Overview of model assumptions

The model followed here builds on the dynamic foraging game described by Rands et al. [26], [27]. In the simpler model described in [26], the decisions made by a pair of individuals are considered. The model uses dynamic programming techniques [28]–[32] to identify the optimal behaviours of a pair of animals, who are both characterised by possessing energy reserves (which defines their ‘state’) which change stochastically as a result of the actions that both individuals take over a series of consecutive decisions. Both members of the pair are able to accurately assess each other’s energetic reserves as well as their own, and their actions are informed by this information. During a period, each individual can choose to conduct one of two actions: either to rest or to forage for the entire period. Both actions incur an energetic cost which depletes the reserves of the individual, but foraging can also lead to the forager finding food (within a stochastic environment), meaning that, on average, it should see a net gain in its energetic reserves if it forages. Energetic reserves are important within the model: it is assumed that if they fall too low, an individual starves to death. It is also assumed that there is an upper limit to the capacity of the reserves, beyond which they cannot be increased further.

As well as the risk of starvation if an individual doesn’t forage, there is also the risk of predation, which depends on the actions of both individuals in the pair. If an individual choses to rest during a period, it incurs a low risk of being predated. If it forages at the same time as its colleague, it incurs a moderate risk of being predated (which could be through increased protection against predation from being in a small group, or enhanced detection, or simply a dilution of risk). If it forages on its own (whilst its colleague rests), it incurs the greatest risk of being predated. Therefore, there is a trade-off within this framework between being predated when foraging, and starving whilst resting. Rands et al. [26] demonstrate that these assumptions can be modelled using a stochastic dynamic game, and show that optimal policies can be calculated, which describe the optimal actions of an individual within a pair: the policies allow an individual to identify the suitable action to conduct given that it knows its own energetic reserves and those of its colleague at a given moment in time. Rands et al. [27] extend these models by considering what occurs when individuals are not identical in the costs they incur for conducting actions, or the amount of energy they gain during a period, or in the risks they face when conducting specific actions.

The model I describe here builds further on this framework. Although Rands et al. [27] considered possible differences between individuals in various parameters, they did not consider what could happen in a dominance interaction, where specific behavioural interactions between the two members of a pair incurred additional costs to one of the members (which I refer to as the ‘subordinate’), similar to the socially-mediated interference costs proposed in [6]. Note that I assume both that the dominance hierarchy has been decided by the pair members prior to the start of the period modelled, and that this hierarchy is adhered to throughout, with no further requirements to maintain it (see [33] for work considering the formation and maintenance of hierarchies). Here, I consider there to be four possible situations where an additional cost can be incurred by the subordinate:

1. when both it and the individual who does not pay a cost (which I refer to as the ‘dominant’) are foraging together (which could for example be a proximity or vigilance cost, or simply a reduction in foraging efficiency due to direct competition from the dominant);
2. when both the dominant and the subordinate are resting (which again could be a proximity cost, or perhaps an energetic cost due to the subordinate expending energy providing a service such as grooming to the dominant);
3. when the subordinate is foraging on its own (which could be through social anxiety at not knowing where the dominant individual is, or through an increased cost of scanning for food, predators, or the dominant);
4. when the subordinate is resting on its own (which again could be through social anxiety at not knowing where the dominant individual is).

In (iii) and (iv), I assume the dominant individual is conducting the opposite behaviour to the subordinate.

Model details

The model considers a dynamic game between pair of players consisting of a dominant and a subordinate individual. This dynamic game builds on solution procedures outlined in [28] and [34], following a state-dependent framework as described in [28]–[31]. General computation methods build on the dynamic game framework for pairs of foragers, outlined in [26] and described in detail in [27], and the reader is referred to the latter for full details of the assumptions and the computational solution process, which are not repeated here. Unless described here, details are identical to those given in [27], and consequently I do not repeat any analysis for the effects of variables other than the socially-mediated costs that are introduced in this paper. To summarise the procedure described in [27] in a brief, using the assumptions about the effects of pair members’ actions as detailed in the overview above, an initial candidate strategy is assumed. This defines all possible actions that each individual should take, given that it knows its own energetic reserves and those of its colleague at a moment in time. Assuming one of the pair members is using the current candidate strategy, a best response can be calculated for its colleague using dynamic programming. To do this, I make an additional initial assumption about how energetic state relates to fitness (where fitness is used as a common currency to compare all possible actions [35]), but this initial assumption about how fitness relates to state is rendered unimportant through strong backwards convergence [28]. Once an optimal response strategy to the current population strategy has been identified, the best response to that strategy could be calculated, and then the best response to that, and so forth, with the aim of identifying an evolutionarily stable strategy (ESS). However, it is difficult to iterate to an ESS using this direct route (e.g. [36], [37]), and I instead used a error-making approach [34], where the candidate strategy is updated at each iterative step by combining the previous candidate strategy with the newly identified best response strategy (weighting the new candidate strategy strongly towards the previous candidate strategy). Using this technique, an ESS is identified through an iterative computational process. For finer detail of the assumptions, please see [26] and [27].

Where the current model differs from that presented in [27] is in the detail of the function denoted H_i(x_i, x_j, t; u_i, u_j, π), which defines the probability that an individual of type i who is alive at the start of time step t, in state x_i (>0), paired with a living colleague of corresponding type j in state x_j (>0) who follows a strategy defined by the candidate strategy π, will survive until the start of the final time step T, if it adopts action u_i and its colleague adopts action u_j in the current time step (assuming that the focal individual i thereafter behaves so as to maximise its chances of surviving until time step T, taking into account errors in decision making). Note that, as with the model described in [27], the candidate strategy π encompasses the candidate responses of both subordinate and dominant individuals within the population (which means that it defines the current ‘best’ action that an individual should take given that it knows its own energy reserves, and those of its colleague: the modelling process considers all possible state combinations of energy reserves for both dominant and subordinate individual, and the candidate strategy therefore includes current ‘best’ actions for all of these).

In order to consider the effects of dominance on behaviour, I replace the H_i(x_i, x_j, t; u_i, u_j, π) function described in [27] with the following set of definitions, which are dependent upon whether the focal individual is dominant or subordinate, and assume that both individuals in a pair are alive at the moment the decision is made (note that the associated functions that describe what occurs when only the focal individual is alive at the decision point, or that describe what happens if no individuals are alive at the decision point, are identical to those presented in [27], and are therefore not described here). Throughout, terms relevant to subordinates are denoted with a subscript s, and terms relevant to dominants are denoted with a subscript d.

If the individual is dominant, I assume

If the focal individual is subordinate, I instead assume

Apart from the novel interference cost terms (described below), terminology here follows [27], and is only briefly summarised here. An individual of type a has a probability m_aR of being predated if it is resting, m_aA if it is foraging alone, and m_aT if it is foraging with its colleague. The function W_a(x_a,x_b,t;π) is the fitness at time t of an individual of type a with energy reserves x_a, and whose colleague has energy reserves x_b, assuming that both individuals follow policy π from that point forward in time. Λ_a(x) defines a ‘chop’ function for an individual of type a as defined in [30], where Λ_a(x)  =  min(S_a,max(x,0)), and S_a is the maximum state value possible for an individual of type a. κ_a(c_a;u) denotes the probability that an individual of type a spends c_a state units of energy during a period if it conducts action u, and γ_a(g_a) denotes the probability that it gains g_a state units of energy during the period if it forages. Both energetic costs and gains were represented within the current model using the same functions described in [27], where the probabilities defined followed a discretised distribution based on normal distributions with defined means and standard deviations.

For simplicity, both dominant and subordinate individual were assumed to have identical probabilities of incurring given gains and costs, and share identical predation risks when conducting particular activities (so m_dT  =  m_sT, etc.). The only difference considered between them was in the extra cost paid by the subordinate individual when it was conducting a paired behaviour that incurred extra energetic costs. These energetic costs are denoted D_dRsR, D_dFsR, D_dRsF and D_dFsF, which represent the extra energetic cost (in state units) incurred for the four possible pairs of behaviours (where the general form D_dUsV represents the cost paid by the subordinate when the dominant conducted action U and the subordinate conducted action V). The dominant is not expected to pay any extra costs for its actions dependent upon the behaviour of the subordinate.

Summary statistics

Once stable behavioural policies for subordinate and dominant individuals had been identified, forward iterations using Markov chain processes [29] were then used to calculate the distribution of a pair’s states within a stable population. Again, full details of the process and assumptions made follow those described in [27]. Having identified a stable distribution of paired states, I then calculated the following summary statistics:

Parameter exploration

I explored the effects of the costs by randomly generating 1,000 sets of other model parameters, and then calculating optimal policies and population distributions for all possible combinations of the four socially-mediated interference costs. Table 1 describes the parameters used within this model, including the use of randomisation to generate differences between parameter sets. For each set of parameters, I considered the sixteen possible scenarios where D_dRsR, D_dFsR, D_dRsF and D_dFsF could each take a value of either 0 or 1 state units, representing a full spectrum of cases where there was a potential cost to be paid by a subordinate individual dependent upon the actions of the dominant member of the pair. After calculating policies and stable population distributions, summary statistics were calculated for each of these sixteen possible scenarios, and exploratory analyses were conducted as detailed below.

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Table 1. Parameters used in model, with values used for model exploration.

https://doi.org/10.1371/journal.pcbi.1002252.t001

Analyses of summary statistics

The model considers four possible interference costs to a subordinate individual, all of which could potentially have a separate effect upon its energetic turnover during a period dependent upon the behaviour of the pair. Therefore, I was interested in the interactions of the costs, as well as each of the costs themselves. To explore this, standard analysis of variance was used to generate F values for the four costs and the eleven possible interactions involving two, three or all four of these. The distribution of the results generated would not fit the standard assumptions necessary for ANOVA, and so I used resampling methods to identify critical F values, following recommendations in [39]. Because I was potentially interested in the effects of interactions, I would have been unable to generate resampled critical F values by the random assortment of results such that the untested costs or interactions were kept correctly assorted. However, my sample population of results was large, and I therefore generated resampled critical F values by freely permuting my entire dataset without restriction, following recommendations in [39]. For each, I used R 2.12.1 [40] to permute the entire dataset without replacement 50,000 times, harvesting the fifteen F values for an ANOVA conducted on each permuted set, and used the quantile function within R to identify the value of the 95% quantile values.

Individual behaviours

All individual increases in dominance cost led to an increase in the amount of foraging behaviour shown by the subordinate (Table 2) – there were also significant interactions between paired costs (and most three- or four-way interactions), although in all cases adding a cost led to an increase in foraging behaviour. Increasing the costs experienced by the subordinate led to increases in most of the individual foraging behaviour shown by the dominant, except for the case where the subordinate only experienced costs when it was resting and the dominant was foraging, which is likely to be a situation when the dominant is not going to be affected by the actions of the subordinate too much.

Paired behaviours

The increases in individual foraging behaviour were also echoed in the paired behaviours (Table 2). Considering all the single costs within the statistical model, increasing any of the costs experienced by the subordinate individual led to an increase in its foraging behaviour, and a decrease in its resting behaviour. The fact that the direction of change is dictated solely by the subordinate individual suggests that the action of the dominant individual is being driven primarily by its foraging partner.

Paired costs led to increases in both individuals foraging together, and (apart from the case where the subordinate always paid a cost when the dominant was foraging), decreases in resting together. The paired interactions when the members of the pair were conducting differing behaviours were mostly non-significant, although there were increases in cases where the dominant rested and the subordinate foraged when the costs experienced by the subordinate occurred when the pair differed in their behaviour.

Synchrony coefficient

As would be expected, pairs become more synchronised when there are costs involved with not being paired, and become less synchronised when there are costs to conducting the same action as each other (Table 2). Considering this alongside the paired behaviour results suggests that although the subordinate is driving the behaviours of the pair, its own actions are therefore partially dictated by the costs that it pays. Note also that this measure (with similar reasoning for the following S statistic) does not discriminate between resting together and foraging together. Therefore, an increase in only one of these paired behaviours may not lead to a corresponding increase in the general level of synchronisation within the pair.

Independence of action

The S statistic decreased in response to a dominance cost when foraging together, and decreased when resting together (Table 2). Resting alone doesn’t incur any more risk of predation than when resting together, and it is therefore feasible that as resting together becomes more costly to the subordinate, it should therefore become more dependent upon the state of its colleague dictating its actions (in this case, avoiding resting together), leading to the decrease in synchrony shown by the synchrony coefficient. The decrease in dependence with an increasing cost of foraging together is echoed in the observation that the subordinate individual should be increasing its foraging regardless of the actions of the dominant.

Repeatability of behaviours

Both the dominant and subordinate individuals tended to increase their repetition of behaviour when there was an extra cost to the subordinate of foraging at the same time as the dominant (Table 3). This is likely to be an effect of foraging being highly synchronised: the subordinate has to forage at the same time as the dominant, and consequently needs the pair to spend more time foraging than the dominant in order to fund the energy it spends (especially, but counterintuitively, whilst foraging). Both individuals tended to reduce their repetition of behaviour when there was an extra cost to the subordinate of resting together. This is likely to be due to the reduction in the amount of time that the subordinate rests overall – an increased likelihood of foraging suggests that a pair of individuals will be swapping between different pairs of behaviours, and is demonstrated in a similar reduction in the mean length of time that pairs of individuals repeated a paired behaviour (Table 3).

The subordinate individual also tended to repeat its own behaviour more often when there was a dominance cost associated with conducting the opposite behaviour to the dominant individual (note here that this means an overall increase in the subordinate repeating a behaviour irrespective of what the dominant is doing, rather than a statement that the subordinate is increasing conducting the opposite behaviour to the dominant). Most of the interactions shown for the subordinate individual also indicate a positive trend. Paired behaviours were also repeated more often when these costs were incurred. These increases are likely to be due to the increase in synchronisation behaviour seen when there is an extra cost to being non-synchronised.

Energetic reserves

As would be expected, incurring an extra cost of dominance to the subordinate meant that its energetic reserves tended to be reduced (Table 3). This was not the case where there was a dominance cost to the subordinate when it rested and the dominant foraged, which may be due to an increase in the subordinate tending to forage in order to avoid this cost. Regardless of which sort of cost was imposed on the subordinate, the dominant tended to gain energetic reserves when there was a cost, which ties in with the increase in subordinate foraging behaviour and corresponding synchronisation by the dominant individual.

Repeatability of difference in energetic reserves

The length of time that the subordinate remained heaviest (when it managed to reach that state of being) was in most cases reduced by imposing a cost of dominance (Table 3). The exception to this followed a similar pattern to the energetic reserves, where imposing a cost when the subordinate rested and the dominant foraged tended to lead to an increase in the length of time that the subordinate remained heaviest. Again, this is likely to be due to the subordinate increasing the amount of time it forages in response to this cost, therefore leading to an increase in its reserves. Increasing the length of time the subordinate individual remained heaviest should logically lead to a decrease in the length of time that the dominant individual remained heaviest, and vice versa. This trend was seen, but was only significant for the situations where the subordinate’s costs were paid for conducting the opposite behaviour to the dominant individual.