of foxes (N.foxes) as a predictor in the model, this represented a gradient between landscapes dominated by incidental predators (boars) and those dominated by predators that performed area-restricted searches (foxes).

The binomial ANCOVA model fitted using MCMC revealed that the mean nest success for the three settling strategies was similar and did not differ (posterior mean intercepts: 54.5%, CI: 44.8–64.4; 54.2%, CI: 49.7–58.9; 50.6%, CI: 46.1–55.2, for cue-providers, cue-users and random-settlers, respectively). However, the comparison of slopes and their reciprocal contrasts showed that although they were not different (at the 95% CI level) between the cue-providers and cue-users, both differed from the random-settlers. Thus, the model was refitted with both cue-providers and cue-users pooled together into a “social” group. Similar conclusions can be inferred from this simplified model: intercepts did not differ (“social”: 54.2%, CI: 50.0–58.4 vs. random-settlers: 50.5%, CI: 46.0–55.1), but the slopes did differ (“social”: -0.173, CI: -0.185–0.162 vs. random-settlers: -0.137, CI: -0.150–0.125). The posterior distribution of the contrast between both estimated slopes did not cover zero, which indicates that the nest survival of random-settlers did not decrease as rapidly as for birds with clustered territories (Fig. 2).

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Fig 2. Probability of nest success in a multi-predator landscape varying in domination of foxes.

Fitted lines and their 95% credible intervals for both “social” (red) and “random” (blue) settlers are shown. The random settling strategy became superior when foxes constituted at least half of the predator community.

https://doi.org/10.1371/journal.pone.0119132.g002

Profitability of Bayesian updating

In the final experiments, we examined whether acquiring information about the social environment and performing Bayesian updating of prior settlement decisions could lead to fitness benefits for birds that bred in clusters as a result of conspecific attraction. We systematically varied the k parameter (range: 2–6, 100 simulations for each value) and modeled the nest success of Bayesian cue-providers and cue-users in a landscape of “pure” predator communities with either 6 boars or 6 foxes. Moreover, in a separate set of simulations (100 each time), we modeled the nest success of naive cue-providers and cue-users and then compared it with that obtained by Bayesian birds at different k values.

The first model considered was a two-way binomial ANOVA with two factors: settling strategy and k parameter. For both boar- and fox-only environments, the cue-providers and cue-users performed equally well in this setup, and there were no differences between them. Thus, the model was refitted with only one predictor (k).

When the landscape was dominated by predators able to perform area-restricted searches, it was profitable for birds to use Bayesian updating of prior settlement decisions. Bayesian cue-providers and cue-users performed better (i.e., reached a higher nest success) than naive birds when the k parameter was 2, 3 or 4; however, when the value was ≥ 5, the fitness benefits of Bayesian updating disappeared (Fig. 3, Table 1).

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Fig 3. Fitted nest success probabilities plotted against the k parameter for boars-only (left panel) and fox-only (right panel) environments.

Points represent posterior means, and lines represent 95% credible intervals. The reference value for k = 0 (in black) denotes the mean success for naive birds which did not perform Bayesian updating of prior settlement decisions. Effects that are not different from the reference are in blue, whereas different effects are shown in red.

https://doi.org/10.1371/journal.pone.0119132.g003

As expected, using a Bayesian approach in the landscape dominated by incidental predators did not provide any benefits. We did not find any relationship between the fitness of birds involved in a conspecific attraction mechanism and the k parameter. Moreover, the nest success of naive cue-providers and cue-users did not differ from those achieved by individuals that performed Bayesian updating (Fig. 3, Table 2).

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Table 2. Parameter values (on the logit scale) of the model relating nest success when boars forage with the threshold value that initiated Bayesian updating of prior settlement decisions.

https://doi.org/10.1371/journal.pone.0119132.t002