Ecological Modelling 193 (2006) 787–795
Birth functions in stage structured two-sex models
Andreas Sundelöf ∗ , Per Åberg
Department of Marine Ecology, Marine Botany, Göteborg University, P.O. Box 461, SE-405 30 Göteborg, Sweden
Received 13 January 2005; received in revised form 8 July 2005; accepted 1 August 2005
Available online 24 October 2005
Abstract
The most commonly used birth function in two-sex demographic models is the harmonic mean birth function. This function
treats all individuals of one sex as identical, i.e., stage specific fecundity is not taken into account. In the analysis presented here,
the harmonic mean birth function is developed to incorporate size and sex specific fecundities. This structured birth function is
compared with the harmonic mean birth function using both a hypothetical population and data for populations of the limpet
Patella vulgata. A general expression to calculate the threshold value where the unstructured and the structured birth functions
coincide is also provided. Using the structured birth function resulted in changes in population dynamics, growth rate, proportion
of males and reproductive output. In conclusion, the choice of birth function is important due to its effects on the deterministic
population characteristics, which in turn may have consequences for the managements of endangered or vulnerable species.
© 2005 Elsevier B.V. All rights reserved.
Keywords: Two-sex model; Harmonic mean; Birth functions; Patella vulgata
1. Introduction
Most demographic models are one-sex models
where it is assumed that both sexes have identical vital
rates or that the dynamics of the population are determined by only one sex (Caswell, 2001). However, there
are many examples of sexual dimorphism in vital rates
such as juvenile mortality (Clutton-Brock et al., 1985)
and fecundity (Clutton-Brock, 1988), which indicate
the need for two-sex models. Male or female domi∗ Corresponding author. Tel.: +46 31 773 27 08;
fax: +46 31 773 27 27.
E-mail addresses: [email protected]
(A. Sundelöf), [email protected] (P. Åberg).
0304-3800/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2005.08.040
nance may on the other hand be large so that there
always are enough individuals of the subordinate sex
and thus sexual dimorphism in survival rates will not
influence the asymptotic population growth, although
it will give rise to skewed sex ratios. There are populations where dominance is weak, i.e., where the
abundance of both sexes is important for the reproductive output, e.g., polygamous populations. Several birth
functions have been used in population modelling and
the harmonic mean function is regarded the least flawed
in human demography (Caswell, 2001). The fertility
functions derived from the harmonic birth function are
fixed clutch sizes times the proportion of males (for
female fertility) or females (for male fertility) and the
maximum per capita (p.c.) birth rate of the popula-
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A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
tion occur at the 1:1 sex ratio (cf., Caswell, 2001). The
harmonic birth function has been applied in several
non-human demographic studies (e.g., Lindström and
Kokko, 1998; Ranta and Kaitala, 1999; Ranta et al.,
1999; Engel et al., 2001) and has further been modified
and applied in polygamous systems (Ranta and Kaitala,
1999), where the maximum reproduction of a population occurs at a skewed sex ratio. However, a limitation
of the harmonic birth function is that the age or stage
structure within each sex is ignored.
Human demographers have developed age structured birth functions and one approach has been to
use separate functions for male and female growth.
The problem with these models has been that the separate sexes may develop different growth rates leading
to extinction of the sex with the lowest growth rate
(Martcheva, 1999). There are examples of age or stage
structured birth functions that do not have this problem like the discrete Birth Matrix-Mating Rule model
(BMMR; Pollak, 1986) and the continuous analogue
of that (Martcheva, 1999), which can be applied to
monogamous populations.
Many marine organisms and wind pollinated terrestrial plants are polygamous in the sense that they
release their gametes (or only male gametes) in the
water column or in the air and as a result of mixing
of gametes many pairs of parents are possible. Often
these organisms also have indeterminate growth and
size-dependent fecundity and it can be argued that large
individuals with larger fecundity will have a larger
clutch size than smaller individuals. However, fertilisation success will also depend on other factors such
as the synchrony of the spawning event and the spatial distribution of individuals (Levitan et al., 1992;
Yund, 2000). The birth functions mentioned above cannot directly be applied to spawning organism with
size-dependent fecundity and in this paper we modify the harmonic birth function, so it can be applied to
such organisms. The aim here is to describe a birth
function for reproduction in demographically structured two-sex populations, compare this function with
the harmonic birth function and illustrate its dynamics
when applied in a matrix population model. It is common to include size specific fecundity or fertility in
one-sex matrix population models (for examples, see
Caswell, 2001). There are also examples of two-sex
models where differential fertility between sexes and
other stages are taken into account (Heide-Jorgensen
et al., 1992). There are other models that include age
structured fecundity, but in an implicit way (Pollard,
1997; Hsu Schmitz and Castillo-Chavez, 2000). However, in this paper, we develop and analyse a model
that can be adopted to either size, age or sex structured
populations, or any combination thereof. We compare
it with the harmonic birth function as well as the harmonic function modified for a harem scenario and this
has to our knowledge not been done before.
The motivation for this work came from our
attempt in developing a demographic model for the
limpet Patella vulgata. This species is a sequential
hermaphrodite, where all individuals are born as neuter,
they nearly all mature as males and at an age of
about three years they change sex and become females
(Ballantine, 1961; Salerna de Mendonca Corte-Real,
1992). This free-spawning and sedentary species has
indeterminate growth and size-dependent fecundity,
and the modified birth function is expected to capture
the main features of the reproduction in P. vulgata as
well as other size and sex structured organisms.
2. Methods
The basic harmonic birth function is given by
Caswell (2001) to be:
B(n) =
2knm nf
nm + n f
(1)
where k is the clutch size and nm and nf are the densities of reproductive males and females, respectively.
This model assumes monogamy and an equal clutch
size of each male and female regardless of its size or
other stage category. Both the model in Eq. (1) and the
modified function described below has the important
assumption that each offspring has one male and one
female parent, thus the fertility functions become (cf.,
Caswell, 2001):
for males:
Fm (n) =
knf
nf + n m
(2)
and for females:
Ff (n) =
knm
nf + n m
(3)
A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
We modify the basic harmonic function by adding
m and f stage classes to each sex, respectively, each
with a stage specific clutch size, ki and kj , respectively.
The resulting structured birth function written in vector
format has the general expression:
B(n) =
2 × km nm × kf nf
km nm + kf nf
(4)
where km , kf are the row vectors of stage specific clutch
sizes and nm and nf are the column vectors of the number of individuals in each stage class for males and
females, respectively. Again from the assumption of
one parent of each sex for each offspring the resulting
fertility functions for each sex and stage class become:
Fmi =
kmi × kf nf
km nm + kf nf
(5)
Ffj =
kfj × km nm
km nm + kf nf
(6)
where the notation is the same as in Eq. (4) except for
i which indicates the specific class of males or females
the fertility function belongs to. In the unstructured
birth function (Eq. (1)), the clutch size is easily biologically interpreted and this clutch size times half the
effective population size equals the reproductive output
from that model. In the structured birth function, k is
stage specific and can be viewed as the mean number
of offspring of both sexes an individual in a given stage
class can have. It can also be viewed as the effective
number of gametes produced by an individual in a given
stage class. Hence, since female and male gametes join
to form zygotes, and subsequent larvae, Eqs. (4)–(6)
must be divided by 2 in order to interpret B(n) as the
effective number of produced larvae.
Theper capita
output from each function, i.e.,
B(n)/( nm + nf ) was used to compare the different models. In this manner, population size will not
be a confounding agent in the comparisons between
the two functions. To test the model with true values, two male and two female classes were used with
the arbitrary possible sizes n = ]0 10[. A few different
combinations of k-values were investigated to reveal
different behaviours of the structured birth function
depending on the relation between male and female
fecundity. In the first case, km = [1 4] and kf = [1 4]
for males and females, respectively. This corresponds
to a uniform development of males and females and
789
where fecundity increases by size. In the second case,
km = [4 7] and kf = [1 1]. This combination of clutch
sizes exemplifies a situation where females are limiting, males have higher fecundity and male fecundity
increases with size. This is qualitatively equivalent to
the biology of P. vulgata, where males produce a larger
number of gametes than females. In the third case, km
and kf were as in case two but population size was
constant while population structure varied. Finally, an
idealized case of the sequential hermaphrodite P. vulgata was simulated, where sex- and size-dependent
fecundity was contrasted with a situation, which was
only sex-dependent.
To fully analyse the structured birth function it was
also subjected to a comparison with a harem model.
To construct the simplest harem model the nf -term in
the denominator of Eq. (1) is divided by a factor, h,
which is the number of females in a harem. In the case
of h > 1, the population is polygamous and for h < 1 the
population is polyandrous (Caswell, 2001).
A matrix population model based on field data for P.
vulgata was used to investigate how the use of different
birth functions may affect the dynamics of a population.
Firstly, the unstructured birth function was used in the
matrix model, implying equal fecundity for both sexes
and individuals of all sizes. Secondly, the structured
birth function was used assuming a reproductive system
with both size- and sex-dependent fecundity.
The matrix population model had four fertile stage
classes, small and large males and small and large
females. Due to the sex change and that nearly all individuals start as males, males are on average smaller
than females. However, the sizes of the different sexes
do overlap since the sex change is gradual through
a cohort spanning 3–6 years (Ballantine, 1961). The
model was parameterised with data for P. vulgata from
two locations in Ireland and two locations on the Isle of
Man (Sundelöf, Jenkins, Delany, Hawkins and Åberg,
unpublished data). Tagged individuals were followed
for 1 year and from these data size specific survival
rates and growth curves of individuals were calculated for each population and further used to calculate
transition probabilities between stage classes. Growth
curves were also used to calculate the size at sex change
(Ballantine, 1961) and clutch sizes. Female clutch sizes
(k-values) were calculated from the number of recruits
in each population and scaled to each size and sex class
by its specific volume. However, male fecundity was
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A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
treated as limiting only in extreme circumstances and
the male clutch sizes (k-values) are set to 108 and 109 ,
respectively, for small and large males (S.J. Hawkins,
personal communication). The matrix model is closed
(only local reproduction and recruitment) and deterministic and these somewhat unrealistic but simplifying properties were chosen to accentuate the effect of
population structure on deterministic dynamics. Consequently, the P. vulgata example should thus be seen
as an application of the structured two-sex model to
real data and not as a good model of the true dynamics
of these P. vulgata populations. The asymptotic population growth rate (λ), sex ratio (proportion male) and
the elasticities of population growth rate to changes
in the projection matrix were calculated for the two
birth functions in each population and were the population variables used to compare the outcome of the
two birth functions. The elasticity is a measure of the
proportional change in λ of a potential proportional
disturbance in the matrix element aij (Caswell, 2001).
To visualize potential differences the elasticity values
were summed over specific matrix regions (1) elasticity of λ when elements of individual growth were
perturbed (G) (2) elasticity of λ when stasis elements
were perturbed (L, loops in the life cycle) (3) elasticity
of λ when fertility elements were perturbed (F). The
summed values were then used in a triangular ordination (for method, see Silvertown et al., 1996).
3. Results
The harmonic birth function given by Eq. (1) has
two dependencies, population density and the sex ratio,
i.e., the proportion of one of the sexes (the p.c. case
is shown in Eq. (10) below). Following our reasoning
of size-dependent fecundity and the resulting differential clutch sizes, the modified harmonic birth function becomes dependent on the population structure,
as well as on the sex ratio and population size (Eq.
(4)). Fig. 1A shows the p.c. birth function for different population structures and sizes, plotted over
the resulting proportion of males. In the birth function for Fig. 1A, k-values are such that larger (older)
individuals have larger fecundity but there is no difference between the sexes. Assuming that increasingly
larger males have increasingly larger k-values and that
females have lower fecundity than males, and do not
experience size-dependent fecundity, and plotting the
p.c. birth function for different structures and sizes,
the proportion of males resulting in the maximum p.c.
reproduction is skewed from 0.5 (Fig. 1B). Also, the
different possible p.c. fertilities for a given proportion
of males is due to the different population structures,
which is shown in Fig. 1C, where the same kx -values
are used but now plotted for a fixed population size
(N = 40). The only characteristic producing variation
in p.c. births for a given proportion of males in Fig. 1C
is the population structure. The resulting variation in
p.c. reproductive output is up to 30% for this particular
mode of reproduction.
3.1. Model comparisons
By ignoring population structure, the unstructured
birth function may either overestimate or underestimate reproductive output compared to a structured birth
function. In Fig. 1D, the p.c. reproduction is plotted for
an unstructured model, the harem model and the structured model. From Fig. 1D, it is clear that for different
proportion males in the population the different models give different predictions of reproductive output.
Setting the structured birth function equal to an unstructured birth function for a given set of clutch sizes solves
for the sex ratio where the unstructured model changes
from overestimating reproduction to underestimating
it, compared to the structured birth function in Eq. (4).
The top boundary of the p.c. graph of the birth function,
Eq. (4), is given by the case when all individuals are in
the largest class for the respective sex, and Eq. (4) thus
reduces to scalars:
1
k m n m kf n f
B(n)
(7)
=
ntot
(km nm + kf nf )(nm + nf )
or
p m km k f n f
1
=
(8)
B(n)
ntot
k m nm + k f nf
where ntot = nm + nf . The unstructured p.c. birth function in our case has the expression:
B(n)
knm nf
1
=
ntot
(nm + nf )2
which becomes:
1
= kpm (1 − pm )
B(n)
ntot
(9)
(10)
A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
791
Fig. 1. (A) The per capita birth function for a set of two male and two female classes with a random population size between 0 and 10 in each
size class. km = kf = [1 4] such that there is a size dependence on fecundity but no difference between sexes. (B) The per capita birth function
for a set of two male and two female classes with a random population size between 0 and 10 in each size class. km = [4 7] and kf = [1 1] such
that there is size- and sex-dependent fecundity. (C) The per capita birth function for a set of two male and two female classes with a random
population size between 0 and 40 in each size class but fixed at a total of N = 40. km = [4 7] and kf = [1 1] such that there is size- and sex-dependent
fecundity. For a given proportion of males the only variation in the model is the population structure. (D) Comparison between a structured and
an unstructured birth function. The solid top curve shows the p.c. births for the unstructured harmonic birth function for a clutch size of 3.25
(k = 3.25). The scatter is the corresponding p.c. births for the structured harmonic birth function, where km = [4 7] and kf = [1 1]. The horizontal
line shows the threshold value of proportion males at which the unstructured model (Eq. (9)) goes from underestimating to overestimating the
structured birth function. Plotted in the middle of the scatter of the structured birth function is the corresponding “harem” model with km = 5.5
and kf = 1.
where pm ≡ nm /(nm + nf ) and (1 − pm ) ≡ nf /(nm + nf ).
The solution of the two birth functions, p∗m , is given
by setting Eq. (8) equal to Eq. (10) and solving for pm
giving two solutions of which p∗m = 1 is trivial but an
intermediate solution is:
p∗m =
kf (k − km )
k(kf − km )
(11)
If kf = km the equation does not have any solutions. This
is rather trivial since the birth functions reduce to being
the same. If, on the other hand, k = km the solutions are
p∗m = 0 and 1. Also, if k = kf then p∗m = 1. For any
other combination, there will be a solution between 0
and 1. There are several ways of calculating the threshold pm -value. It is important to sort out each sex’s
contribution of gametes or clutch sizes before comparing the two birth functions since the p∗m -value will
depend strongly on the k, km and kf . Eq. (11) states
which part of the parameter space either of the functions will predict the larger value. The relation between
p∗m and the effect on the dynamics is not straight forward since it will depend on several other variables (the
skewness of the p.c. B-function, the variation around
the mean and also how the sex ratio is changed deterministically in the populations). However, in general
the effect on population dynamics will be greater the
further p∗m is from 0.5. In Fig. 1D, k = 3.25, km = 7
and kf = 1, which means that the k in Eq. (10) is
the mean clutch size of the whole population. Thus,
k = (kf + km )/4.
A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
792
Table 1
Deterministic growth rate (λ), proportion male (pm ) and calculated threshold proportion of males (p∗m ) for four different populations of P. vulgata
calculated using matrix population models with either an unstructured or a structured birth function
Bullens Bay
Garrettstown
Port Saint Mary
Derbyhaven
λ unstructured
λ structured
pm unstructured
pm structured
p∗m
1.318
1.868
0.786
0.9
1.199
1.452
0.870
0.983
0.599
0.750
0.093
6.1E-05
0.535
0.663
0.220
0.188
0.3012
0.2599
0.0016
0.1041
For details see text. Observed mean proportion of males were in all populations around 0.5–0.6.
3.2. P. vulgata case study
The population structures of P. vulgata found in
the field samplings were all male dominated which is
expected for a long-lived sequentially hermaphroditic
organism. The estimated mortalities showed a dependence on size of the individual, where larger individuals
had lower mortality. Growth rates of the populations in
Garrettstown and Bullens Bay were more sensitive to
changes in fertility and stasis when the structured birth
function was used but for the populations in Port Saint
Mary and Derbyhaven the response was the opposite
(Fig. 2). For both birth functions, the population growth
rate was larger than 1 in the two Irish populations but
smaller than 1 in the two populations in the Isle of Man
(Table 1). In all populations, the growth rate was closer
to unity when the structured birth function was used. In
all populations, a similar trend was seen for the proportion of males where it was closer to a 1:1 sex ratio in the
case of the structured birth function (Table 1). Moreover, the calculated p∗m -values (Table 1) were smaller
than the equilibrium proportion of males in the simulated populations. The only exception is in the case of
the Derbyhaven population simulated with the unstructured birth function. The extremely small p∗m -value for
PSM is an effect of the low recruitment rate, which is
the only determinant of the unstructured birth function.
The unstructured birth function will thus underestimate
fertility drastically for all proportions of male down to
0.0016, compared to the structured birth function, and
hence lead to a lower growth rate of this population.
The p∗m for Derbyhaven falls between the equilibrium
proportion of males for the unstructured and structured
birth functions simulations.
4. Discussion
Fig. 2. Elasticity of population growth rate summed over matrix
regions of individual growth (G) stasis (L) and fertility (F). Two
symbols represent each population, a filled symbol for the results
with the structured birth function and an open symbol for the results
with the unstructured birth function. (, ) Bullens Bay, (䊉, )
Garrettstown, (♦, ) Port Saint Mary, (, ) Derbyhaven. Note that
the contribution of fertility to lambda in the Derbyhaven case with
the unstructured birth function is zero. The equilibrium population
structure of this population lacks males, and is totally dominated by
survival of large females, hence the lack of contribution from reproduction.
Surprisingly large changes to the deterministic
dynamic characteristics of a closed population were
found by replacing an unstructured birth function with
a more realistic structured birth function. Although
we have not fitted the structured model to time series
data, the more realistic structured reproductive model
demonstrates that the choice of birth functions has
large effects on the intrinsic dynamics. The stable
skewed sex ratios in the studied limpet populations
are a result of the demographic rates and the fertility
A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
function resulting in a stable size structure. The structured birth function produced population growth rates
and sex ratios, which were closer to unity than those
produced by the unstructured model. Observed mean
proportion of males are for all populations approximately 0.5–0.6. The results from the simulations of the
different birth functions show that the structured birth
function performs better than the unstructured one. The
fact that proportions of males in the field are strictly
larger than any of the calculated p∗m -values again suggests that the structured birth function is to prefer
over the unstructured birth function for this species.
Similarly, for every pair being compared (structured
versus unstructured model), the elasticity of population growth rate to changes in stasis was higher in the
population with the lower rate of population growth.
The two populations in Isle of Man were dominated
by large females and these populations expressed low
growth rates which were sensitive to changes in stasis.
As observed by Silvertown et al. (1996) to manage a
rapidly declining trend in such a population the elasticity analysis alone is misleading since the need for
recruitment is most likely greater than an increase in
stasis. For a sequential hermaphrodite, this is even more
crucial since new recruits will be of the opposite sex,
thus constituting a prerequisite for reproduction. However, the main conclusion from the elasticity analysis is
that the elasticity can change substantially depending
on which birth function is used. Since elasticity analyses often are used as an aid in management of species
the choice of birth function could thus have large effects
on management plans.
We show that the use of a structured birth function
strongly influences the dynamics of populations and
the elasticity of population characters. Current literature suggests that the common practice today to calculate production from populations in two-sex models
is either dominance models or unstructured birth functions (Lindström and Kokko, 1998; Ranta and Kaitala,
1999; Ranta et al., 1999; Armsworth, 2001; Engel et al.,
2001). The current work highlights the need for knowing whether an organism expresses size-dependent
fecundity or not when using a two-sex model. A harem
situation, for example, is mimicked in the structured
birth function, although in a more general manner. In
Fig. 1D, the line within the scatter is the p.c. births from
the unstructured model, modified for the harem situation. Although the harem function could be viewed
793
as a good approximation for reproduction of a system like this, none of the variation in reproduction due
to potential size-dependent fecundity in individuals is
accounted for. This variation may be great and is dependent on the difference in clutch sizes between sizes and
sexes. The greater the difference in clutch sizes, the
greater the variation in calculated reproductive output
depending on the population structure. If, for example, the clutch sizes differ as in the example in Fig. 1D
(km = [4 7] and kf = [1 1]) the harem function will perform fairly well at high proportions of male. However,
if proportion of males is low the harem function will
be decent at best. Again, if the clutch sizes differ even
more (for example, km = [2 9] and kf = [1 1]) the situation is even worse. We suggest a change of focus on
the issue of production where size-dependent fecundity
should be central. A structured birth function should be
applied whenever a pronounced size-dependent fecundity is observed and the reproductive output of the
population is of particular interest.
Several studies have recently investigated the population dynamic properties of monogamous two-sex
population models (Martcheva, 1999) and differences
between monogamous and polygamous two-sex population models (Lindström and Kokko, 1998; Ranta and
Kaitala, 1999; Ranta et al., 1999). Ranta and Kaitala
(1999) conclude from a model using harmonic birth
functions that polygamous and monogamous individuals can co-exist only if the polygamous individuals are less fecund or have lower survival than the
monogamous individuals. The framework they use is an
unstructured harmonic birth function where the clutch
size of females is scaled for polygamy. Other studies confirm the differences in population dynamics
between polygamous and monogamous models (e.g.,
Legendre et al., 1999). However, they all use unstructured birth functions, although many of the studies
would be applicable to structured populations with size,
age or sex specific life-history characters that potentially could change the outcome of the simulated models. In the structured harmonic birth function polygamy
and monogamy turn up as special cases.
Populations of organisms that express sizedependent fecundity have earlier been modelled mainly
with one-sex models or unstructured two-sex models
(Åberg, 1992; Lindström and Kokko, 1998; Ranta and
Kaitala, 1999; Ranta et al., 1999; Engel et al., 2001).
However, the fertility of individuals in populations of
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A. Sundelöf, P. Åberg / Ecological Modelling 193 (2006) 787–795
organisms that express size-dependent fecundity (many
aquatic and terrestrial invertebrates, aquatic algae and
terrestrial plants) will evidently depend on population
structure. Size-dependent fertility is often included in
one-sex matrix models but under the assumption that
one sex is limiting (e.g., Åberg, 1992). Reproductive
output becomes increasingly more dependent on population structure the larger the difference in fecundity
between sexes and sizes of individuals. Thus, the need
for a structured birth function is greater for models of
species expressing great sexual dimorphism and sizedependent fecundity.
Studies of stage structured populations of species
with complex life cycles, where fecundity is typically size-dependent (Engel et al., 2001), could gain
from using a stage structured birth function. For many
species it may, however, be difficult to get the data
needed for a detailed demographic population model,
but still one should be aware of the possibility of either
under- or overestimating the total reproduction in a
population when using an unstructured birth function
compared to a structured one. The importance of modelling the correct birth function for a population may
have important implications for conservation biology
and formulation of population viability analysis (PVA)
as stated by studies concerning mating systems other
than stage structured systems (Legendre et al., 1999).
The calculation of p∗m states at which point the unstructured birth function predicts lower number of p.c. births
compared to the structured one. The calculation of p∗m
also determines how skewed the birth function is without doing any of the numerical simulations. Thus, it
may also help in evaluating past and present PVA’s.
The model presented here is parameterised for P.
vulgata and becomes specific to this particular lifehistory but it can easily be adapted to fit different life
histories by changing primary sex ratio or size specific fecundity. It was not necessary to include the
primary sex ratio in the present model since all recruits
of P. vulgata are born neuter, turn male at about 1
year of age and at about three years of age individuals turn female (Ballantine, 1961). However, for other
organisms where the sex determining system is known
the primary sex ratios can typically be modelled by
the random meeting of X and Y reproductive cells in
zygote formation with the addition of a recruit mortality term. In earlier models of reproduction, the primary
sex ratio has been assumed to be 1:1 or close to it (e.g.,
from observed primary sex ratios in human populations (Pollak, 1986; Martcheva, 1999). However, the
modelling of primary sex ratios different from 1:1 can
easily be incorporated as a constant, or function, into
Eq. (7) and has no implications on the generality of the
function described here.
Acknowledgements
We wish to thank Jane Delaney and Stuart Jenkins for providing us the field data. Steve Hawkins is
thanked for general discussions about P. vulgata biology and Per Jonsson and two anonymous reviewers for
valuable comments on the manuscript. This work has
been a part of the European project DELOS (EVK3CT-2000-00041). The work was further funded by
Kapten Carl Stenholms donations fund.
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