Content uploaded by Tomas de Camino
Author content
All content in this area was uploaded by Tomas de Camino on Oct 27, 2014
Content may be subject to copyright.
vol. 172, no. 1 the american naturalist july 2008
On Net Reproductive Rate and the Timing of
Reproductive Output
T. de-Camino-Beck
1,
*
and M. A. Lewis
1,2,†
1. Department of Mathematical and Statistical Sciences, University
of Alberta, Edmonton, Alberta T6G 2G1, Canada;
2. Department of Biological Sciences, University of Alberta,
Edmonton, Alberta T6G 2G1, Canada
Submitted January 30, 2007; Accepted February 5, 2008;
Electronically published May 28, 2008
Online enhancement: appendix.
abstract: Understanding the relationship between life-history pat-
terns and population growth is central to demographic studies. Here
we derive a new method for calculating the timing of reproductive
output, from which the generation time and its variance can also be
calculated. The method is based on the explicit computation of the
net reproductive rate ( ) using a new graphical approach. Using
R
0
nodding thistle, desert tortoise, creeping aven, and cat’s ear as ex-
amples, we show how and the timing of reproduction is calculated
R
0
and interpreted, even in cases with complex life cycles. We show that
the explicit formula allows us to explore the effect of all repro-
R
0
ductive pathways in the life cycle, something that cannot be done
with traditional analysis of the population growth rate (l). Addi-
tionally, we compare a recently published method for determining
population persistence conditions with the condition and
R
1 1
0
show how the latter is simpler and more easily interpreted biolog-
ically. Using our calculation of the timing of reproductive output,
we illustrate how this demographic measure can be used to under-
stand the effects of life-history traits on population growth and
control.
Keywords: matrix model, net reproductive rate, biological control,
generation time, reproductive output, persistence.
Persistence of biological populations is the result of the
evolution of life-history patterns that have consequences
for population growth. Early theoretical work by Cole
* E-mail: tomasd@math.ualberta.ca.
†
E-mail: mlewis@math.ualberta.ca.
Am. Nat. 2008. Vol. 172, pp. 128–139. 䉷 2008 by The University of Chicago.
0003-0147/2008/17201-42375$15.00. All rights reserved.
DOI: 10.1086/588060
(1954) showed how changes in reproductive schedules can
have consequences in population growth rate. Cole’s work
established the foundations of demographic analysis—
based on the estimation of parameters such as growth rate,
net reproductive rate, and length of a generation—and
provided the foundation for theoretical and empirical
work on population demography (e.g., Cole 1960; Mur-
doch 1966; Charnov and Schaffer 1973; Goodman 1975).
Since Cole’s work, substantial theoretical advances have
been achieved, in particular with the use of matrix models
popularized and summarized in a book by Caswell (2001).
Matrix models provide an intuitive modeling strategy
where the life cycle of the organisms can be described ex-
plicitly in terms of life ages or stages. Demographic param-
eters of population growth rate (l), net reproductive rate
(the lifetime reproductive output of an individual; ), and
R
0
generation time (the time it takes for the population to
increase by ) have also been derived for matrix models
R
0
(Caswell 2001), providing a direct connection between life
cycle structure and these demographic parameters.
In general, most of the analysis in matrix models has
been based on the numerical calculation of l and the
sensitivity of l to model parameters. In particular, elas-
ticity or sensitivity analysis of l to age/stage transitions in
matrix models has become a standard method in demo-
graphic analysis in ecology, management, and evolution
(Silvertown et al. 1993; Franco and Silvertown 1996; Pfister
1998; Oli and Dobson 2003, 2005; Gaillard et al. 2005;
Shea 2004). In all but the simplest matrix models, cal-
culation of the population growth rate and elasticity re-
quires numerical estimates, thus restricting results to the
particular population parameter estimates.
Although total reproductive output and timing of re-
production have an impact on population growth and
persistence, parameters like and generation time have
R
0
not been well studied using matrix models. The calculation
of is thought to be too complicated to perform ana-
R
0
lytically (Hastings and Botsford 2006b), and its numerical
calculation for stage-structured models (Cushing and
Zhou 1994) has rarely been made, in favor of the calcu-
lation of l and elasticity/sensitivity analysis of l. The dif-
Reproductive Rate and Reproductive Output 129
Figure 1: Simple matrix and its associated graph. There is a2 # 2 A
directed edge in the graph for every entry in the matrix. In the graph,a
ij
for a transition , the edge is directed from node j to node i.a
ij
ficult calculation of generation time in stage-structured
models (Cochran and Ellner 1992; Lebreton 2005) had
dissuaded demographers from using as a measure ofR
0
the impacts of reproduction and survival on population
growth and persistence, primarily because the interpre-
tation of without an estimate of generation time canR
0
be misleading (Birch 1948; Cole 1960; Caswell 2001). Here
we show that both and mean generation time can beR
0
directly calculated for stage-structured models.
In a previous article (de-Camino-Beck and Lewis 2007),
we developed a graphical method for calculating ex-R
0
plicitly, and we showed how it can be applied in the context
of biological control. In this article, we derive a formula
for calculating the timing of reproductive output based on
the equation. This formula allows for the calculationR
0
of the timing of reproductive output, the mean generation
time, and the mean generation time variance for stage-
structured models. We start by giving the theoretical back-
ground in matrix models and life cycle graphs. Then, we
briefly describe the graph-theoretic method to calculate
and the timing of reproductive output. Later, we showR
0
with four examples how and timing of reproductiveR
0
output are calculated, and we discuss the implications for
demographic analysis. Additionally, we compare the con-
dition with a persistence condition for matrix mod-R
1 1
0
els derived by DeAngelis et al. (1986) and recently re-
derived by Hastings and Botsford (2006b), and we discuss
its implication in the management and control of
organisms.
Net Reproductive Rate, Generation Time,
and Persistence
In this section, we briefly introduce matrix models. First,
we show how to calculate the net reproductive rate using
a graphical approach. Next, we show that the concept of
the net reproductive rate can be extended to determine
not only the total number of offspring produced but also
their distribution over the life span of the parent. Then,
we describe the persistence condition proposed by De-
Angelis et al. (1986) and Hastings and Botsford (2006b),
which we will call the Hastings-Botsford persistence con-
dition, and compare this with .R 1 1
0
Matrix Models
Stage-structured models are population dynamics models
where life cycle stages are explicitly defined. Stages can be
size classes or life forms (e.g., larvae, juvenile, seeds, seed-
ling, adult plants). In matrix form, a stage-structured
model is defined as
n p An ,(1)
t⫹1 t
where is a vector of stages at time t and is annAn # n
t
projection matrix. Each entry, , in the matrix representsa A
ij
the contribution from stage j in time t to stage i in time
. The projection matrix can also be represented ast ⫹ 1 A
a life cycle graph where each node in the graph corresponds
to a stage and each arrow represents transitions, (fig. 1).a
ij
The matrix is composed of survivorship and fecundityA
transitions. These can be decomposed into a transition
matrix and a fecundity matrix . Transition matrix en-TF
tries, , describe survivorship, the probability of survivalt
ij
from stage j to i. The fecundity matrix entries, , describef
ij
fecundities, the maximum reproductive output from stage
j to i. This decomposition is not mathematically unique
but is uniquely determined by the biology of the organism.
Net Reproductive Rate R
0
Given an initial distribution of stages , the number ofn
0
offspring produced by the individuals initially present is
22
……
Fn ⫹ FTn ⫹ FT n ⫹ p F(I ⫹ T ⫹ T ⫹ )n
00 0 0
⫺1
p F(I ⫺ T) n .
0
(2)
The first term on the left-hand side of the first line of the
equation represents first-year fecundity, the second term
represents fecundity following a year of survival, and the
third term represents fecundity following two years of sur-
vival and so forth. The matrix is referred to as
⫺1
F(I ⫺ T)
the next-generation matrix (Cushing and Zhou 1994; Li
and Schneider 2002).
The net reproductive rate, , is the average number ofR
0
offspring that a single reproducing individual can produce
130 The American Naturalist
Figure 2: A shows the transition matrix and its decomposition in sur-
vivorship and fecundities. In B, fecundities are multiplied by . Then,
⫺1
R
0
using rule A in figure 3, self-loops (gray) are eliminated, yielding the
graph in C. Node 1 is eliminated by multiplying the two edges in gray
(rule in fig. 3C to produce the graph in D). The resulting loops are added
together (rule in fig. 3B to obtain E). Applying step 5 in E to the single
node graph yields .R
0
Figure 3: Graph reduction rules. A, Self-loop elimination with . B,b ! 1
Parallel path elimination. C, Node elimination. Rules A and B show
elimination of paths, and rule C shows the elimination of node 2. Graph
reduction is done by repeatedly applying these rules until only one node
is left.
over its lifetime and is mathematically defined as the dom-
inant eigenvalue of the next-generation matrix
⫺1
R p r[F(I ⫺ T)], (3)
0
where denotes the dominant eigenvalue. Anr(7) R 1 1
0
implies , and implies (Li and Schneiderl 1 1 R ! 1 l ! 1
0
2002). Thus, conditions on model parameters that satisfy
also guarantee that and vice versa. However,R
1 1 l 1 1
0
the calculation of using formula (3) is not always al-R
0
gebraically straightforward. A recently discovered method
provides a simple alternative (de-Camino-Beck and Lewis
2007). This uses graph reduction methods on the life cycle
graph to calculate . The calculation starts with the de-R
0
scription of the projection matrix as a life cycle graph,
similar to the one shown in figure 2A. Once the life cycle
graph has been specified, the procedure is as follows. (1)
Identify survivorship and fecundity transitions. (2) Mul-
tiply all fecundity transitions in the graph by . (3)
⫺1
fR
ij 0
Eliminate survivorship self-loops, using rule A in figure 3.
(4) Reduce the graph using the graph reduction rules de-
fined in figure 3 until only nodes with fecundity self-loops
are left. When a node is eliminated, all pathways that go
through that node have to be recalculated. (5) When only
one node with a single self-loop is left, eliminate the final
node by setting the self-loop equal to 1 and solve this
equation for .R
0
Note that the resulting formula is the same regardlessR
0
of the order by which nodes are eliminated. As an example,
consider the hypothetical graph in figure 2B. After apply-
ing the graph reduction method as shown in figure 2B–
2E, the resulting isR
0
f
22
R p
0
(1 ⫺ t )
\
22
reproduction from adult to adult
tf
21 12
⫹ .(4)
(1 ⫺ t )(1 ⫺ t )
\
11 22
reproduction going through seed bank
The population persists if . The formula showsR 1 1 R
00
two reproductive pathways. If either is 11, then the pop-
ulation persists. The first pathway accounts for the repro-
duction of plants that start as adults and produce repro-
ducing adults immediately the next year. The second
pathway represents seeds that go to the seed bank that
take at least an extra year to become adults again. The
terms are interpreted as the average time an
⫺1
(1 ⫺ t )
ii
individual stays in stage i.
Typically, the net reproductive rate formula can be writ-
Reproductive Rate and Reproductive Output 131
ten explicitly as the sum of fecundities for m pathways as
follows:
…
R p R ⫹ R ⫹⫹R ,(5)
012 m
where is the fecundity for a pathway that ends in re-R
i
production. We will call each a fecundity pathway. Be-R
i
cause the pathways are summed, the population will persist
if any . From the previous example, the first pathwayR
1 1
i
is , and the second isR p f /(1 ⫺ t ) R p tf/[(1 ⫺
122 22 2 2112
. If , , or the sum is 11, then thet )(1 ⫺ t )] RR R⫹ R
11 22 1 2 1 2
population will persist.
This graph reduction method for calculating (de-R
0
Camino-Beck and Lewis 2007) relates to a different es-
tablished method for calculating the eigenvalues l of the
projection matrix . There, each entry of the life cycleA a
ij
graph is multiplied by , and the graph reduction steps
⫺1
l
of figure 3 are used to yield a characteristic polynomial
for the eigenvalues l (Caswell 2001). The difference be-
tween the methods (multiplying by [step 2] rather
⫺1
fR
ij 0
than multiplying each by [above]) is suf-
⫺1
a p t ⫹ f l
ij ij ij
ficient to yield the reproductive rate rather than l.InR
0
other words, is the dominant eigenvalue of the next-R
0
generation matrix , while l is the dominant
⫺1
F(I ⫺ T)
eigenvalue of the projection matrix (Caswell 2001).T ⫹ F
The population growth rate, l, can be calculated with the
graph method as well. However, the procedure yields a
higher-degree polynomial where the interpretation is not
intuitive. For example, the characteristic polynomial for
the eigenvalues associated with the life cycle graph in figure
2is
2
l ⫺ (t ⫹ t ⫹ f )l ⫹ [t (t ⫹ f ) ⫺ tf] p 0, (6)
11 22 22 11 22 22 21 12
which has dominant eigenvalue
1
2
冑
l p t ⫹ t ⫹ f ⫹ (t ⫹ t ⫹ f ) ⫺ 4[t (t ⫹ f ) ⫺ tf].
11 22 22 11 22 22 11 22 22 21 12
{}
2
(7)
While this eigenvalue gives the population persistence
condition as , there is no intuitive interpretation ofl
1 1
the condition, unlike the fecundity pathways used for
. When the number of stages in the life cycle is threeR
0
or more, it is difficult or impossible to calculate the dom-
inant eigenvalue explicitly, and numerical methods must
instead be used.
Timing of Reproductive Output
Since the work by Cole (1954), timing of reproductive
output and its implications for fitness have been the sub-
ject of widespread study (Beckerman et al. 2002; Ranta et
al. 2002; Oli and Dobson 2003; Coulson et al. 2006). In
this section, we show our calculation of how can beR
0
extended to yield the total number of offspring produced
as well as their distribution over the life span of the parent.
To explore timing of reproduction in a stage-structured
model using , the time that it takes to go through eachR
0
pathway has to be included in the formula. Rather thanR
0
grouping terms according to fecundity pathway terms,
equation (5), in the net reproductive rate, formula can be
grouped according to the number of time steps taken to
complete a pathway. For example, equation (4) can be
rewritten as
2
…
R p f (1 ⫹ t ⫹ t ⫹ )
022 2222
22
……
⫹ tf(1 ⫹ t ⫹ t ⫹ )(1 ⫹ t ⫹ t ⫹ )(8)
21 12 11 11 22 22
p f
22
\
onetimesteptoreproduction
⫹ (ft ⫹ tf)(9)
22 22 21 12
\
two time steps to reproduction
2
…
⫹ (ft ⫹ tft⫹ tft) ⫹
22 22 21 12 11 21 12 22
\
threetimestepstoreproduction
…
p S ⫹ S ⫹ S ⫹ .(10)
123
If the population is growing ( ), offspring producedl 1 1
after several time steps will contribute less to population
growth than those produced immediately. This motivates
us to connect to l by using the current value of futureR
0
reproduction. The current value of future reproductive
output is assigned on the basis of the contribution that
the future reproductive output will make to the overall
population growth. Formally, we define and start
⫺1
t p l
by multiplying each pathway that takes n time steps to
complete by the weight . From the example above, path-
n
t
way takes two time steps to completeS p (ft ⫹ tf)
222222112
( ); thus, the pathways with its associated weights aren p 2
. Thus, means , and
22
S t p (ft ⫹ tf)tl1 1 t ! 1
222222112
the reproductive output two steps hence is diminished by
the factor . Summing across all possible values of n yields
2
t
the present value of all future reproduction
⬁
n
S(t) p S t .(11)
冘
n
np1
By analogy with generating functions in probability the-
ory, we also refer to this as the -generating function.R
0
Equation (11) can also be found by taking the z transform
of the life cycle graph with . For details of applying
⫺1
z p t
132 The American Naturalist
a z transform, see Caswell (2001, chap. 7). For life cycle
graphs, a more compact representation of equation (11)
arises from simply multiplying each survivorship and fe-
cundity term by t. For example, applying this method to
equation (4) yields
2
f t tft
22 21 12
S(t) p ⫹ . (12)
(1 ⫺ t t)(1⫺ t t)(1 ⫺ t t)
22 11 22
Because equation (11) is a Taylor series representation
for the function , the number of direct descendants inS(t)
n time steps, , can be recovered as the Taylor seriesS
n
coefficients from the nth derivative of :S(t)
n
1 dS
S p (0). (13)
n
n
n! dt
The net reproductive rate can be recovered from byS(t)
recognizing that it is the total number of new individuals
produced that are direct descendants of the original in-
dividual, and hence
⬁
R p S(1) p 冘 S .
0 n
np1
Generation time can be interpreted as the length of time
it takes the population to increase by a factor of . InR
0
stage-structured models, the calculation of generation time
is difficult since stage is not related to age and individuals
can stay in a stage for a long period of time (Cochran and
Ellner 1992; Lebreton 2005). There are several approximate
methods for estimating generation time (Caswell 2001).
The simplest is given by
log R
0
˜
T p .(14)
log l
The mean generation time can be understood exactly
by considering a randomly selected direct descendant of
the original reproducing individual. The year in which the
descendant is produced is a random variable T with mean
and variance :
2
mj
TT
S (1)
m p ,
T
R
0
2
S (1) S (1) S (1)
2
j p ⫹⫺ .(15)
T
2
RRR
00 0
See the appendix in the online edition of the American
Naturalist (“Generation Time Derivation”) for a detailed
derivation. The mean generation time , equation (15),m
T
is an alternative to the previous approximation (eq.
˜
T
[14]). The relationship with equation (14) can be found
by writing the population at time t in terms of indi-N
t
viduals from previous time steps using the renewal equa-
tion,
⬁
N p SN ,(16)
冘
tnt⫺n
np1
with given and for . A population growingNNp 0 n ! 0
0 n
at rate l takes the form . Substitution of this form
t
N p cl
t
into equation (16) yields the Euler-Lotka equation for the
population growth rate,
⬁
⫺n ⫺1
1 p S l p S(l ). (17)
冘
n
np1
Hence, the population growth rate is given by
. Consider now an entire population repro-
⫺1
S(l ) p 1
ducing at time . Then,
˜
T
˜
0, n ( T
S p .(18)
n
˜
{
R , n p T
0
The Euler-Lotka equation gives , and therefore
˜
⫺T
1 p R l
0
equation (14) follows. In other words, is an approxi-
˜
T
mation to the true generation time, , which is foundm
T
under the assumption that there is a single, episodic re-
productive event at time rather than continual repro-
˜
T
duction. Equation (15) not only gives an exact generation
time, , but also yields the variability in the generationm
T
time, .
2
j
T
Caswell (1989), in the first edition of his book, applies
a similar approach, using the z transform of a graph with
no disjoint loops to obtain . However, Caswell usesR
0
equation (14) to calculate generation time. We go further
by deriving the -generating function from which theR
0
timing of reproductive output and formulas for the mean
generation time and the generation time variance can be
derived.
Hastings-Botsford Persistence Conditions and R
0
As shown in “Net Reproductive Rate ,” the formula forR
0
provides a threshold condition for population growthR
0
( ), which is easier to calculate than the populationR 1 1
0
growth rate threshold . Also, it can be understoodl 1 1
biologically as a parent’s total contribution to the next
generation arising from each of the possible fecundity
pathways (see eq. [5]). It is natural to ask whether there
are any other simple threshold conditions for population
growth other than and . Indeed, there is aR 1 1 l 1 1
0
condition, first derived by DeAngelis et al. (1979, 1986)
and rederived and applied by Hastings and Botsford
(2006a, 2006b). However, these Hastings-Botsford con-
ditions have two limitations that do not arise in the anal-
ysis of . First, for a given model, there may be manyR
0
Reproductive Rate and Reproductive Output 133
conditions to check as opposed to a single one ( ).R 1 1
0
Second, the conditions cannot be interpreted biologically
in a manner similar to the condition .R
1 1
0
The Hastings-Botsford conditions are strict mathemat-
ical conditions for population persistence. For an n # n
matrix , the Hastings and Botsford (2006a, 2006b) per-A
sistence condition states that a population will persist if
and only if or if any principal minor,a
1 1
ii
m⫹1
(⫺1) det (J) 1 0, (19)
where is an m-dimensional principal submatrix ofJ
including . The principal minors of areQ p A ⫺ IQ Q
the determinants of the submatrices obtained by deleting
the same rows and columns in .n ⫺ m Q
In other words, when the population persists ( ),R
1 1
0
at least one of the self-loops exceeds 1 or one of thea
ii
principal minors of is positive (eq. [19]). When theA ⫺ I
population does not persist ( ), all self-loops are
!1,R ! 1
0
and all the principal minors are negative. While the
Hastings-Botsford persistence condition is mathematically
analogous to , their analytical formula comes at aR
1 1
0
cost. Unlike , the condition has limited biologicalR 1 1
0
interpretation.
Consider the example from figure 2. On the basis of
the Hastings-Botsford conditions, the population will per-
sist if or or ifaa
1 1
11 22
33
(⫺1) det (Q) p (⫺1) det (A ⫺ I)
3
p (⫺1) [(a ⫺ 1) (a ⫺ 1) ⫺ aa] 1 0,
11 22 21 12
in other words, if
aa
21 12
1 1. (20)
(1 ⫺ a )(1 ⫺ a )
11 22
To compare this condition with , we rewrite theR 1 1
0
Hastings-Botsford condition in equation (20) using the
survivorship and fecundity decomposition shown in figure
2A. The population will persist if , or, givent ⫹ f
1 1
22 22
,ift ⫹ f ! 1
22 22
tf
21 12
1 1. (21)
(1 ⫺ t )[1 ⫺ (t ⫹ f )]
11 22 22
Now, consider calculated for the same example inR
0
equation (4). The population will persist if
ftf
22 21 12
R p ⫹ 1 1. (22)
0
1 ⫺ t (1 ⫺ t )(1 ⫺ t )
22 11 22
These two conditions are equivalent (i.e., they hold or
fail under identical conditions on the parameters), even
though they are expressed differently. This can be shown
by multiplying both sides of equation (21) by [1 ⫺
. Equation (22) is the sum of fecundity(t ⫹ f )]/(1 ⫺ t )
22 22 22
pathways that yield the lifetime reproductive output. There
is no similar interpretation for equation (21).
In this simple example, the condition on the sign of
principal minors (eq. [19]) reduces to a single condition.
However, as indicated by Hastings and Botsford (2006b),
for large matrices, all the principal minors need to be
calculated. In many cases, and in all examples given by
Hastings and Botsford (2006b), it is sufficient to check
condition (19) for the largest submatrix (i.e., ) inJ p Q
equation (19). However, it is possible to come up with
counterexamples where this condition is not sufficient and
all principal minors must be checked (see “Creeping
Aven”).
Examples
To further illustrate the calculation of , the timing ofR
0
reproductive output, and the mean generation time ,m
T
we reconsider Hastings and Botsford (2006b) examples of
the persistence of nodding thistle (“Nodding Thistle”) and
the desert tortoise (“Desert Tortoise”). We also include
examples that demonstrate more complicated life cycles
and show how the calculation of is straightforward andR
0
simple. For creeping aven (“Creeping Aven”), the full set
of Hastings-Botsford inequalities (26 in all) must be
checked, whereas the condition is easy to calculateR
1 1
0
as a single inequality. An example of invading cat’s ear is
presented in the appendix (“Cat’s Ear, Jury Test, and ”).R
0
For nodding thistle, desert tortoise, and creeping aven, we
also calculate the timing of reproductive output and the
mean and variance of the generation time, on the basis of
parameter estimates from the literature.
Each of the examples considered in this section has also
been previously analyzed by other authors using eigenvalue
analysis (Doak et al. 1994; Shea and Kelly 1998; De Kroon
et al. 2000; Weppler et al. 2006). In these analyses, the
population growth rate l was numerically calculated, and
conclusions were drawn about the impact of model pa-
rameters on l (elasticity analysis).
Nodding Thistle
Population Persistence. The nodding thistle (Carduus
nutans) matrix model (Shea and Kelly 1998) provides a
good example of the calculation and analysis of , timingR
0
of reproduction, and the calculation of generation time.
This thistle causes damage to grazing lands in New Zealand
and Australia, and the life cycle graph is shown in figure
4A. Using graph reduction, we obtain
134 The American Naturalist
Figure 4: Life cycle graphs of nodding thistle (A), desert tortoise (B), and creeping aven (C). Fecundity transitions are multiplied by to
⫺1
fR
ij 0
proceed with the graph reduction. Survivorship transitions are labeled . The life cycle of Geum reptans (C) is as described by Weppler et al. (2006).t
ij
Node 1 p seedling, node 2 p juvenile, node 3 p small adult, node 4 p medium adults, and node 5 p large adults. Clonal reproduction fecundities
are f
33
, f
34
, and f
35
. Sexual reproduction is f
13
, f
14
, and f
15
.
large plant reproduction
=
R p f ⫹ tf⫹ tf⫹ ttf
0 22 3223 4224 324324
\
pathways without the seed bank
large plant reproduction
=
tf⫹ tt f ⫹ ttf ⫹ ttt f
21 12 21 32 13 21 42 14 32 43 21 14
⫹ .
1 ⫺ t
11
(23)
\
pathways that visit the seed bank
From equation (23), it is clear that the population will
persist if any of the terms or the sum of any term in R
0
is 11. There are two stages, seed bank and small plants
(nodes 1 and 2), that are common in all pathways and
that could be used for control. Reduction in these tran-
sitions would maximize the number of pathways affected
by control. The fecundities from node 4 (large plants)
f
i4
to other plants appear in four of the eight fecundity path-
ways, but occurs in only two of them. This would suggest
f
i3
that large plants should also be targets of control. In con-
trast to the results of Shea and Kelly (1998), these obser-
vations are qualitative and do not require numerical es-
timates. These results are consistent with previous work
(Jongejans et al. 2006). Previous studies used elasticity
analysis from parameter estimates for populations in Aus-
tralia, New Zealand, and its native range in Eurasia to
determine strategies for control. However, inspection of
the formula reveals that thistle has many pathways thatR
0
Reproductive Rate and Reproductive Output 135
Figure 5: Number of new individuals obtained after n time iterations,
starting with one reproductive individual. Iterations are calculated using
equation (13). A, Nodding thistle data taken from Shea and Kelly (1998).
B, Desert tortoise data taken from Doak et al. (1994). C, Creeping aven
data taken from Weppler et al. (2006).
lead to reproduction. Higher elasticity of matrix transitions
indicates higher impact on l, but this does not ensure that
a reduction in the transition will be sufficient to ensure
. On the other hand, if total eradication is desired,l
! 1
then each pathway or sum of pathways in equation (23)
has to be
!1. See the appendix (“Hastings-Botsford Per-
sistence Conditions for Nodding Thistle”) for comparison
of the condition with the Hastings-Botsford con-R
1 1
0
dition for nodding thistle.
Timing of Reproduction. Using parameter estimates from
Shea and Kelly (1998; their table 2), the number of new
individuals over time (fig. 5A), starting with one repro-
ducing individual, is calculated using equation (13). As
seen in figure 5A, this thistle has short generation times
with low variance, and most of the reproductive output
occurs in the first 2 years of the life cycle. It can also be
seen that for this particular set of parameter estimates,
reproduction seems to become negligible after the fourth
year of introduction of the original individual. For an
effective control strategy, it would be reasonable to ensure
the application of control measures for at least the duration
of the timing of reproduction (approximately 4 years).
We also estimated generation time and variance using
equation (15). For Midland, , ,R p 4.1116 m p 1.9789
0 T
and , and for Argyll, ,
2
j p 0.5375 R p 1.8113 m p
T 0 T
, and . There are large differences in
2
1.6909 j p 0.3429
T
but not in timing of reproductive output. This suggestsR
0
that environmental differences change the overall output
but do not change reproductive strategies. For comparison,
the generation time calculated with equation (14) for
˜
T
both sites is 1.779 and 1.630, respectively.
Desert Tortoise
Population Persistence. Using graph reduction, we calcu-
lated for the desert tortoise (Gopherus agassizii) matrixR
0
model (Doak et al. 1994). The calculated formula isR
0
pathway to reproduction at age 6 and higher
=
R p
0
ttttt f
21 32 43 54 65 16
(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )
22 33 44 55 66
pathway to reproduction at age 7 and higher
=
tttttt f
21 32 43 54 65 76 17
⫹
(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )
22 33 44 55 66 77
tttttttf
21 32 43 54 65 76 87 18
⫹
(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )(1 ⫺ t )
22 33 44 55 66 77 88
.
\
pathway to reproduction at age 8 and higher
(24)
The matrix for the desert tortoise is an Usher matrix,
a stage-structured matrix where all reproduction goes to
a newborn stage. When individuals can stay in one stage
i, such as in the desert tortoise, a term will appear
⫺1
(1 ⫺ t )
ii
in the fecundity pathway where stage i is involved. Rela-
beling equation (24) with the same notation as that used
136 The American Naturalist
by Hastings and Botsford (2006b), the condition R 1 1
0
is written as
aaaaaf
10 21 32 43 54 5
R p
0
(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )
11 22 33 44 55
aaaaaaf
10 21 32 43 54 65 6
⫹
(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )
11 22 33 44 55 66
aaaaaaaf
10 21 32 43 54 65 76 7
⫹ 1 1.
(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )(1 ⫺ a )
11 22 33 44 55 66 77
(25)
In this particular case, the condition and the firstR
1 1
0
Hastings-Botsford persistence condition are identical
(Hastings and Botsford 2006b). Each term in equation (24)
corresponds to a pathway from birth to reproduction (as
indicated by the parentheses in the equation). As indicated
before, the population will persist if any of the terms or
the sum of any term in is
11. If the population startsR
0
with one newborn individual, then the first reproduction
would occur at stage 6, as indicated by the first pathway
in the equation. Each pathway shares the survivorship until
stage 6. Doak et al. (1994) found using numerical elasticity
analysis that survival of adult females contributed the most
to population growth rate. This can also be seen from
simple observation of equation (24), where the terms
lie in the denominator, allowing to grow quickly1 ⫺ tR
ii 0
as approaches 1. With , we also note that at the sametR
ii 0
time, the pathway to adult female ( ) must bettttt
21 32 43 54 65
maintained if maximizing reproductive output is the goal.
Timing of Reproduction. Using the matrix from Doak et
al. (1994; their table 5), we calculated the timing of re-
productive output. Figure 5B shows the number of off-
spring over time, when the population starts with one
reproducing individual. The maximum number of new
individuals resulting from one new individual peaks at
about 15 years, with , mean generation timeR p 0.696
0
, and variance . Using equation
2
m p 19.209 j p 67.443
TT
(14), the mean generation time is . The high
˜
T p 19.872
variance indicates that individuals can stay for long periods
in one stage before moving to the next one.
The high variance and spread out reproductive output
(fig. 5B) are a consequence of individuals being able to
stay for long periods in one stage class. If adults are pro-
tected, in principle, this would improve reproductive out-
put. However, improved adult tortoise survival does not
guarantee survival of new generations. Thus, conservation
strategies that substantially exceed the mean generation
time (20 years) would be required if persistence of this
species needs to be ensured.
Creeping Aven
Population Persistence. Creeping aven (Geum reptans)isa
clonal perennial rosette that grows in alpine conditions.
The life cycle for creeping aven is shown in figure 4C
(Weppler et al. 2006).
Using graph reduction, the calculated isR
0
vegetative reproduction
=
tf tttf ttf
43 34 43 54 45 34 43 54 35
R p f ⫹⫹ ⫹
033
2
1 ⫺ t (1 ⫺ t )(1⫺ t )(1⫺ t )(1 ⫺ t )
44 44 45 44 45
sexual reproduction
=
tt f tttf
21 32 13 21 32 43 14
⫹⫹
[
1 ⫺ t (1 ⫺ t )(1 ⫺ t )
22 22 44
tttttf tttt f
21 32 43 54 45 14 21 32 43 54 15
⫹⫹,
2
]
(1 ⫺ t )(1 ⫺ t )(1⫺ t )(1⫺ t )(1 ⫺ t )(1 ⫺ t )
22 44 45 22 44 45
\
sexual reproduction
(26)
where and is
⫺1
t p (tt ) /[(1 ⫺ t )(1 ⫺ t )] (1 ⫺ t )
45 45 54 44 55 45
the time an individual spends looping between stages 4
and 5. The population will persist if any of the terms or
the sum of any terms in is
11. Transitions occur inRt
0 ii
almost all fecundity pathways. This means that the time
an individual stays in one stage could have a large impact
on and on population growth. Terms appear
⫺2
R (1 ⫺ t )
044
because in those pathways, stage 4 is visited twice. Weppler
et al. (2006) used elasticity analysis of l to determine which
transitions contribute most to population growth. They
found that had the greatest contribution, and our anal-t
ii
ysis of is consistent with their observation. However,R
0
we use only the analytical solution to and no numericalR
0
estimates. Weppler et al. (2006) determined that for creep-
ing aven, both sexual and clonal reproduction seem to be
equally predominant. Looking at the pathways in the R
0
equation reveals that there are many pathways that lead
to both types of reproduction. However, vegetative repro-
duction pathways are shorter, which suggests faster repro-
duction through clonal reproduction. Additionally, the
equation for (eq. [26]) shows that if transition ,Rf
1 1
033
then population will persist, regardless of longer repro-
ductive pathways. This leads us to conclude that vegetative
reproduction may be predominant when is large.f
33
This analysis goes beyond a simple condition for de-
termining persistence. The formula shows explicitly allR
0
the pathways for vegetative or sexual reproduction,
whereas the Hastings-Botsford condition simply tests for
persistence. The full Hastings-Botsford conditions (eq.
[19]) comprise 26 inequalities that must be individually
evaluated. These can be simplified to three inequalities,
any one of which, if satisfied, guarantees persistence (see
Reproductive Rate and Reproductive Output 137
“Hastings-Botsford Persistence Conditions for Creeping
Aven” in the appendix). We have not been able to evaluate
these inequalities directly to , although this may beR
1 1
0
possible.
Timing of Reproduction. Using transition entries from
Weppler et al. (2006; their table 1), we calculated the net
reproductive rate and mean generation time. As an ex-
ample, we consider parameter values for only one of the
sites (“vandret da Porchabella”) for 2000–2001. The num-
ber of new descendants over time is shown in figure 5C.
This figure indicates that the largest reproductive output
is predominant in the first year. This is consistent with
our previous analysis of transitions in the formula.fR
33 0
From the second year, there is a mixture of sexual and
clonal reproduction that spreads out for longer than 20
years, resulting in a large variance. This analysis suggests
two phases of reproduction: short-term clonal reproduc-
tion and longer-term reproduction with a mixture of sex-
ual and clonal. The first phase yields the potential for fast
short-term growth under good environmental conditions,
whereas the second phase would “time-average” possible
variations in long-term environmental conditions. The
dominance of these phases may be determined by envi-
ronmental stochasticity and other environmental factors.
With , the mean generation time isR p 1.581 m p
0 T
with variance . Equation (14) results in
2
6.446 j p 96.912
T
a generation time of .
˜
T p 6.912
Cat’s Ear
The example of cat’s ear can be found in “Cat’s Ear, Jury
Test, and ” in the appendix. This case is unusual in thatR
0
it leads to a quadratic equation for the net reproductive
rate . We do not provide a proof, but we believe thatR
0
the quadratic equation results when there are disjoint re-
productive pathways (reproductive pathways that are in-
dependent of each other). This is the only life cycle we
know of that does this, and we include the example for
completeness.
Discussion
Matrix models have established a formal foundation for
the study of the effect of life-history changes in population
dynamics, given a biological description of life cycle events.
It has been widely shown that elasticity/sensitivity analyses
of l are powerful measures of the consequences of life-
history events on population growth rate (Caswell 2001).
The development by de-Camino-Beck and Lewis (2007)
of a method for calculating the reproductive rate, , ex-R
0
plicitly allows for analysis to complement demographic
studies. In this article, we take this a step further, deriving
the calculation of the timing of reproductive output, which
further allows for the calculation of mean and variance of
the generation time. Once the life cycle has been described
as a matrix model, can be easily calculated from theR
0
graph representation, even for complex life cycles. The
analytical formula can be used to perform qualitativeR
0
analysis of life cycle structure and persistence. When ma-
trix transition estimates exist, the timing of reproductive
output can be used to study early versus late reproduction
strategies as well as the mean generation time to determine
how spread reproduction occurs, and generation time var-
iance can be used to study differences in reproductive
strategies. From the next-generation matrix shown in “Net
Reproductive Rate ,” the per generation reproductiveR
0
value can be calculated as the left eigenvector of this ma-
trix, in a fashion similar to that described for the per year
reproductive value (left eigenvector of the projection ma-
trix) by Caswell (2001).
Although we do not fully address evolutionary questions
about semelparity and iteroparity, like the ones posed by
Cole (1954), in this article, we believe that the -gen-R
0
erating function and the timing of reproduction will assist
in understanding these types of life histories. With , itR
0
is possible to study how changes in life cycle structure
affect fecundity pathways and the timing of reproduction.
For instance, it can be seen from the -generating func-R
0
tion that when , ; hence, the value of late re-l k 1 t K 1
productive output is very low, and early reproduction has
a stronger impact on population persistence (the weight
becomes increasingly smaller as n becomes large). It
n
t
can also be seen that when an individual can stay a long
time in one stage, becomes large, which in turn
⫺1
(1 ⫺ a )
ii
results in a more spread out timing of reproduction.
When we compare our analysis using to the previousR
0
analysis using l, several themes emerge. First, qualitative
conclusions regarding the effect of key model parameters
on population persistence are similar between the two
analyses. However, depiction of as a sum of fecundityR
0
pathways allows us to explore the effects of secondary (less
key) pathways on population persistence. The reproduc-
tion through these pathways can actually be sufficient to
allow the population to persist. These results can be missed
using typical sensitivity/elasticity analyses of l. Second, the
explicit formula allows a simple (nonnumerical) anal-R
0
ysis of the role of model parameters on population per-
sistence. This contrasts with the numerical methods
needed to analyze the dependence of l to model param-
eters. The fecundity pathways analysis is different fromR
0
a method known as loop analysis (van Groenendael et al.
1994). Loop analysis shows life cycle loop contributions
to population growth rate, but these loops do not nec-
essarily correspond to directed fecundity pathways, and it
depends on the calculation of elasticities. In contrast, the
138 The American Naturalist
analysis presented here focuses on directed reproduc-R
0
tion pathways and does not require numerical estimates
of vital rates. Last, while we know that and theR
1 1
0
Hastings-Botsford conditions are mathematically equiva-
lent, the condition can be interpreted biologicallyR
1 1
0
and intuitively in terms of fecundity pathways, whereas
the Hastings-Botsford conditions cannot. In all cases but
one (creeping aven), we show how to algebraically connect
the two conditions. From the formula, it is easy to seeR
0
that if any term (pathway) is 11, or if the sum of com-
bination of terms is
11, then the population will persist.
In terms of control of invading organisms, analysis,R
0
as shown in the examples, complements sensitivity/elas-
ticity analyses of l. The analytical solution of can beR
0
used to understand fecundity pathways and assist in the
experimental design of demographic studies or assist in
finding the best short- and long-term control strategies
for pest species. As shown in the thistle example (“Nodding
Thistle”), one strategy could be targeting transitions that
affect the most fecundity pathways. A second factor to
consider is how spread out the timing of reproduction is.
Targeting transitions that affect early reproduction may
result in reduction of population growth rate; however, if
control efforts are not maintained for a long period and
if the timing of reproductive output indicates that there
is still late reproduction, a reinvasion may occur.
Acknowledgments
We thank C. Bampfylde, C. Jerde, A. McClay, M. Wonham,
and the Lewis Lab for discussions, comments, and revi-
sions on the manuscript. We also thank anonymous re-
viewers for valuable comments on the manuscript. T.d.-
C.-B. was supported by Mathematics of Information
Technology and Complex Systems, a University of Alberta
Studentship, a Natural Sciences and Engineering Research
Council (NSERC) Discovery Grant, and an NSERC Col-
laborative Research Opportunity Grant. M.A.L. gratefully
acknowledges support from an NSERC Discovery Grant,
an NSERC Collaborative Research Opportunity Grant, and
a Canada Research Chair.
Literature Cited
Beckerman, A., T. Benton, E. Ranta, V. Kaitala, and P. Lundberg.
2002. Population dynamic consequences of delayed life-history
effects. Trends in Ecology & Evolution 17:263–269.
Birch, L. 1948. The intrinsic rate of natural increase of an insect
population. Journal of Animal Ecology 17:15–26.
Caswell, H. 1989. Matrix population models. 1st ed. Sinauer, Sun-
derland, MA.
———. 2001. Matrix population models: construction, analysis, and
interpretation. 2nd ed. Sinauer, Sunderland, MA.
Charnov, E., and W. Schaffer. 1973. Life-history consequences of
natural selection: Cole’s result revisited. American Naturalist 107:
791–793.
Cochran, M., and S. Ellner. 1992. Simple methods for calculating
age-based life-history parameters for stage-structured populations.
Ecological Monographs 62:345–364.
Cole, L. 1954. The population consequences of life history phenom-
ena. Quarterly Review of Biology 29:103–137.
———. 1960. A note on population parameters in cases of complex
reproduction. Ecology 41:372–375.
Coulson, T., T. Benton, P. Lundberg, S. Dall, B. Kendall, and J. Gail-
lard. 2006. Estimating individual contributions to population
growth: evolutionary fitness in ecological time. Proceedings of the
Royal Society B: Biological Sciences 273:547–555.
Cushing, J., and Y. Zhou. 1994. The net reproductive value and
stability in matrix population models. Natural Resource Modeling
8:297–333.
DeAngelis, D., C. Travis, and W. Post. 1979. Persistence and stability
of seed-dispersed species in a patchy environment. Theoretical
Population Biology 16:107–125.
DeAngelis, D., W. Post, and C. Travis. 1986. Positive feedback in
natural systems. Springer, Berlin.
de-Camino-Beck, T., and M. A. Lewis. 2007. A new method for
calculating net reproductive rate from graph reduction with ap-
plication to the control of invasive species. Bulletin of Mathe-
matical Biology 69:1341–1354.
De Kroon, H., A. Plaisier, and J. Vangroenendael. 1987. Density de-
pendent simulation of the population dynamics of a perennial
grassland species, Hypochaeris radicata. Oikos 50:3–12.
De Kroon, H., J. van Groenendael, and J. Ehrlen. 2000. Elasticities:
a review of methods and model limitations. Ecology 81:607–618.
Doak, D., P. Kareiva, and B. Kleptetka. 1994. Modeling population
viability for the desert tortoise in the western Mojave desert. Eco-
logical Applications 4:446–460.
Franco, M., and J. Silvertown. 1996. Life history variation in plants:
an exploration of the fast-slow continuum hypothesis. Philosoph-
ical Transactions of the Royal Society B: Biological Sciences 351:
1341–1348.
Gaillard, J., N. Yoccoz, J. Lebreton, C. Bonenfant, S. Devillard, A.
Loison, D. Pontier, and D. Allaine. 2005. Generation time: a re-
liable metric to measure life-history variation among mammalian
populations. American Naturalist 166:119–123.
Goodman, D. 1975. Theory of diversity-stability relationships in ecol-
ogy. Quarterly Review of Biology 50:237–266.
Hastings, A., and L. Botsford. 2006a. Persistence of spatial popula-
tions depends on returning home. Proceedings of the National
Academy of Sciences of the USA 103:6067–6072.
———. 2006b. A simple persistence condition for structured pop-
ulations. Ecology Letters 9:846–852.
Jongejans, E., A. Sheppard, and K. Shea. 2006. What controls the
population dynamics of the invasive thistle Carduus nutans in its
native range? Journal of Applied Ecology 43:877–886.
Kot, M. 2001. Elements of mathematical ecology. Cambridge Uni-
versity Press, Cambridge.
Lebreton, J. 2005. Age, stages, and the role of generation time in
matrix models. Ecological Modelling 188:22–29.
Li, C., and H. Schneider. 2002. Applications of Perron-Frobenius
theory to population dynamics. Journal of Mathematical Biology
44:450–462.
Murdoch, W. 1966. Population stability and life history phenomena.
American Naturalist 100:5–11.
Reproductive Rate and Reproductive Output 139
Murray, J. 1993. Mathematical biology. 2nd ed. Springer, Berlin.
Oli, M., and F. Dobson. 2003. The relative importance of life-history
variables to population growth rate in mammals: Cole’s prediction
revisited. American Naturalist 161:422–440.
———. 2005. Generation time, elasticity patterns, and mammalian
life histories: a reply to Gaillard et al. American Naturalist 166:
124–128.
Pfister, C. 1998. Patterns of variance in stage-structured populations:
evolutionary predictions and ecological implications. Proceedings
of the National Academy of Sciences of the USA 95:213–218.
Pico, F., N. Ouborg, and J. van Groenendael. 2004. Influence of selfing
and maternal effects on life-cycle traits and dispersal ability in the
herb Hypochaeris radicata (Asteraceae). Botanical Journal of the
Linnean Society 146:163–170.
Ranta, E., D. Tesar, and A. Kaitala. 2002. Environmental variability
and semelparity vs. iteroparity as life histories. Journal of Theo-
retical Biology 217:391–396.
Shea, K. 2004. Models for improving the targeting and implemen-
tation of biological control of weeds. Weed Technology 18:1578–
1581.
Shea, K., and D. Kelly. 1998. Estimating biocontrol agent impact with
matrix models: Carduus nutans in New Zealand. Ecological Ap-
plications 8:824–832.
Silvertown, J., M. Franco, I. Pisanty, and A. Mendoza. 1993. Com-
parative plant demography: relative importance of life-cycle com-
ponents to the finite rate of increase in woody and herbaceous
perennials. Journal of Ecology 81:465–476.
van Groenendael, J., H. Dekroon, S. Kalisz, and S. Tuljapurkar. 1994.
Loop analysis: evaluating life-history pathways in population pro-
jection matrices. Ecology 75:2410–2415.
Weppler, T., P. Stoll, and J. Stocklin. 2006. The relative importance
of sexual and clonal reproduction for population growth in the
long-lived alpine plant Geum reptans. Journal of Ecology 94:869–
879.
Associate Editor: Catherine A. Pfister
Editor: Donald L. DeAngelis
