## 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
\begin{equation}
\label{sfm}
\frac{w'(y_p^*)}{w(y_p^*)} - \frac{1}{y_p^*} = 0\;.
\end{equation}
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,
\begin{equation}
\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 \; .
\end{equation}
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
\begin{equation}
\label{yq-extra-structs}
\frac{w'(y_p^*)}{w(y_p^*)} -
\frac{1 + q'(y_p^*)}
{y_p^* + q(y_p^*)} = 0 \; ,
\end{equation}
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
\begin{equation}
\label{yq-linear-yp}
\frac{w'(y_p^*)}{w(y_p^*)} -
\frac{1 + kn^{-1}}
{y_p^*} = 0 \; .
\end{equation}
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
\begin{equation}
\label{yq-log-yp}
\frac{w'(y_p^*)}{w(y_p^*)} -
\frac{1 + kn^{-1}y_p^{*-1}}
{y_p^*} = 0 \; .
\end{equation}
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.