Resource transfers and evolution: helpful offspring and sex allocation

Abstract

In some vertebrates, offspring help their parents produce additional offspring. Often individuals of one sex are more likely to become "helpers at the nest". We analyze how such sex-biased offspring helping can influence sex-ratio evolution. It is essential to account for age-structure because the sex ratios of early broods influence how much help is available for later broods; previous authors have not correctly accounted for this fact. When each female produces the same sex ratio in all broods (as assumed in all previous analyses of sex-biased helping), the optimal investment strategy is biased towards the more-helpful sex. When a female has facultative control over the sex ratio in each brood and each helper of a given sex increases the resource available for offspring production by a fixed amount, the optimal strategy is to produce only the more-helpful sex in early broods and only the less-helpful sex in later broods. When there are nonlinear returns from helping, i.e., each helper increases the amount of resource available for reproduction by an amount dependent upon the number of helpers, the optimal strategy is to maximize resource accrual from helping in early broods (which may involve the production of both sexes) and then switch to the exclusive production of the less-helpful sex in later broods. The population sex ratio is biased towards the more-helpful sex regardless of whether the sex ratio is fixed or age-dependent. When fitness returns from helping exhibit environmental patchiness, females are selected to produce only males on some patches and only females on others, and the population sex ratio may be biased in either direction. We discuss our results in light of empirical information on offspring helping, and we show via meta-analysis that there is no support for the claim of Griffin et al. [Griffin, A.S., Sheldon, B.C., West, S.A., 2005. Cooperative breeders adjust offspring sex ratios to produce helpful helpers. Amer. Nat. 166, 628-632] that parents produce more of the helpful sex when that sex is rare or absent.

Figure 1

Lifetime output with linear returns from helping and two broods (z = 2).Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. The constraints π₁, the maximal ${\widehat{\pi }}_2$, and the minimal ${\stackrel{.}{\pi }}_2$ are shown as solid lines. Accessible combinations of F_L and M_L are shaded and are bounded by (dotted) lines with slope −c_f/c_m. Strategy MM produces only males in both broods, FF produces only females in both broods, MF produces only males in the first brood and only females in the second, and FM produces only females in the first brood and only males in the second.

Figure 2

Lifetime output with a fixed investment ratio k and linear returns from helping. Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. There is a single bout of helping (α = 2) and a maximum of ten broods (z = 10). π_L is the constraint on lifetime output (solid curve). ϕ_min and ϕ_max are the minimum and maximum fitness isoclines (dashed). $\widehat{\beta }$ is the equilibrium fitness bisector and is the equal investment line. The point formula is a global ESS and is convergence stable.

Figure 3

(A). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. There is a single bout of helping (α = 2), a maximum of 10 broods (z = 10), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. π_L is the constraint on lifetime output (solid curve). ϕ_min and ϕ_max are the minimum and maximum fitness isoclines (dashed). $\widehat{\beta }$ is the equilibrium fitness bisector and ε is the equal investment line. There are diminishing returns in both sexes, a = b = 0.95. (B). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are increasing returns in both sexes, a = b = 1.02. (C) Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.99, and increasing returns in males, b = 1.03. (D). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.98, and increasing returns in males, b = 1.03.

Figure 3

(A). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. There is a single bout of helping (α = 2), a maximum of 10 broods (z = 10), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. π_L is the constraint on lifetime output (solid curve). ϕ_min and ϕ_max are the minimum and maximum fitness isoclines (dashed). $\widehat{\beta }$ is the equilibrium fitness bisector and ε is the equal investment line. There are diminishing returns in both sexes, a = b = 0.95. (B). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are increasing returns in both sexes, a = b = 1.02. (C) Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.99, and increasing returns in males, b = 1.03. (D). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.98, and increasing returns in males, b = 1.03.

Figure 3

(A). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. There is a single bout of helping (α = 2), a maximum of 10 broods (z = 10), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. π_L is the constraint on lifetime output (solid curve). ϕ_min and ϕ_max are the minimum and maximum fitness isoclines (dashed). $\widehat{\beta }$ is the equilibrium fitness bisector and ε is the equal investment line. There are diminishing returns in both sexes, a = b = 0.95. (B). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are increasing returns in both sexes, a = b = 1.02. (C) Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.99, and increasing returns in males, b = 1.03. (D). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.98, and increasing returns in males, b = 1.03.

Figure 3

(A). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. There is a single bout of helping (α = 2), a maximum of 10 broods (z = 10), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. π_L is the constraint on lifetime output (solid curve). ϕ_min and ϕ_max are the minimum and maximum fitness isoclines (dashed). $\widehat{\beta }$ is the equilibrium fitness bisector and ε is the equal investment line. There are diminishing returns in both sexes, a = b = 0.95. (B). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are increasing returns in both sexes, a = b = 1.02. (C) Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.99, and increasing returns in males, b = 1.03. (D). Lifetime output with a fixed investment ratio k and nonlinear linear returns from helping. Assumptions as in 3(A) except that there are diminishing returns in females, a = 0.98, and increasing returns in males, b = 1.03.

Figure 4

(A) Lifetime output with linear returns from helping and conditional sex ratios. Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. Points on the boundary follow the same labeling convention as Fig. 1. Lifetime output is M_L, F_L. β is the fitness bisector and ϕ is the effective fitness isocline (dashed). (A) Accessible combinations of F_L and M_L are shaded and are bounded by dotted lines with slope −c_f/c_m. (B). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only males in the first two broods and mixture in the third brood. (C). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only females in the first brood, both sexes in the second brood, and only males in the third brood. (D). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A). At equilibrium, all females produce only males in the first two broods and only females in the third (MMF).The equilibrium fitness bisector $\widehat{\beta }$ passes through this point, and the equilibrium fitness isocline $\widehat{\varphi }$ intersects the set of accessible strategies only at this point. is the equal-investment line.

Figure 4

(A) Lifetime output with linear returns from helping and conditional sex ratios. Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. Points on the boundary follow the same labeling convention as Fig. 1. Lifetime output is M_L, F_L. β is the fitness bisector and ϕ is the effective fitness isocline (dashed). (A) Accessible combinations of F_L and M_L are shaded and are bounded by dotted lines with slope −c_f/c_m. (B). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only males in the first two broods and mixture in the third brood. (C). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only females in the first brood, both sexes in the second brood, and only males in the third brood. (D). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A). At equilibrium, all females produce only males in the first two broods and only females in the third (MMF).The equilibrium fitness bisector $\widehat{\beta }$ passes through this point, and the equilibrium fitness isocline $\widehat{\varphi }$ intersects the set of accessible strategies only at this point. is the equal-investment line.

Figure 4

(A) Lifetime output with linear returns from helping and conditional sex ratios. Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. Points on the boundary follow the same labeling convention as Fig. 1. Lifetime output is M_L, F_L. β is the fitness bisector and ϕ is the effective fitness isocline (dashed). (A) Accessible combinations of F_L and M_L are shaded and are bounded by dotted lines with slope −c_f/c_m. (B). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only males in the first two broods and mixture in the third brood. (C). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only females in the first brood, both sexes in the second brood, and only males in the third brood. (D). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A). At equilibrium, all females produce only males in the first two broods and only females in the third (MMF).The equilibrium fitness bisector $\widehat{\beta }$ passes through this point, and the equilibrium fitness isocline $\widehat{\varphi }$ intersects the set of accessible strategies only at this point. is the equal-investment line.

Figure 4

(A) Lifetime output with linear returns from helping and conditional sex ratios. Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. Points on the boundary follow the same labeling convention as Fig. 1. Lifetime output is M_L, F_L. β is the fitness bisector and ϕ is the effective fitness isocline (dashed). (A) Accessible combinations of F_L and M_L are shaded and are bounded by dotted lines with slope −c_f/c_m. (B). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only males in the first two broods and mixture in the third brood. (C). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A) except that each female produces only females in the first brood, both sexes in the second brood, and only males in the third brood. (D). Lifetime output with linear returns from helping and conditional sex ratios. Assumptions as in 4(A). At equilibrium, all females produce only males in the first two broods and only females in the third (MMF).The equilibrium fitness bisector $\widehat{\beta }$ passes through this point, and the equilibrium fitness isocline $\widehat{\varphi }$ intersects the set of accessible strategies only at this point. is the equal-investment line.

Figure 5

Lifetime output with many broods and linear returns from helping. There is a single bout of helping (α = 2), the maximum number of broods is z = 30, and a = b = 1.Males are more helpful than females (τ_m=1.0, τ_f=0.5), and c_f=c_m=γ_m=γ_f=l_f1=l_m1=1. The set of accessible lifetime strategies is shaded, and the optimal lifetime strategy is (${\widehat{M}}_L$, ${\widehat{F}}_L$). $\widehat{\beta }$ is the equilibrium fitness bisector, and $\widehat{\varphi }$ is the equilibrium fitness isocline (dashed). ε is the equal-investment line.

Figure 6

The relationship between the effect size for helping (r_help) and the effect size for change of sex ratio (r_sex). Circles denote r_sex “offspring” estimates based on comparing the aggregate of offspring produced by pairs with helpers and the aggregate of offspring produced by pairs without helpers. Triangles denote r_sex “brood” estimates based on comparing sex ratios produced by pairs with helpers and sex ratios produced by pairs without helpers. Numbers denote species as listed in Table 1.