Resource availability and adjustment of social behaviour influence patterns of inequality and productivity across societies

Life-cycle

I assume a population of asexually-reproducing haploid individuals subdivided into a very large number of groups (as in Wright’s (1931) infinite island model), which I suppose vary in resource availability. Generations are non-overlapping, and following (Rodrigues & Gardner, 2012; Rodrigues & Gardner, 2016), I suppose that patches are of two types: high-quality resource-rich patches and low-quality resource-poor patches (although the resource availability of a patch may change from one generation to the next, as discussed below). Each patch contains a high-quality and a low-quality breeder (cf. Rodrigues & Gardner, 2013a; Rodrigues & Gardner, 2016). The fecundity of both breeders is assumed to be very large, albeit different. More specifically, in any given generation, the high-quality breeder (denoted by ‘H’) in a resource-rich patch (denoted by ‘R’) gives birth to f_HR(x_HR, x_LR) offspring, while the low-quality (denoted by ‘L’) gives birth to f_LR(x_LR, x_HR) offspring (where f_HR(x_HR, x_LR) ≥ f_LR(x_LR, x_HR). The fecundity of each breeder depends on the level of expression of a social behaviour by the high-quality individual (denoted x_HR) and by the low-quality individual (denoted x_LR). Equivalent fecundities and levels of expression of the social behaviour for individuals in a resource-poor patch (denoted by ‘P’) are denoted f_HP(x_HP, x_LP) and f_LP(x_LP, x_HP), x_HP and x_LP. I describe the social behaviour in more detail below.

Following social interactions and reproduction, a fraction 1 – d of the offspring remain in their natal patch, while a fraction d disperse to a random patch in the population. Dispersers survive dispersal with probability 1 – c, where c is the mortality cost of dispersal. Offspring then compete for the available breeding vacancies created by the death of the adult generation. Offspring are identical in quality and therefore all have equal chances of becoming a high- or a low-quality breeder. Patches may then undergo changes in their resource availability. Resource-rich patches remain resource-rich with probability p_R→R, and become resource-poor with probability p_R→P = 1 − p_R→R; resource-poor patches remain resource-poor with probability p_P→P, and become resource-rich with probability p_P→R = 1 − p_P→P. After these ecological changes the life-cycle repeats. Model notation is summarised in Table 1.

Methodology

I am interested in the evolution of a social trait that is costly to the actor and is either helpful or harmful to the recipient. I assume (as stated above) that this trait is expressed at a level that is conditional upon an individual’s state (i.e., its quality, i.e., ‘H’ or ‘L’, and the resource availability of its patch, i.e., ‘R’ or ‘P’). I use the neighbour-modulated approach to kin selection (Taylor & Frank, 1996; Frank, 1998; Rousset, 2004; Rousset & Ronce, 2004; Rodrigues & Gardner, 2013a) to derive the inclusive fitness effect of the trait of interest.

The inclusive fitness effect of the trait of interest depends on two key quantities: relatedness and reproductive value. Relatedness gives a measure of the value of social partners pertaining to the degree to which they share genes in common (Hamilton, 1964). Reproductive value gives a measure of the value of social partners pertaining to the degree to which they are able to project copies of their genes into the gene pool of future generations (Fisher, 1930; Taylor, 1990; Grafen, 2006). I then conceptualise the fitness effect of the trait in terms of Hamilton’s rule, which gives the condition for the evolution of an alternative allele (Hamilton, 1964; Charnov, 1977). Finally, given Hamilton’s rule, I can obtain evolutionarily stable levels of the traits of interest, which occur whenever the left-hand side (LHS) of Hamilton’s rule is zero, and therefore there is selection neither for an increase nor a decrease in trait value. To solve the model, I use a combination of analytical and numerical methods, a detailed description of which can be see in the Appendix S1.

First, I determine the relatedness coefficients among group members. For this purpose, I define a set of recursion equations describing changes in the genetic structure of groups from one generation to the next (Taylor, 1992; Rousset, 2004; Rodrigues & Gardner, 2012; Rodrigues & Gardner, 2016). More specifically, the relatedness coefficients depend on the state of the patch in the previous generation, i.e., the probability p_γ|σ, that a patch currently in state σ was in state γ in the previous generation; the probability φ_γ that both adults are philopatric; the probability U_Hγ² (or $U_{\mathrm{L}\gamma }^2$) that they are both offspring of a high-quality (or low-quality) breeder; and the probability 2U_HγU_Lγ that one is an offspring of a high-quality breeder and the other is an offspring of a low-quality breeder. If they are siblings, their relatedness is one, otherwise their relatedness is r_γ, with γ ∈{R, P}. Collecting these terms together, we obtain the following recursion equations (1)${\displaystyle r_{\sigma }^{\prime }=\sum _{\gamma \in \left\{\mathrm{R},\mathrm{P}\right\}}{p}_{\gamma |\sigma }{\phi }_{\gamma }\left(U_{\mathrm{H}\gamma }^2+U_{\mathrm{L}\gamma }^2+2U_{\mathrm{H}\gamma }{U}_{\mathrm{L}\gamma }{r}_{\gamma }\right),\sigma \in \left\{\mathrm{R},\mathrm{P}\right\},}$ which can be solved for equilibrium (i.e., $r_{\sigma }^{\prime }=r_{\sigma }$), in order to derive the relatedness coefficients in each patch type (see Appendix S1 for details).

Second, I determine the reproductive value of breeders (denoted by v), where reproductive value is defined as the contribution of a breeder to the gene pool of future generations (Fisher, 1930; Grafen, 2006). The reproductive value of a breeder is given by the elements of the left eigenvector associated to the leading eigenvalue of the fitness matrix w, which is (2)${\displaystyle \mathbf{w}={\left(w_{\rho \pi ⟶\eta \gamma }\right)}_{4\times 4}.}$ Each element of this matrix denotes the number of offspring of a focal breeder that secure a breeding place in the next generation, according to the state of the breeder (i.e., her quality ρ, and her patch type π), and according to the breeding state of her offspring (i.e., the offspring quality η, and the offspring patch type γ; see Appendix S1 for details).

Third, I determine the neighbour-modulated fitness of a focal breeder of each possible state. To do this, I sum the number of successful offspring the focal breeder has, weighting each successful offspring by its reproductive value. This weighted total is then divided by the reproductive value of a random breeder in the focal class, yielding (3)${\displaystyle W_{\rho \pi }=\frac{\sum _{\gamma \in \left\{\mathrm{R},\mathrm{P}\right\}}\sum _{\eta \in \left\{\mathrm{H},\mathrm{L}\right\}}{w}_{\rho \pi ⟶\eta \gamma }{v}_{\eta \gamma }}{v_{\rho \pi }}.}$ Fourth, I determine the expected fitness of a randomly chosen focal breeder, who might be of any state. This is given by (4)${\displaystyle \overline{w}=\sum _{\pi \in \left\{\mathrm{R},\mathrm{P}\right\}}\sum _{\rho \in \left\{\mathrm{H},\mathrm{L}\right\}}{u}_{\rho \pi }{v}_{\rho \pi }{W}_{\rho \pi },}$ where u_ρπ is the frequency of individuals in state ρ (i.e., either high- or low-quality) and in patch type π (i.e., either resource-rich or resource-poor), which is given by the right eigenvector associated to the leading eigenvalue of matrix w (see Appendix S1 for details). Finally, the fitness effect of a social behaviour enacted by a focal actor in state α in a patch with resource availability σ is given by the slope of an individual’s expected fitness ($\overline{w}$) on its breeding value for that level of expression (g_ασ). This is given by (5)${\displaystyle \frac{\mathrm{d}\overline{w}}{\mathrm{d}{g}_{\alpha \sigma }}=\sum _{\pi \in \left\{\mathrm{R},\mathrm{P}\right\}}\sum _{\rho \in \left\{\mathrm{H},\mathrm{L}\right\}}{u}_{\rho \pi }{v}_{\rho \pi }\frac{\mathrm{d}{W}_{\rho \pi }}{\mathrm{d}{g}_{\alpha \sigma }}.}$ Note that social interactions unfold among individuals in the same patch, and therefore a behaviour expressed in resource-rich patches (or in resource-poor patches) does not affect individuals in resource-poor patches (or in resource-rich patches). Thus, in Eq. (5), the slope of fitness of individuals in resource-rich patches (or in resource-poor patches) on the breeding value of individuals in resource-poor patches (or in resource-rich patches) is zero.

Hamilton’s rule

If the fitness effect, given by Eq. (5), is greater than zero (i.e., $\mathrm{d}\overline{w}∕\mathrm{d}{g}_{\alpha \sigma }>0$), then a slightly higher value of the trait evolves, a condition that is often known as Hamilton’s rule (Hamilton, 1964; Charnov, 1977; Rodrigues & Gardner, 2013a). Hamilton’s rule governing the behaviour of a quality-α breeder in a type-σ patch is given by (6)${\displaystyle -\underset{\genfrac{}{}{0ex}{}{\text{primary}}{\text{cost}}}{\underbrace{C_{\alpha \sigma }{V}_{\alpha \sigma }}}+\underset{\genfrac{}{}{0ex}{}{\text{primary}}{\text{benefit}}}{\underbrace{B_{\alpha \sigma }{V}_{\rho \sigma }{r}_{\sigma }}}-\underset{\text{kin competition cost}}{\underbrace{\left(B_{\alpha \sigma }-C_{\alpha \sigma }\right)w_{\sigma }^{\varphi }\left(v_{\alpha \sigma }^{\varphi }+v_{\rho \sigma }^{\varphi }{r}_{\sigma }\right)}}>0,}$ where: C_ασ is the fecundity cost paid by the quality-α actor in the type-σ patch; V_ασ (or V_ρσ) is the reproductive value of the actor’s (or recipient’s) offspring; B_ασ is the benefit enjoyed by the recipient; r_σ is the relatedness between the actor and recipient; $w_{\sigma }^{\varphi }$ is the probability that a native random offspring remains and wins a breeding site in the focal type-σ patch; and $v_{\alpha \sigma }^{\varphi }$ (or $v_{\rho \sigma }^{\varphi }$) is the philopatric component of the actor’s (or recipient’s) expected reproductive value (i.e., the reproductive value derived from offspring that remain in the local patch). In the Appendix S1, I provide additional information on how to calculate these quantities.

Hamilton’s rule immediately yields an inclusive fitness interpretation of the behaviour, which can be partitioned into a primary cost and benefit of the behaviour and a kin competition cost. First, the actor’s behaviour costs her C_ασ offspring, with each offspring representing a decrement V_ασ in her reproductive value. Second, the actor’s behaviour improves the fecundity of the recipient by an amount B_ασ, with each additional offspring representing an increment V_ρσ in reproductive value, an increment that must be depreciated by the relatedness r_σ between social partners. Finally, the behaviour leads to the production of B_ασ − C_ασ additional offspring, with a fraction $w_{\sigma }^{\varphi }$ of these offspring remaining and displacing other offspring in the local patch. The displacement of offspring destroys a portion $v_{\alpha \sigma }^{\varphi }$ of the actor’s reproductive value, and a portion $v_{\rho \sigma }^{\varphi }$ of the partner’s reproductive value, where the partner’s portion must be depreciated by the coefficient of relatedness r_σ.

Hamilton’s rule gives the condition for the evolution of a social trait, which is either a helping trait when the benefit is greater than zero (i.e., B_ασ > 0), or a harming trait when the benefit is less than zero (i.e., B_ασ < 0). If Hamilton’s rule holds, then the mutant expressing a slightly higher value of the trait spreads to fixation. To analyse the action of kin selection, one can set the left-hand side of Hamilton’s rule to zero, and re-arrange the resulting equation in the form of C_ασ∕B_ασ = A_ασ. The variable $A_{\alpha \sigma }=\left(r_{\sigma }{V}_{\rho \sigma }-w_{\sigma }^{\varphi }\left(v_{\alpha \sigma }^{\varphi }+v_{\rho \sigma }^{\varphi }{r}_{\sigma }\right)\right)∕\left(V_{\alpha \sigma }-w_{\sigma }^{\varphi }\left(v_{\alpha \sigma }^{\varphi }+v_{\rho \sigma }^{\varphi }{r}_{\sigma }\right)\right)$ is the potential for helping (Rodrigues & Gardner, 2013a). Helping evolves if the potential for helping is positive (A_ασ > 0), whilst harming evolves if the potential for helping is negative (A_ασ < 0), providing the cost (C_ασ) is sufficiently small. Of particular interest for the analysis is the coefficient of temporal correlation in resource availability (denoted by τ). At one extreme, when the coefficient of temporal correlation is 1 (i.e., τ = 1), the amount of locally available resources remains constant from one generation to the next. At another extreme, when the coefficient of temporal correlation is −1 (i.e., τ = − 1), the amount of locally available resources fluctuates from one generation to the next. The potential for helping and the other model variables depend on the coefficient of temporal correlation, as we shall see below.

Competitive effort

Above, we have seen how to analyse the action of kin selection in terms of ratios of costs and benefits, which I have assumed constant. More generally, a behaviour will influence the relative fecundity of each actor, and this in turn will influence the variables, such as relatedness and reproductive value, that mediate the action of kin selection. Thus, to understand the level of helping and harming at evolutionary equilibrium, we need to specify how behaviour and fecundity are intertwined.

The particular behavioural function specifying the fecundity f_ρσ of a focal individual in state ρ in a type-σ patch will determine both the costs and the benefits of the behaviour. Let x_ρσ be the phenotype of a focal individual in state ρ in a focal type-σ patch, and x_σ the vector of phenotypes of all individuals in the focal group, in a population where the average trait values are given by z_ρσ. I can then specify that social interactions mediate the fecundity of the focal individual through a personal component Ψ_ασ, which is a function of the focal individual’s phenotype x_ασ, and a group component Θ_σ, which is a function of the phenotype of all group members, as defined by x_σ. If the two components, personal and group, are multiplicative, then the fecundity cost of the behaviour is determined by the additive inverse of the effect of the focal individual’s phenotype on the personal component of its fecundity, i.e., C_ασ =  − (∂Ψ_ασ∕∂x_ασ)Θ_σ. That is, an increase in the focal individual’s phenotype will allow her to acquire a larger share of the resources available in the local patch. Moreover, the fecundity benefit is determined by the effect of the actor’s phenotype on the group component of the focal individual’s fecundity, i.e., B_ασ = (∂Θ_σ∕∂x_ασ)Ψ_ασ. That is, an increase in the focal individual’s phenotype also causes a decrease in the amount of resources available in the local patch.

Here, I focus on a particular form of social interactions, where x_ασ is defined as the competitive effort of a focal individual (c.f. Frank, 1994). In the appendix, I explore alternative behavioural functions, which, as we shall see below, give the same qualitative results (Appendix S1). I assume that competitive effort increases the rate or efficiency of resource acquisition by an individual, but decreases the total amount of resources available to the group. We can interpret greater competition as a harming (selfish) strategy because an individual gains a direct benefit at a cost to her social partner; while we can interpret reduced competition as a helping (cooperative) strategy because an individual pays a direct cost to provide a benefit to her group mate. I assume that the quality of individuals, denoted by q_ρσ, varies, with high-quality individuals holding an intrinsic fecundity advantage relative to low-quality individuals. I assume that social interactions have an additive effect on the baseline fecundity of individuals, with social interactions either amplifying or alleviating the initial inequality between individuals. Furthermore, I assume that there are more resources available in resource-rich patches than in resource-poor patches. Thus, μ_R > μ_P, where μ_R (or μ_P) is the amount of resource in resource-rich (or resource-poor) patches. The fecundity function of a focal class-ρ individual in a type-σ patch is then given by (7)${\displaystyle f_{\rho \sigma }={\mu }_{\sigma }\left(q_{\rho \sigma }+\frac{x_{\rho \sigma }}{\sum _{\eta \in \left\{\mathrm{H},\mathrm{L}\right\}}{x}_{\eta \sigma }∕n_{\sigma }}\left(1-\sum _{\eta \in \left\{\mathrm{H},\mathrm{L}\right\}}\frac{x_{\eta \sigma }}{n_{\sigma }}\right)\right),}$ where Ψ_ασ = x_ρσ, ${\Theta }_{\sigma }=\left(1-{\sum }_{\eta \in \left\{\mathrm{H},\mathrm{L}\right\}}{x}_{\eta \sigma }∕n_{\sigma }\right)∕{\sum }_{\eta \in \left\{\mathrm{H},\mathrm{L}\right\}}{x}_{\eta \sigma }∕n_{\sigma }$, and n_σ is the number of individuals in the patch, which in the model is two across all patches. My aim is to determine the optimal competitive effort for high- and low-quality individuals in resource-rich and in resource-poor patches (i.e., $z_{\mathrm{HR}}^{\mathrm{\ast }}$, and $z_{\mathrm{LP}}^{\mathrm{\ast }}$; and $z_{\mathrm{HP}}^{\mathrm{\ast }}$, and $z_{\mathrm{LP}}^{\mathrm{\ast }}$). These are the competitive efforts at which the LHS of Hamilton’s rule for each trait is neither positive nor negative, and which I check for convergence stability (Christiansen, 1991; Eshel, 1996; Taylor, 1996; see Appendix S1 for details), where convergence stability means that the evolutionary trajectories of the traits under selection will always converge towards the singular strategies. While I have not tested the convergence stability of the general model, I have found no instances where the singular strategies fail the convergence stability test. This suggests that the singular strategies of the specific competitive effort behavioural functions presented above are always convergence stable.

Coefficients of inequality

Above, I have described how fecundity may vary both within and between groups. Here, I define the coefficients of inequality (denoted by G) to provide an operational measure of variation in fecundity both within and between groups. First, I define the coefficients of inequality within resource-rich and resource-poor patches, and I then define the coefficient of inequality between resource-rich and resource-poor patches.

The coefficient of within-group inequality, denoted by G_ω,σ, is defined as G_ω,σ = 1 − f_Lσ∕f_Hσ, with σ ∈ {R, P}. Thus, when both breeders produce an equal number of offspring (i.e., f_Lσ = f_Hσ), the group is strictly egalitarian and the coefficient of within-group inequality is zero (i.e., G_ω,σ = 0). By contrast, when the fecundity of low-quality individuals is negligible relative to the fecundity of high-quality individuals (i.e., f_Lσ ≪ f_Hσ), the group is extremely unequal and the coefficient of within-group inequality is one (i.e., G_ω,σ = 1). The coefficient of between-group inequality, denoted by G_β, is defined as G_β = 1 − F_L∕F_H, where F_σ = f_Lσ + f_Hσ is the productivity of type-σ patches. If the productivity of the two patch types is identical (i.e., F_L = F_H), then between-group inequality is zero (i.e., G_β = 0). By contrast, if the productivity of resource-poor patches is negligible in relation to the productivity of resource-rich patches (i.e., F_L ≪ F_H), then between-group inequality is one (i.e., G_β = 1).

Finally, I specify the baseline levels of both within-group inequality, denoted by $G_{\omega ,\sigma }^0$, and between-group inequality, denoted by $G_{\beta }^0$, by excluding the effects of social interactions. That is, I calculate the initial levels of inequality assuming fully selfish individuals (i.e., x_ησ = z_ησ = 1). Thus, from Eq. (7), I find that the initial level of within-group inequality is given by $G_{\omega ,\sigma }^0=1-{\mu }_{\sigma }{q}_{\mathrm{L}\sigma }∕{\mu }_{\sigma }{q}_{\mathrm{H}\sigma }$, while the initial level of between-group inequality is given by $G_{\beta }^0=1-{\mu }_{\mathrm{L}}\left(q_{\mathrm{HL}}+q_{\mathrm{LL}}\right)∕{\mu }_{\mathrm{H}}\left(q_{\mathrm{HH}}+q_{\mathrm{LH}}\right)$. The difference between initial and evolved levels of inequality gives the effect of social interactions on inequality, as we shall see below.

Behaviour in resource-rich patches

Let us first focus on how selection shapes the behaviour of individuals in resource-rich patches. As shown in Fig. 1C, low-quality individuals always help more than high-quality individuals. If we look at Fig. 1A, we see that both high- and low-quality breeders value offspring of the social partner equally (same r_H). However, as shown in Fig. 1B, the intensity of kin competition experienced by high-quality breeders is higher than the intensity of kin competition experienced by low-quality breeders. More specifically, because high-quality breeders produce more offspring, high-quality breeders have more offspring competing for the available breeding sites than low-quality breeders. Hence, $v_{\mathrm{HR}}^{\varphi }$ is relatively higher than $v_{\mathrm{LR}}^{\varphi }$. As a result, the high-quality breeder is selected to suppress the reproduction of her low-quality partner to improve the chances that her philopatric offspring obtain a breeding site. By contrast, the low-quality breeder is selected to raise offspring of the high-quality social partner, as the low-quality breeder experience relatively lower kin competition costs.

Looking again at Fig. 1C, we see that average helping peaks when the environment is relatively stable (τ ≈ 1). More specifically, we find that selection for helping initially increases as the environment becomes unstable, but it rapidly starts to decrease as the instability of the environment increases further. The decrease in selection for helping with environmental instability is explained by a monotonic decrease in relatedness with environment instability (Fig. 1A). That is, as the environment becomes unstable, the probability that the focal resource-rich patch was a resource-poor patch increases, and therefore the probability that unrelated immigrants from resource-rich patches (rather than related philopatric individuals) recolonize the focal patch increases. The peak in the selection for helping, by contrast, is explained because the intensity of kin competition does not fall linearly with temporal instability. In particular, the philopatric component of reproductive value shows an initial sharp decline with temporal instability (Fig. 1B), which explains the initial increase in selection for helping with temporal instability.

Behaviour in resource-poor patches

Let us now turn the attention to the action of selection in resource-poor patches. As shown in Fig. 1F, low-quality individuals always help more than high-quality individuals (a pattern also observed in resource-rich patches). Again, this is because low-quality individuals experience less kin competition than high-quality individuals (i.e., $v_{\mathrm{HP}}^{\varphi }>v_{\mathrm{LP}}^{\varphi }$; Fig. 1E).

Figure 1F also shows that the average helping peaks when the environment is relatively unstable (τ ≪ 1); specifically: in unstable environments, both high- and low-quality individuals help more; in stable environments, high-quality individuals harm low-quality individuals, and low-quality individuals help high-quality individuals less than in unstable environments. This is because both relatedness and the intensity of kin competition depend on the coefficient of temporal stability (Figs. 1D and 1F). In unstable environments, group mates are more likely to be close relatives because the probability that the focal patch was resource-rich in the previous generation is higher, and therefore the probability that the two breeders are both philopatric is also higher. On the other hand, there is a disproportional large influx of unrelated immigrants arriving at the local patch, which erodes the intensity of kin competition. Thus, unstable environments tend to favour the immediate fitness returns generated by helping behaviour among group mates, even if this gives rise to some additional amount of kin competition. In stable environments, group mates are relatively more likely to be unrelated because of low productivity in the local patch over consecutive generations, which means that the focal patch has been colonised by successive generations of unrelated immigrant individuals. On the other hand, if good fortune strikes, the local patch may become resource-rich, and therefore the next generation of resident individuals may be relatively more successful than the current generation. Thus, stable environments tend to favour lower helping among group mates, such that local offspring are more likely to win breeding spots in the next generation.

Competitive effort

I now focus on the stable levels of competitive effort. In resource-rich patches, I find that as the environment becomes more stable, the competitive effort of high-quality individuals falls sharply, while the competitive effort of low-quality individuals falls more slowly (Fig. 2A). As a result, when the environment becomes more stable, the difference between the competitive effort of high- and low-quality individuals decreases. High-quality individuals invest less in competitive effort as the environment becomes more stable because environmental stability leads to relatively higher relatedness among social partners. Since in temporally stable environments, high- and low-quality individuals are more closely related (Fig. 1A), high-quality individuals are selected to share a bigger portion of the available resources with low-quality individuals.

Let us now look at the optimal competitive effort in resource-poor patches (Fig. 2B). Here, I find that as the environment becomes more stable, the competitive effort of high-quality individuals rises sharply, while the competitive effort of low-quality individuals rises more slowly (Fig. 2C). As a result, when the environment becomes more stable, the difference between the competitive effort of high- and the low-quality individuals rises. Owing to the steep increase in the competitive effort of high-quality individuals, both the fecundity of the high- and low-quality individuals falls (Fig. 2D).

In summary, environmental stability decreases the difference between the competitive effort of high- and low-quality individuals within resource-rich groups, while it reinforces the difference between the competitive effort of high- and low-quality individuals within resource-poor groups (compare Fig. 2A with Fig. 2B).

Group productivity and inequality

How do optimal competitive strategies influence individual, group productivity, and inequality? As shown in Fig. 2C, in resource-rich patches both the fecundity of high- and low-quality individuals rises as the environment becomes more stable. However, the fecundity of low-quality individuals rises more rapidly than the fecundity of high-quality individuals. As a result, within-group inequality among social partners falls as the environment becomes more stable in resource-rich patches (Fig. 3B). In resource-poor patches, both the fecundity of high- and low-quality individuals falls as the environment becomes more stable, as shown in Fig. 2D. However, the fecundity of low-quality individuals falls more rapidly than the fecundity of high-quality individuals. As a result, within-group inequality rises as the environment becomes more stable in resource-poor patches (Fig. 3B).

Overall, I find that both in resource-rich and resource-poor patches group productivity and within-group inequality are negatively correlated (compare Figs. 3A with 3B). Thus, while temporal stability leads to higher group productivity in resource-rich patches, in resource-poor patches it leads to lower group productivity (Fig. 3A). As a result, temporal stability leads to higher inequality between groups (Fig. 3C). In other words, in temporally stable environments, groups in resource-rich habitats become relatively more productive (i.e., the total number of offspring produced in each of the resource-rich habitats) than groups in resource-poor habitats as a result of their evolved social behaviours.

Finally, I find that social interactions always reduce the initial levels of within-group inequality, when I compare the initial level of inequality with the level of inequality at behavioural equilibrium (Fig. 3B). This is partially because social interactions have an additive effect on baseline fecundity. As a result, because the baseline fecundity of low-quality individuals is relatively smaller, an additive effect has a disproportional effect on the fecundity of low-quality individuals. By contrast, the change in the level of inequality between types of patches depends on the coefficient of temporal correlation (Fig. 3C). When the environment is relatively stable (τ > 0), between-group inequality rises relative to the initial levels of inequality. By contrast, when the environment is relatively unstable (τ < 0), between-group inequality falls relative to the initial levels of inequality.

Early-life acquisition of individual quality

My main model assumes that each patch accommodates exactly one high- and one low-quality individual. This mode of quality acquisition echoes scenarios in which social interactions mediate the state (or quality) of each group member, such as in the hierarchical structures observed in social insects or mammals. Alternatively, quality may be acquired early on during development, in which case patches need not have precisely one high-quality and one low-quality individual. Here, I modify my model to consider this scenario. In particular, I assume that offspring become high-quality with probability Q and low-quality with probability 1 – Q, and, for simplicity of analysis, I assume that Q = $\frac{1}{2}$. Hence, the composition of each patch is determined by a random variable. More specifically, patches can be either pure patches and accommodate two high- or two low-quality individuals (i.e., HH or LL patches), or mixed patches and accommodate one high- and one low-quality individual (i.e., HL patches). The frequency of each patch type, denoted by p_ij, is determined by the binomial distribution $p_{ij}=\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)Q^k{\left(1-Q\right)}^{n-k}$, where k is the number of high-quality individuals in the patch, n = 2 is the number of individuals per patch, and $ij\in \left\{\mathrm{HH},\mathrm{HL},\mathrm{LL}\right\}$ (see Appendix S1 for more details).

I find that early-life acquisition of quality leads to patterns of investment in competitive effort that are similar to those of the main model (compare Figs. 2 with 4). More specially, I find that in resource-rich patches competitive effort is lowest when the environment is relatively stable, and rises as the environment becomes unstable (Fig. 4A). Within resource-rich patches, however, there are noticeable differences in the average levels of competitive effort that strongly depend on patch composition. In particular, the highest and lowest levels of competitive effort are observed in mixed patches, where high-quality individuals express the highest levels of competitive effort and low-quality individuals express the lowest levels of competitive effort (Fig. 4A). Within pure patches, competitive effort is highest among low-quality breeders and lowest among high-quality breeders (Fig. 4A). Turning the attention to resource-poor patches, I find that competitive effort is highest when the environment is relatively stable, and falls as the environment becomes unstable (Fig. 4B). As in resource-rich patches, the highest and lowest levels of competitive effort are observed in mixed patches, where high-quality individuals express the highest levels of competitive effort, and low-quality breeders express the lowest levels of competitive effort (Fig. 4B). Within pure patches, competitive effort is highest among high-quality breeders and lowest among low-quality breeders (Fig. 4B).

These patterns of competitive effort lead to patterns of group productivity and inequality as a function of temporal correlation that are similar to those found in the main model (compare Figs. 3 with 5). In resource-rich patches, productivity is highest when the environment is relatively stable, and decreases as environmental instability increases (Fig. 5A). In resource-poor patches, productivity is lowest when the environment is relatively stable, and increases as environmental instability increases (Fig. 5A). Both in resource-rich and resource-poor patches, inequality is inversely correlated with patch productivity (Figs. 5A and 5B). In resource-rich patches, inequality is lowest when the environment is relatively stable, and it increases as environmental instability increases (Fig. 5B). In resource-poor patches, inequality is highest when the environment is relatively stable, and decreases as environmental instability increases (Fig. 5B). As a result, I find that inequality between patch types is highest when the environment is relatively stable, and decreases as environmental instability increases (Fig. 5C).

The effect of patch size

My main model assumes that each patch accommodates exactly two breeders. More generally, group size can be much larger than two, a factor that affects the genetic structure of the population, and therefore has implications for the evolution of social behaviour. Here, I contemplate this scenario by assuming that group size can be greater than two. As in the main model, I assume that half of the group members are high-quality and half are low-quality. I find that as patch size increases the relatedness between group mates decreases, which leads to higher levels of competitive effort. As a result, the amount of public good available to group members decreases, and therefore inequality increases (see Figure I1 in the Appendix S1). Despite this, I find that the patterns of competitive effort as a function of the coefficient of temporal correlation remain similar to those obtained in the main model.