## Abstract

The purpose of this paper is twofold, technological and scientific. The technological is to demonstrate how professional scientific publishing can be done with web technologies (namely, HTML5, CSS3 and Javascript). My hope is to demonstrate that with web technologies, one can achieve all the typesetting and styling tricks that are familiar from the professional literature (multiple columns, math, hyphenation, citations and bibliography generation, etc.), while allowing for much greater flexibility, such as adjusting to different screen sizes and devices (e.g., desktop, mobile devices, or printed on paper), as well as displaying the same text in different alternative ways (by changing the accompanying CSS and Javascript, but not the document itself; i.e., separation of content and presentation). I try to achieve this goal convincingly by using a concrete real-life example of my own scientific research and writing. The rest of the text deals with the scientific.

The seminal work on evolution of offspring size by Smith.Fretwl.74 has had lasting impact in the four decades since its publication. However, a major flaw in the Smith-Fretwell model (hereafter, SFM) still remains. Namely, its prediction that optimal offspring size should not depend on maternal size or total reproductive effort. This prediction strongly contrasts with the widespread observation of variation in offspring size with different aspects of the maternal phenotype and reproductive allocation. In this note, I explore why this incongruity between theory and observation has persisted so long, despite various modifications to the theory. I suggest that SFM can be brought to agree with observations, by realizing that the original formulation implicitly includes the ideal assumption that reproductive effort relates only to direct material costs, and therefore, is equal or proportional to clutch mass or the product of offspring size and number. When explicitly considering additional (overhead) costs of reproduction, variation in optimal offspring size with maternal phenotype is readily obtained. I provide a modified expression for optimal offspring size that includes the effect of such overhead costs. In addition, I identify functional forms of such overhead costs that facilitate variation in optimal offspring size.

## Introduction

Offspring size is a fundamental life-history trait Roff.02 that – being a clear expression of the continuity of the phenotype Wstbrh.03 – bridges two separate individual lifecycles – that of the mother and that of the offspring. Despite 40 years since its publication, the work of Smith.Fretwl.74 is still the standard point of reference for both empirical and theoretical work on the evolution of offspring size. The Smith-Fretwell model (hereafter SFM), states that maternal fitness is the product of offspring number (e.g., number of eggs) and size-dependent performance of the offspring. The model is summarized by $F(y_p) = nw(y_p)$, where $y_p$ represents the size of offspring (or propagules; e.g., eggs or seeds), $F$ is maternal fitness, $n$ offspring number, and $w$ is the offspring size-performance curve. The formulation of SFM further assumes $n(y_p) = E/y_p$, where $E$ denotes total reproductive effort (i.e., the total amount of resources that a mother can divide among many small or few large offspring). The expression for optimal offspring size (i.e., the value of $y_p$ that maximizes $F$), denoted by $y_p^*$, is then $$\label{sfm} \frac{w'(y_p^*)}{w(y_p^*)} - \frac{1}{y_p^*} = 0\;.$$ This expression is similar to Charnov's (Charnv.76) marginal value theorem.

It is clear from that optimal offspring size depends on the shape of the performance curve, $w(y_p)$, and many later studies explored how this curve changes in different environments or circumstances Temme.86, Lloyd.87,Haig.90, Schltz.91, Morris.98, Hendry.Day.etal.01b.

However, although SFM predicts a single optimal value that, independent of total reproductive effort ($E$ or $n$), empirical data shows much variation in offspring size in many cases. For example, variation within and among females Kaplan.Cooper.84, Reznck.Yang.93, correlations between egg size and total reproductive effort Fox.Czesak.00, Caley.Schwrz.etal.01, Beck.Beck.05, Nasutn.Robrts.etal.10, and trends of increasing or decreasing offspring size with maternal size or age Landa.92a, Fox.Czesak.00, Kamler.05, Marshl.Keough.08, Kndsvt.Rsnthl.etal.11.

This disagreement between theory and observation becomes even further pronounced by contrasting with the other half of the size-number tradeoff, namely with Lack's model of most productive clutch size Lack.54 . Lack's model has been successfully amended to account for effects of parental quality and parental survival, and thus brought into accord with empirical observations Roff.02. In contrast, the single optimal offspring size prediction of SFM has been much more persistent, despite attempts to modify SFM by incorporating effects of maternal survival Winklr.Wallin.87.

Two possible ways to resolve this nagging incongruity have been previously pursued. First, if offspring number is constrained, increased reproductive effort translates to larger offspring Begon.Parker.86. Similarly, morphological constraint can lead to an increase in offspring size with maternal size Marshl.Hepel.etal.10, Nasutn.Robrts.etal.10.

A second type of resolution deals with how the offspring size-performance curve ($w$ in may vary among females. In particular, if offspring of larger females experience stronger sib competition Parker.Begon.86, or if larger females secure better oviposition sites Hendry.Day.etal.01b. This second path also relates to studies that try to explain variation in offspring size within females or clutches Kaplan.Cooper.84, Temme.86, Mcgnly.Temme.etal.87, Kaplan.Cooper.88, Haig.90, Schltz.91 (e.g., due to bet-hedging, when faced with spatio-temporal environmental stochasticity).

However, as also pointed out by recent studies Marshl.Hepel.etal.10, Kndsvt.Rsnthl.etal.11, Jrgnsn.Auer.etal.11, these kinds of explanation are limited, in the sense of either being taxon-specific (and thus not applicable in general), or in fact predicting opposite patterns of variation than those actually observed Hendry.Day.03. Because positive correlation between female size and offspring size is taxonomically widespread (e.g., insects, Fox.Czesak.00; marine invertebrates, Marshl.Keough.08, Marshl.Hepel.etal.10, Nasutn.Robrts.etal.10; fish, Reznck.Yang.93, Rolnsn.Htchng.10, Jrgnsn.Auer.etal.11, Kndsvt.Rsnthl.etal.11; plants, Sakai.Harada.01, Sakai.Sakai.05), we should try to look for an explanation through more general aspects of reproductive allocation and offspring provisioning.

Ultimately, offspring or maternal performance and offspring provisioning are cumulative quantities, and their analysis requires a dynamical approach. Indeed, more recent work aims at better incorporation of physiological, developmental and behavioral processes Sargnt.Taylor.etal.87, Snervo.99, Marshl.Keough.08, Uller.While.etal.09, Segers.Tbrsky.11,Sakai.Harada.01, Kflawi.06, Jrgnsn.Auer.etal.11, Marshl.Hepel.etal.10, Kndsvt.Alonzo.etal.10, Kndsvt.Bonsal.etal.11. For example, Jrgnsn.Auer.etal.11 apply a model of size-dependent mortality and growth to live-bearers, and conclude that offspring-size-female-size correlation may arise because prenatal offspring mortality is the same as maternal mortality, which may be lower for larger females. However, as discussed above for sib competition and similar explanations, such an explanation cannot hold in general.

A second model, by Kndsvt.Bonsal.etal.11, applies stochastic dynamic programming to describe the maternal energy budget during reproduction, in order to obtain patterns of variation in offspring size and number with maternal age and size. Their model provides a general conclusion that variation in offspring size with maternal size or age depends on survival costs experienced by the mother.

However, these models, although incorporating some dynamic aspects of performance, do not provide a full dynamical description of the entire lifecycle (either abstracting the adult phase via a fixed parameter for total reproductive effort, or relying on the phenomenological size-performance curve of to describe offspring performance). Moreover, it is important to understand the reason to why SFM does not predict variation in offspring size, so to better isolate their causes. Surprisingly, earlier but somewhat unnoticed work Kzlwsk.96a found that optimal offspring size should depend on maternal size, by applying a dynamic energy allocation model that combines both dynamical description of offspring performance and explicit effects of provisioning finite-sized offspring.

Therefore, resolution of the optimal offspring size problem and the related incongruity between theory and observation can be divided into two tasks. The first is to clarify why SFM produces an optimal value that does not depend on maternal phenotype or reproductive effort. The second task is to provide a model for optimal offspring size that incorporates a full and detailed dynamic description of offspring and maternal performance throughout the entire lifecycle.

In this note, I concentrate on the first task, while the second is described in detail in a second publication Filin.15. I develop a modified version of SFM that explicitly accounts for inefficiencies and overhead loss of maternal reserves during reproduction. I show that previous attempts to modify SFM have, nonetheless, produced the SFM prediction, exactly because of not incorporating such overhead costs of reproduction. When such costs are considered, I readily obtain that the three-way tradeoff between offspring size, offspring number and maternal survival, leads to optimal covariation of offspring size with offspring number, total reproductive effort, and maternal phenotype.

## The model

Previously, Winklr.Wallin.87 tried to modify SFM to account for effects of maternal survival. Following their analysis, I can represent maternal fitness as $F(y_p,n) = nw(y_p) + S(Y-ny_p)$, where $n$ and $y_p$ are offspring number and size, respectively, and $S$ is maternal survival that depends on maternal reserves, following the reproduction event. Maternal reserves prior to reproduction is given by $Y$, and the product $ny_p$ represents clutch mass or reproductive effort, i.e., the amount of reserves that were expended in order to produce $n$ offspring of size $y_p$.

The optimal offspring size and number (denoted by $y_p^*$ and $n^*$, respectively) are easily obtained by setting the derivatives of $F$ to zero. In this case, I, in fact, obtain the SFM expression for optimal offspring size (). This result is similar to that of Winklr.Wallin.87, namely, inclusion of maternal survival in SFM does not affect the optimal offspring size.

Reproductive effort, however, usually entails additional energetic costs, beyond those that directly translate to offspring size Hrshmn.Zera.07. Such overhead costs may include, for example, reproductive support structures (Kawano.Hara.95; e.g., flowers), external structures of eggs or seeds (e.g. egg capsules; Nasutn.Robrts.etal.10), or respiration costs during offspring provisioning Sakai.Harada.01. Moreover, such costs may not necessarily scale as the product $ny_p$, and exhibit other functional forms.

I can, therefore, conclude that there is an implicit assumption in SFM (as well as later studies, such as Winklr.Wallin.87). These models assume an ideal situation, where the amount of maternal reserves expended in reproduction relates solely to direct costs of producing the clutch mass (i.e., $n$ offspring of size $y_p$). By introducing overhead costs on top of the direct costs, $ny_p$, the expression for fitness can be rewritten as $F = nw(y_p) + S(Y - ny_p - y_q)$, where $y_q$ represents overhead costs, i.e., expended maternal reserves that do not translate to offspring mass. In general, $y_q$ may depend on offspring size, offspring number, clutch mass, and additional parameters.

I now obtain the following expression for optimal offspring size, $$\label{sfm-with-yq} \frac{w'(y_p^*)}{w(y_p^*)} - \frac{1+n^{-1}\partial y_q /\partial y_p} {y_p^* + \partial y_q /\partial n} = 0 \; .$$ This expression has a similar structure to . The first term relates to offspring performance. The second term arises from the three-way tradeoff between offspring size and number, which determine current reproduction, and future reproduction (maternal survival).

The important difference between and , is that, in my expression, the second term depends not only on offspring size, but also on offspring number (through the explicit appearance of $n^{-1}$ in the numerator, and possible dependence of overhead costs, $y_q$, on $n$, $y_p$, or clutch mass, $ny_p$). Clearly, when there is no overhead costs ($y_q = 0$), degenerates to , and I obtain the SFM result. However, nonzero overhead costs may potentially cause optimal offspring size to vary with offspring number or reproductive effort, in sharp contrast to the prediction of SFM.

## Results: allometry of overhead costs

It is easy to verify that if overhead costs are constant (i.e., a fixed parameter) or depend solely on maternal phenotype (i.e., $Y$ in this model), still yields the SFM solution (), i.e., optimal offspring size is independent of maternal phenotype, offspring number, or reproductive effort. However, if overhead costs depend also on offspring size and/or number, the resulting optimal offspring size may potentially vary (overhead costs, $y_q$, appear in through the terms $n^{-1}\partial y_q /\partial y_p$ and $\partial y_q /\partial n$).

The question that arises is, therefore, which functional forms of $y_q(y_p,n)$ are biologically meaningful, and what is their influence on optimal offspring size. One obvious source of overhead costs is additional structures of propagules, such as egg casings or dispersal structures Sakai.Kkzawa.etal.98, Nasutn.Robrts.etal.10, that do not eventually translate into offspring initial mass. The form of overhead costs can be written in this case as the product of offspring number, $n$, and overhead cost per offspring, $q(y_p)$, where $q$ is some arbitrary function of offspring size. Plugging that into , I obtain the expression $$\label{yq-extra-structs} \frac{w'(y_p^*)}{w(y_p^*)} - \frac{1 + q'(y_p^*)} {y_p^* + q(y_p^*)} = 0 \; ,$$ where $q'(y_p) = dq/dy_p$. Clearly the dependence on offspring number disappears in this case and there is a single optimal offspring size, independent of total reproductive effort, although different than the value predicted by SFM (compare with ).

A similar situation occurs when $y_q$ depends solely on clutch mass, i.e., the product $ny_q$ (i.e., $y_q = y_q(ny_p)$). In fact, it is easy to verify that the extra terms introduced by cancel each other and in this case we obtain the SFM expression for optimal offspring size (). This is a strong conclusion, because it means that any cost that relates solely to the total transfer of mass between female and progeny (such as metabolic cost associated with this mass transfer), cannot explain variation in offspring size.

Given that these two obvious sources of overhead costs fail to produce any variation in offspring size, what other allometric relation for overhead costs can be expected? One additional universal source of energetic costs during offspring production and provisioning is respiration Sakai.Harada.01. This sort of cost requires an explicit consideration of the time dimension. Offspring production and provisioning is not instant, but occurs over a duration that is likely to increase with offspring size, $y_p$. Maternal respiration costs accumulate over longer periods, as offspring size increases.

For a given offspring size, this duration can be reduced by increasing the per-offspring rate of provisioning (either by reducing number of offspring or increasing total rate of provisioning). However, because there is a physiological maximum for such per-offspring provisioning rate (what Sakai.Harada.01 call terminal-stream-limitation), this can only have a limited effect.

As an illustration, taking the simplest form of such overhead costs, i.e., $y_q=ky_p$ (where $k$ here is a fixed parameter), yields $$\label{yq-linear-yp} \frac{w'(y_p^*)}{w(y_p^*)} - \frac{1 + kn^{-1}} {y_p^*} = 0 \; .$$ If propagule or offspring exhibit exponential growth during production or provisioning, overhead costs may take the form $y_q=k\ln(y_p)$, for which optimal offspring size is given by $$\label{yq-log-yp} \frac{w'(y_p^*)}{w(y_p^*)} - \frac{1 + kn^{-1}y_p^{*-1}} {y_p^*} = 0 \; .$$ In both cases, the effect of overhead costs diminishes as offspring number, $n$, increases, leading to variation in offspring size, as offspring number (or reproductive effort) increases.

demonstrates typical curves of optimal offspring size against offspring number using the expressions in and . In both cases optimal offspring size is below the value predicted by SFM (), and increases with offspring number, approaching the SFM value asymptotically. Positive correlation between offspring size and total reproductive effort is also shown (). The figures were prepared using GNU R Rsoftware.

## Discussion

In this note I clarified why the Smith-Fretwl.74 prediction that optimal offspring size is independent of reproductive effort has been so persistent. By assuming that cost of reproduction relates solely to direct costs of producing the clutch mass, Smith.Fretwl.74, as well as later work Winklr.Wallin.87, could not have observed the effect of overhead costs of reproduction. As I presented in this work, overhead costs of reproduction are a crucial element of maternal performance that has not been considered previously in relation to the offspring-size problem.

Using the general expression that I derived (), I identified circumstances, in which variation in offspring size is expected, as well as cases where it should not occur. In particular, respiration costs that result in overhead costs that are dependent on offspring size will typically result in positive correlation between offspring size and offspring number, clutch mass or total reproductive effort (see ). Moreover, the pattern of increase in offspring size is concave down, as typically observed in empirical data Kamler.05, Nasutn.Robrts.etal.10.

It is reasonable to expect that several sources of overhead costs in offspring production should occur simultaneously. For example, costs associate with construction of egg casings or dispersal structures (as discussed above) can operate in addition to maternal metabolic costs. Moreover, respiration costs may also depend on the maternal phenotype (for example, on maternal body mass), introducing yet another source of correlation between offspring size and maternal phenotype. The further explores these additional points.

I note that clutch mass is often used in empirical studies as a measure of reproductive effort Roff.02, Caley.Schwrz.etal.01, Nasutn.Robrts.etal.10. However, my analysis and results here suggest that clutch mass underestimates the full energetic expenditure during reproduction. Overhead costs of reproduction have important consequences to life-history, and should be explicitly addressed.

Finally, the model in this note is an elaboration of the Smith-Fretwell model, which is rather simplistic, and does not include much biological detail. The shortcomings of SFM and similar models have been previously discussed in detail Brnrdo.96, Marshl.Keough.08. In this last paragraph I would just elaborate on one. Namely, the breakdown of total offspring size into functional components, i.e., structure vs. reserves Koojmn.10, Filin.09. Given that both maternal and offspring size are composed of these two distinct functional components, an additional dimension of potential variation, i.e. division of total size between size components, must enter into consideration. This is further explored in great detail in a second publication Filin.15.