The evolution of simultaneous progressive provisioning revisited: extending the model to overlapping generations

Abstract

“Simultaneous progressive provisioners” feed their offspring gradually as they develop—and typically feed more than one offspring simultaneously (SIM) at a time. In contrast, “sequential mass provisioners” supply offspring one after another (SEQ). Utilizing individual-based simulations, Field (Nature 404:869–871, 2005) compared the lifetime reproductive success of these strategies in different scenarios. Accordingly, SEQ should evolve in the majority of cases—SIM only has an evolutionary benefit if offspring depend on their mothers’ protection until adulthood even past the provisioning period. However, this is only one potential explanation for the evolution of SIM. Here, we present an alternative mechanism for solitary individuals with overlapping generations. We propose an analytical model (comprising Field’s former approach) utilizing growth rate instead of lifetime reproductive success as a measure of fitness. Our model shows that multiplicative geometric effects on fitness would typically compensate for the demographic disadvantages of SIM (due to prolonged dependency) and consequently support the evolution of SIM over SEQ for a wide range of life history parameters. The optimal level of SIM (i.e., the optimal number of eggs laid simultaneously) is determined by offspring development time, survival rates, and foraging efficiency of the mother. Only extreme values of these demographic parameters would favor a transition to SEQ behavior. Our model provides a coherent explanation of selective favoring of SIM over SEQ that may also contribute to understanding why SIM is the dominant strategy among social insect species.

Significance statement

Workers in social insects typically feed several offspring simultaneously while solitary species with parental care—apart from a few exceptions—provision brood cells one after another. The provisioning pattern might play a prominent role in the evolutionary pathway to higher social organization. Based on a novel theoretical approach, we show that geometric growth benefits increase selection pressure towards simultaneous progressive provisioning in species with generation overlap. Such geometric benefits may specifically emerge in seasonal eusocial species. This result alters former assessment of causal mechanisms and extends findings focusing on solitary insects. It adds a new and reasonable explanation for the dominance of simultaneous provisioning among social species.

Appendix

Proposition 1:

Lifetime reproductive success (LRS) is always decreasing with clutch size b if b > D _L /f.

Proof:

If b > D _L /f, factor q is independent of b: $q={s}^{D_P}$. LRS is $R=w\frac{b{s}^{f\cdotp b}}{1-{s}^{f\cdotp b}}$ with w = (q + (1 − q)(1 − μ)). Calculating the derivative with respect to b yields

$$ {R}^{\prime }(b)=-w\frac{s^{fb}\left({s}^{bf}-bf \ln (s)-1\right)}{{\left({s}^{bf}-1\right)}^2}=-w\frac{s^{fb}\left({s}^{bf}- \ln \left({s}^{bf}\right)-1\right)}{{\left({s}^{bf}-1\right)}^2}=w\frac{y\left(1-y+ \ln (y)\right)}{{\left(y-1\right)}^2} $$

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with y = s ^bf and 0 < y < 1. Function g(y) = 1 − y + ln(y) decides on the sign of R ^′(b). At the interval borders, g(y → 0) =  − ∞ and g(1) = 0. g ^′(y) = 1/y − 1 is always >0 for 0 < y < 1. Thus, g(y) is always <0 for 0 < y < 1 and R ^′(b) is always negative.

Proposition 2:

If b ≤ D _L /f, lifetime reproductive success is always decreasing with clutch size b for μ = 0 and always increasing for μ = 1.

Proof:

If b ≤ D _L /f, factor q depends on b: $q={s}^{D_P+{D}_L-bf}$. LRS is $R={w}^{\prime}\mu \frac{b}{1-{s}^{f\cdot b}}+\left(1-\mu \right)\frac{b{s}^{fb}}{1-{s}^{f\cdot b}}$ with ${w}^{\prime }={s}^{D_L+{D}_P}$. If μ = 0, the arguments from A1 can be applied to show that R ^′(b) is always negative, too. If μ = 1,

$$ {R}^{\prime }(b)={w}^{\prime}\frac{1-y+y \ln (y)}{{\left(1-y\right)}^2} $$

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Function $\overset{\sim }{g}(y)=-y+y \ln (y)+1$ decides on the sign of R ^′(b). $\overset{\sim }{g}\left(y\to 0\right)=1$ and $\overset{\sim }{g}(1)=0$. ${\overset{\sim }{g}}^{\prime }(y)=\mathit{\ln}(y)$ is always negative, and thus, $\overset{\sim }{g}(y)$ is always positive for 0 < y < 1. Thus, R ^′(b) is always positive.

Proposition 3:

There is no local relative maximum of lifetime reproductive success for b ≤ D _L /f, and lifetime reproductive success is maximized for b = 1 or b = D _L /f.

Proof:

In general, the derivative of LRS R(b) is

$$ {R}^{\prime }(b)={w}^{\prime}\mu A-\left(1-\mu \right)B $$

with $A=\frac{1-y+y \ln (y)}{\left(1-y\right)2}$ and $B=-\frac{y\left(1-y+\mathit{\ln}(y)\right)}{{\left(y-1\right)}^2}$

LRS R(b) is increasing as long as R ^′(b) > 0. We further analyze this condition:

$$ {R}^{\prime }(b)>0\iff {w}^{\prime}\mu A-\left(1-\mu \right)B>0\iff \frac{w^{\prime}\mu }{1-\mu }>\frac{B}{A} $$

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h(y) = B/A is always increasing for increasing y within the interval 0 < y < 1 (Fig. 4).

Fig. 4

Function h(y) = B/A in dependence of y.

Thus, there must be a value y ₀ such that if y < y ₀, then $\frac{B}{A}<\frac{w^{\prime}\mu }{1-\mu }$. It is not necessary to calculate y ₀ —we further just make use of its existence. As y = s ^bf, the derivative of LRS R ^′ is always greater than 0 (and R(b) is increasing) if b > b ₀ with ${b}_0= \ln \frac{y_0}{f \ln (s)}$. If b is smaller than this threshold, R(b) is decreasing. In general, R(b) is either always decreasing from b = 1 to b = D _L /f or it is decreasing from b = 1 to b = b ₀ and increasing from b = b ₀ to b = D _L /f. In either case, LRS cannot have a local maximum.

Corollary 4: Critical values of offspring dependency μ promoting the transition from SEQ to SIM can be calculated analytically from Eq. (5) for any parameter combination:

$$ {\mu}_{crit}\left({D}_L,{D}_P,s,f\right)=\frac{L}{s^{D_L+{D}_P}K+L} $$

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with $K=\frac{1}{1-{s}^f}-\frac{D_L}{1-{s}^{f{D}_L}}$ and $L=\frac{D_L{s}^{f{D}_L}}{1-{s}^{f{D}_L}}-\frac{s^f}{1-{s}^f}$.

This directly follows from A3. A transition between SEQ and SIM as optimal strategies will occur when

$$ R(1)=R\left({D}_L/f\right) $$

This can be used to determine threshold values for any parameter in dependence of the others.

Corollary 5:

$$ \underset{s\to 1}{\mathit{\lim}}{\mu}_{crit}\left({D}_L,{D}_P,s,f\right)=1/2 $$

This limit can easily been calculated by applying the Limit[] function of the computer algebra system Mathematica (Wolfram Research Inc. 2016) to Eq. (12).

Proposition 4:

Worker number W _max at the beginning of sexual production affects final cumulative sexual number S(T) at time t = T after the beginning of sexual production as a linear multiplicative factor in the colony model provided by Oster and Wilson (1978): S(T) ∼ W _max

Proof:

The dynamic equation for the number of sexuals S(t) at time t after the onset of sexual production is

$$ \frac{\partial S(t)}{\partial t}=c{W}_{max}{e}^{-\mu t}-\nu S(t) $$

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with worker efficiency rate c, worker mortality rate μ, and sexual mortality rate ν. The solution to this equation is

$$ S(t)=\frac{c{W}_{max}}{\mu -\nu}\left({e}^{-\nu t}-{e}^{-\mu t}\right) $$

resulting in

$$ S(T)\sim {W}_{max} $$

at the end of the season when t = T. Calculation has been checked with computer algebra system Mathematica (Wolfram Research Inc. 2016).