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Journal of Animal Ecology 2010, 79, 109–116
doi: 10.1111/j.1365-2656.2009.01601.x
Stochastic demography and population dynamics in the
red kangaroo Macropus rufus
Niclas Jonzén1*, Tony Pople2, Jonas Knape1 and Martin Sköld3
1
Department of Theoretical Ecology, Ecology Building, Lund University, SE-223 62 Lund, Sweden; 2Invasive Plants and
Animals, Biosecurity Queensland, Department of Primary Industries and Fisheries, GPO Box 46, Brisbane, Qld 4001,
Australia; and 3Department of Economics, Statistics and Informatics, Örebro University, SE-701 82 Örebro, Sweden
Summary
1. Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity
is defined as the proportional change in population growth rate caused by a proportional change
in a vital rate. Deterministic analyses may not necessarily be informative because large variation in
a vital rate with a small deterministic elasticity may affect the population growth rate more than a
small change in a less variable vital rate having high deterministic elasticity.
2. We analyse a stochastic environment model of the red kangaroo (Macropus rufus), a species
inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital
rates.
3. Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate.
4. Even if changes in average rainfall have a larger impact on population growth rate, increased
variability in rainfall may still be important also in long-lived species. The elasticity with respect to
the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of
vital rates.
5. Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction
of c. 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by
more than c. 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how
environmental stochasticity translates into population growth rate.
Key-words: climate, elasticity, mammals, matrix models
Introduction
Life-history evolution and population dynamics are two central fields of ecology where the original theories were deterministic but we have gained tremendously in general
understanding by considering stochastic models (Lande,
Engen & Sæther 2003). The relevance of models capturing
the random variation experienced by natural populations
goes beyond ecological theory and is motivated by the need
to evaluate the population consequences of human impact
such as harvesting (Lande, Engen & Sæther 1995; Jonzén,
Ripa & Lundberg 2002) as well as climate change (Boyce
et al. 2006). The latter will expose organisms to novel
environmental conditions and potentially affect the magni*Correspondence author. E-mail: [email protected]
tude and frequency of environmental events (Karl, Knight &
Plummer 1995; Easterling et al. 2000) that shape the life
history and demography of species.
Large terrestrial mammalian herbivores are widespread
and some of them are found in strongly fluctuating environments such as rainfall-driven savannas and grasslands
(Nowak 1991). Despite living in stochastic environments and
thus showing fluctuating vital rates (Gaillard et al. 2000), the
demography and population dynamics are often studied
using deterministic matrix models (Caswell 2001). Deterministic analyses of long-lived species have shown that the elasticity of population growth rate to adult survival is stronger
than for juvenile survival and fecundity (Gaillard, FestaBianchet & Yoccoz 1998; Lebreton 2006). However, adult
survival is known to be less variable than juvenile survival
(Gaillard et al. 1998) and large annual variation in an element
2009 The Authors. Journal compilation 2009 British Ecological Society
110 N. Jonze´n et al.
calculations, we assume that rainfall is the only driver of population
change through its effect on survival and reproduction. There are of
course other external factors that will affect the population growth
rate, but this is still a reasonable assumption given the lack of significant limiting interactions with predators and ⁄ or parasites within
Australia’s sheep rangelands (Caughley, Shepherd & Short 1987;
Pople & Grigg 1998). The annual survival rate was modelled as a
logistic function of annual rainfall (R):
sk ¼
ck eak þbk R
;
1 þ eak þbk R
1
(a)
0·8
0·6
0·4
0·2
STOCHASTIC DENSITY-INDEPENDENT MODEL
We used a demographic model with three age classes (0–12 months,
12–24 months and 24+) and a 1-year time step between t and t + 1.
In drought conditions, red kangaroo populations often show in order
what is typical of large mammals (Hanks 1981): decreased juvenile
survival, delayed age at sexual maturity, reduced fecundity and
decreased adult survival. Such a sequence has been found throughout
Australia (Frith & Sharman 1964; Newsome 1965; Shepherd 1987;
Pople 1996).
Food supply as measured by pasture biomass or approximated by
rainfall is correlated with the rate of increase (Caughley, Bayliss &
Giles 1984; Bayliss 1985, 1987; Cairns & Grigg 1993). To simplify
P(reproduction|rainfall)
0
0
Materials and methods
ðeqn 1Þ
where k refers to either adults (k = a) or juveniles (k = j). Due to
data limitations, we only implemented two survival schedules: juvenile survival from the first- to the second-age class (when the young
become independent),and adult survival between the later-age classes
(Fig. 1a, Table 1). Details on the estimation of survival are given in
Tenhumberg et al. (2004).
The reproductive biology of red kangaroos has been well studied
(see review by Tyndale-Biscoe & Renfree 1987). Breeding may, if
environmental conditions are good, occur year round and females
give birth to c. 1Æ5 young per year. As we are only modelling the
female segment of the population, the number of female offspring
produced per reproducing female is 0Æ75. The proportion of females
giving birth at a given age is, however, dependent on the environment. Frith & Sharman (1964) studied how the proportion of mature
females at a given age varied between areas differing in the amount of
annual rainfall. We used data from the study by Frith and Sharman
to interpolate the (increasing) proportion of mature females within
the three age classes as a function of annual rainfall (Fig. 1b,
Appendix S1, Supporting Information).
To generate variability in the vital rates, we sampled annual rainfall with replacement from a 123-year-long time series on annual precipitation in Menindee at the edge of Kinchega National Park (32 S,
142 E). As a comparison, we also drew annual rainfall from a lognormal distribution, with parameters estimated from the Menindee
data set. However, the results were not affected by the alternative
procedures and we therefore present the analysis based on resampling
only. Rainfall statistics are given in Fig. 2.
Survival
with a small elasticity may affect the population growth rate
more than a small change in an element having high elasticity
but less variability (Sæther 1997). Still, both empirical (e.g.
Benton, Grant & Clutton-Brock 1995; Morris et al. 2008)
and theoretical (Orzack & Tuljapurkar 1989) work suggests
that long-lived species are often less sensitive to variability in
vital rates than short-lived species. How sensitive they are to
changes in environmental variability has, however, not been
thoroughly investigated (but see Davis, Pech & Catchpole
2002). Hence, there is clearly a need for quantitative analyses
to establish how variation in the environment affects population growth rate by changing means and variability in demographic rates (Sæther 1997).
Understanding and predicting variation in population
growth rate is also a challenge to population management. In
South Australia, aerial surveys are conducted annually to
estimate population abundance of the red kangaroo and the
western grey kangaroo (Macropus fuliginosus, Desmarest;
Grigg et al. 1999). The counts in a given year are used to set
the harvest quota in the following year. A less costly alternative would be to decrease the survey frequency and to predict
population abundance using a population model, data from
previous surveys, and relevant covariates such as rainfall
(Hauser, Pople & Possingham 2006). However, if population
managers are going to rely on model predictions rather than
survey data in some years, it becomes crucial to understand
how stochastic rainfall translates into population growth rate
(Jonzén et al. 2005), which is what we are addressing here.
In this study, we make use of the approach developed by
Tuljapurkar and co-workers (e.g. Tuljapurkar 1990; Tuljapurkar, Horvitz & Pascarella 2003) to calculate the elasticity
of the stochastic growth rate with respect to the mean and
variability in vital rates (i.e. survival and fecundity). We
apply the methods to a stochastic environment model (Lebreton 2006) of the red kangaroo (Macropus rufus, Desmarest),
a species inhabiting the arid and semi-arid zone of Australia,
where the environment is characterized by unpredictable and
highly variable rainfall. The elasticities with respect to the
vital rates are compared to the elasticities with respect to the
mean and variability of rainfall and we also evaluate how the
stochastic growth rate changes with harvesting and changes
in rainfall patterns.
1
100
200
300
400
500
600
700
800
100
200
300
400
500
Rainfall (mm)
600
700
800
(b)
0·8
0·6
0·4
0·2
0
0
Fig. 1. (a) Survival of individuals of age above (solid line) or below
(dotted line) 12 months as a logistic function of annual rainfall.
Parameter values are given in Table 1. (b) Probability of reproduction at age 12–24 months (dotted line) and 24+ months (solid line)
as a function of annual rainfall.
2009 The Authors. Journal compilation 2009 British Ecological Society, Journal of Animal Ecology, 79, 109–116
Stochastic demography and population dynamics 111
Table 1. Parameter values used in the logistic survival functions
STOCHASTIC GROWTH RATE AND ELASTICITIES
Survival parameters
Value(s)
Interpretation
aj
aa
bj
ba
cj
ca
)3
0Æ1
0Æ014
0Æ01
0Æ8
0Æ99
Juvenile survival constant
Adult survival constant
Rain effect on juvenile survival
Rain effect on adult survival
Maximum juvenile survival
Maximum adult survival
Finally, we also tested the performance of a demographic
model having 13-age classes, starting from newborn and each
class covering 3 months except from the final group, which
includes all animals being 36 months or older (when all animals
are likely to be sexually mature), and we let the time step
between time t and t + 1 be 3 months. However, the results
did not change and we therefore only report results using the
simple model having three age classes.
To summarize, we let N(t) be the 3 · 1 population vector at time t
and X(t) is the 3 · 3 stochastic transition matrix. The mapping from
time t ) 1 to time t is given by
NðtÞ ¼ XðtÞNðt 1Þ;
ðeqn 2Þ
where the temporal variation is due to stochastic rainfall affecting
survival rates and the probability of reproduction at a given age.
Rainfall (mm)
800
(a)
400
Frequency
1900
1920
1940
Year
1960
1980
2000
XðtÞ0 vðtÞ
vðt 1Þ ¼ XðtÞ0 vðtÞ ;
1
ðeqn 5Þ
of population structure vectors and reproductive vectors respectively,
where kk1 denotes ‘1-norm’, i.e. summation of the absolute value of
all elements and u(0) = v(T) = 1 ⁄ 3Æ(1, 1, 1) (or some other arbitrary
initial vectors).
All model properties were obtained using numerical simulations in
MATLAB (The MathWorks Inc. 2000). We were interested in separating the elasticity of ks with respect to the mean lij and the standard
deviation rij of vital rate Xij:
@ log ks
;
@ log lij
@ log ks
¼
:
@ log rij
ESl
ij ¼
ðeqn 6Þ
ðeqn 7Þ
where Æv(t), u(t)æis a scalar product and Cij is the perturbation of the
Xij element. We get the elasticity of ks with respect to the mean of Xij
by setting Cij(t) = lij. Similarly, by setting Cij(t) = Xij(t) ) lij, we
get the elasticity with respect to the standard deviation.
5
0
0
100
200
300 400 500
Rainfall (mm)
600
700
800
(c)
0·5
ACF
ðeqn 4Þ
X
1 T vi ðtÞCij ðtÞuj ðt 1Þ
;
T!1 T
kðtÞhvðtÞ; uðtÞi
t¼1
10
0
–0·5
–1
0
XðtÞuðt 1Þ
;
kXðtÞuðt 1Þk1
uðtÞ ¼
Dij ¼ lim
(b)
15
1
We also generated the sequences
The sum of these two elasticities equals the so-called stochastic
elasticity (Tuljapurkar et al. 2003), which confounds the effects of
changing mean and variability of the vital rates (Haridas & Tuljapurkar 2005). The full elasticity matrix was computed numerically following Tuljapurkar (1990):
200
20
ðeqn 3Þ
ESr
ij
600
0
1880
For the model given by eqn 2, we computed the stochastic growth
rate ks from
1
jNðtÞj
ln
ðCaswell 2001and references thereinÞ:
lnðks Þ ¼ lim
t!1 t
jNð0Þj
2
4
6
Time lag (years)
8
10
Fig. 2. (a) Time series, (b) histogram and (c) autocorrelation function
of total annual rainfall in Menindee. The dotted lines in (c) indicates
the approximate 95% confidence interval. The average annual rainfall is 242 mm, the standard deviation is 107 mm and the coefficient
of variation 0Æ44.
THE EFFECT OF HARVESTING AND CLIMATE CHANGE
Red kangaroos are harvested for meat and skins and, since 1984,
each of the mainland States has offered annual harvest quotas that
are a constant proportion (10–20%) of the estimated population
size (Pople & Grigg 1998; Grigg & Pople 2001). Modeling studies
have suggested that harvest rates of c. 10–20% of kangaroo populations are sustainable in the long term (Caughley 1987; Hacker,
McLeod & Druhan 2003; Pople 2003) but, in some years and areas,
red kangaroo populations face annual harvest fractions above 30%
(Pople 2006). We were interested in contrasting elasticity patterns
between an unharvested and a harvested population. In the harvested population, we decreased the annual survival rate of the oldest age class by 30%. Primarily adults are harvested as harvesters
are paid according to carcass and skin size. The harvest tends to be
male biased, so a harvest rate of 30% for a female population is
particularly conservative.
2009 The Authors. Journal compilation 2009 British Ecological Society, Journal of Animal Ecology, 79, 109–116
112 N. Jonze´n et al.
Australian rainfall varies substantially, especially in the dry and
semi-dry areas inhabited by the red kangaroo. Projections of future
rainfall patterns are uncertain with models simulating either increase
or decrease across large areas of Australia (Whetton et al. 2005). To
study the effect of possibly changing rainfall patterns due to climate
change, we increased or decreased the observed values of annual
rainfall by a constant times the mean value and, again, sampled
from data with replacement. An alternative procedure would be to
multiply each value by a constant. The latter procedure does not
only change the average rainfall but also the variance. Because it is
not known which procedure is most consistent with future global
change, we evaluated both methods. To get a general idea about
how harvest affected the stochastic growth rate under different rainfall scenarios, we varied the annual harvest fraction between 0 and
0Æ4 and, for each combination of harvest fraction and average rainfall, we estimated the stochastic growth rate.
Finally, we note that anthropogenic modification of the biosphere
is projected to increase climatic variability, resulting in more variable rainfall and more frequent extreme temperatures (Karl et al.
1995; Easterling et al. 2000).We therefore estimated the elasticity of
ks to a change in the standard deviation of annual rainfall (see
Appendix D in Haridas & Tuljapurkar (2005) for the numerical procedure). For a comparison, we also estimated the elasticity of ks to a
change in mean rainfall.
SERIAL AUTOCORRELATION
There is no evidence of serial autocorrelation in the rain data and
decreasing the prediction interval makes no difference as the autocorrelation with lag 1 month is only c. 0Æ1. However, as environmental
serial autocorrelation can have a sizeable effect on population growth
rate (Tuljapurkar & Haridas 2006) and we want to make sure that
our results are robust to any misspecification of the autocorrelation
structure, we nevertheless simulated rainfall from a process where log
of rainfall was assumed to follow an AR(1), having the observed (on
a log scale) mean and variance as expectations and let the autoregression parameter be 0Æ8. That did not change the elasticity patterns.
Results
The stochastic rainfall translates into variation in vital rates
and especially juvenile survival, and the reproduction in age
classes 2 and 3 fluctuates considerably (Fig. 3). The adult survival does not fluctuate much, reaching 5% of the maximum
possible variance given its mean (see Morris & Doak 2004),
compared to 19% in juvenile survival. The age distribution
fluctuates heavily but, on average, c. 30% of the population
belong to age class 1, 10% to age class 2 and 60% are found
in age class 3 (Fig. 4).
In Fig. 5, we present the elasticities with respect to the
l
r
mean, ESij , and the variability, ESij , of the matrix elements.
The stochastic growth rate is far more sensitive to a proportional change in the mean value when compared to the standard deviation of a matrix element. Summing up all the
standard deviation elasticities, we get an estimate of the effect
of variability on population growth rate:
Tr ¼
X
r
ESij ¼ 0026:
ij
There is coupling between the summed elasticities with
respect to the mean (Tl) and variability (Tr) of the matrix elements such that Tl + Tr = 1 (Haridas & Tuljapurkar
2005). Hence the value of Tr found here is relatively small.
The stochastic growth rate (with no harvest) was estimated to
be 1Æ056, which should be compared to the deterministic
growth rate derived from the average matrix (k = 1Æ068),
where the elements of the average matrix are the expected
survival and reproductive rates. To what extent this difference
is negligible or important depends on the context. The variability in vital rates does, however, induce variation in the population growth rate (range = 0Æ50–1Æ61, CV = 20%), which is
important for population management (see Discussion).
(b) x 104
2·5
(a)
15 000
1·5
1
5000
0·5
0
0
0·5
Juvenile survival
0
0
1
(c) x 104
3
Frequency
Frequency
2
10 000
(d)
2
0·5
Adult survival
1
x 104
1·5
2
1
1
0
0
0·5
0·5
Fecundity age class 2
1
0
0
0·5
Fecundity age class 3
1
Fig. 3. Histogram of (a) juvenile annual
survival rate, (b) adult annual survival rate,
fecundity (number of female offspring per
female) in (c) age class 2 and (d) age class 3.
2009 The Authors. Journal compilation 2009 British Ecological Society, Journal of Animal Ecology, 79, 109–116
Stochastic demography and population dynamics 113
0·8
0·7
Age distribution
0·6
0·5
0·4
0·3
0·2
Fig. 4. An example of the temporal fluctuation
in age structure. Age class 1 (open circles):
mean = 0Æ28, SD = 0Æ047, CV = 0Æ17. Age
class 2 (filled triangles): mean = 0Æ11,
SD = 0Æ039, CV = 0Æ35. Age class 3 (filled
circles): mean = 0Æ60, SD = 0Æ076, CV =
0Æ13.
0·1
0
900
910
In an unharvested population survival in age class 3 has
l
the highest ES followed by survival from age class 1 to 2,
fecundity in class 3, survival from class 2 to 3 and fecundity in
age class 2. Harvesting 30% of age class 3 increases the magl
nitude of ES of survival of the two youngest age classes and
of fecundity in the final age class, but the relative ranking of
the elasticities remains (Fig. 5). With
the survival
harvesting
l
in age class 3 again has the highest ESij , but is now followed
by
in age class 3. The latter also has the highest
fecundity
Sr Eij .
In accordance with previous simulation studies, the model
kangaroo population analysed here can sustain an annual
920
930
940
950
960
Time (years)
970
980
990
1000
harvest fraction of c. 20% (Fig. 6). However, if average rainfall drops by more than c. 10%, any level of harvesting may
be unsustainable, as even an unharvested population cannot
maintain itself. A caveat is that there is no density dependence so there is no harvest compensation to halt any longterm declines. As can be seen in Fig. 6, a given increase in
average rainfall has less effect than a given decrease on the
stochastic growth rate. Hence, we need to reduce harvesting
if the average annual rainfall declines, but we should not
increase harvesting to the same extent if the average annual
rainfall were to increase. This is a result of Jensen’s inequality
(Ruel & Ayres 1999) and the concave relationship between
Fig. 5. Proportional change of the stochastic
growth rate for a proportional change in the
mean value (a, b) or the standard deviation
(c, d) of survival (a, c) and fecundity (b, d).
The solid lines represent an unharvested
population and the dotted line a population
where 30% of the individuals aged 24
months or more are harvested.
2009 The Authors. Journal compilation 2009 British Ecological Society, Journal of Animal Ecology, 79, 109–116
114 N. Jonze´n et al.
20
1·2
1·2
1·2
10
1·1
1·1
1·1
5
1
1
0·9
1
0
0·9
0·9
0·8
0·7
0·6
0·8
1
–5
0·7
0·6
0·5
0·4
0·3
5
6
0·
0·
–20
0
0·2
0·
–15
7
0·
8
–10
0·9
Change in average rainfall (%)
15
0·05
0·1
0·15
0·4
0·2
0·3
0·25
0·3
0·35
0·4
Harvest fraction
Fig. 6. The effect of harvesting (annual fraction of age class 3) and a
change in the average annual rainfall on the stochastic growth rate.
rainfall and the stochastic growth rate. The latter can be seen
by noting that – for a given harvest fraction – the distance
between contour lines increases with increasing rainfall
(Fig. 6). The concave relationship also implies that increased
variability in vital rates will decrease the stochastic growth
rate, as shown by the negative standard deviation elasticities
(Fig. 5c,d).
Reducing the average rainfall by multiplying the observed
mean with a constant gives very similar results. Compared to
the results reported in Fig. 6, a change in average rainfall does
not change the stochastic growth rate to the same extent.
However, the overall picture is almost identical (not shown).
Finally, we evaluated the effect on the stochastic growth
rate of a proportional disturbance of the standard deviation
in annual rainfall. In contrast to the weak effect of a proportional disturbance of the variability in vital rates, the elasticity of ks to a change in the standard deviation of annual
rainfall turned out to be far more important and was estimated to )0Æ10. This should, however, be compared with the
elasticity of ks to a change in mean rainfall, which was estimated to 0Æ55. Harvesting 30% of age class 3 increased the
elasticities with respect to the mean and the standard deviation of rainfall to 0Æ78 and )0Æ14 respectively. The increase in
the elasticity with respect to the mean caused by harvest
mainly stems from an increase in the elasticity with respect to
juvenile survival. The decrease of the elasticity with respect to
the standard deviation stems from changes of roughly equal
magnitude in the elasticities with respect to juvenile survival,
adult fecundity and adult survival, the first two negative and
the later positive.
Discussion
In most studies on elasticity in random environments, the socalled stochastic elasticity, ES, is calculated (Caswell 2001).
However, ES confounds the effects of changing mean and
variability of the vital rates (Haridas & Tuljapurkar 2005)
and we have therefore, following Tuljapurkar et al. (2003),
separated the effects of perturbing the mean and variability
of the matrix elements. Applying this framework to a model
population of the red kangaroo, a classical example of population dynamics strongly affected by stochastic rainfall
(Caughley et al. 1987), we show that the elasticity to variability in vital rates is very low compared to the elasticity to the
mean values. This may seem surprising considering the
importance of the highly variable rainfall for kangaroo
demography, but as the coefficient of variation of adult survival rate is rather small (Fig. 3), we should not expect a proportional change in the standard deviation to be very
important in relation to a proportional change in the mean
(see Fig. 5; see also Morris et al. (2008)). A change in the standard deviation of juvenile survival could be pronounced, but
its effect is hampered by the relatively stable and productive
third age class (Fig. 4).
The weak effect of variability in vital rates in the red
kangaroo is in accordance with a recent study (Morris
et al. 2008) showing that short-lived species are predicted
to be more strongly affected by increased variability in
vital rates relative to long-lived species. To calculate the
elasticity of the stochastic growth rate with respect to the
mean and variability in vital rates is motivated by the fact
that climate models suggest changes in not only average
conditions but also in variability (Karl et al. 1995; Easterling et al. 2000). Hence, we should map from changes in
the interesting environmental variables via vital rates to
population change (Boyce et al. 2006). As noted by Morris et al. (2008), the data needed are rarely available and
our study is important in showing that the generally low
elasticity to vital rate variability in long-lived species
(Morris et al. 2008) does not necessarily imply that those
species are insensitive to changes in environmental variability. The overall effect of a change in the variability of
rainfall is a combination of mean and standard deviation
elasticities for all vital rates affected by rainfall (Haridas
& Tuljapurkar 2005), and the estimated elasticity with
respect to the standard deviation of rainfall is comparable
to the elasticities with respect to the mean of all vital rates
but the survival in age class 3. Hence ignoring the effect
of increased variability in rainfall is not motivated by
referring to the low standard deviation elasticities. However, one should bear in mind that the elasticity with
respect to the mean rainfall is five times the elasticity with
respect to the standard deviation of rainfall, i.e. keeping
track of the average conditions is far more important in a
long-lived species such as the red kangaroo. This is in line
with some previous empirical studies on long-lived species
(e.g. Benton et al. 1995; Morris et al. 2008).
Even though long-lived species in general seem to be less
sensitive to environmental fluctuations than short-lived
species, there are likely to be interspecific differences in how
they respond to environmental stochasticity. The ability of a
life history to handle stochasticity is dependent on the
amount of environmental variation, how it affects the vital
rates and on the details of the life history, as well as on the
generation time (Orzack & Tuljapurkar 1989). To contrast
2009 The Authors. Journal compilation 2009 British Ecological Society, Journal of Animal Ecology, 79, 109–116
Stochastic demography and population dynamics 115
the case of the red kangaroo to another long-lived mammal,
consider the red deer (Cervus elaphas) on Rum in Scotland.
This population also shows weak effects of stochasticity on
elasticity patterns, but has a different life history than the red
kangaroo in that there are trade-offs between current fecundity, future fecundity and survivorship (Benton et al. 1995).
Incorporating these costs of reproduction gives negative
elasticities with respect to survival in most age classes.
Several other studies have highlighted the importance of
covariance among vital rates (e.g. Sæther & Bakke 2000)
and failing to take that into account can lead to misleading conclusions (Morris & Doak 2004; Coulson, Gaillard
& Festa-Bianchet 2005). Vital rates in red kangaroos are
positively correlated due to the effect of rainfall but
female red kangaroos show intriguing adaptations to
unpredictable fluctuations between periods of dry and wet
conditions that do not require the inclusion of trade-offs.
For instance, females can delay the progress of pregnancy
such that they can rapidly replace a young that died early
or left the pouch (Tyndale-Biscoe & Renfree 1987), and
the cost of reproduction is therefore very low.
It is reasonable to assume that the low standard deviation
elasticities reported here are the results of an evolutionary
past where the plasticity in vital rates was under strong selection. The reproductive strategy in the red kangaroo is especially well adapted to unpredictable fluctuations, and there is
much more to be gained (in fitness terms) by increasing the
average of vital rates rather than further minimizing the fluctuations (see also Gaillard & Yoccoz 2003). However, an
increased harvest fraction makes the population more sensitive to a decrease in average rainfall, mainly through increasing the importance of juvenile survival, or to an increase in
the variability of rainfall through increasing the sensitivity of
adult fecundity and juvenile survival. Hence, to maintain a
sustainable harvesting strategy, it is important to keep track
of the nature of changes in climate and their effects on vital
rates.
In conclusion, the red kangaroo system illustrates an
example where climate effects are indirect and mediated by
the intrinsic processes of birth, death and age structure
(Krebs & Berteaux 2006). By making a rigorous analysis that
explicitly partial out the effect of mean and variability in vital
rates due to stochastic rainfall, we have shown that red kangaroos are likely to be affected not only by any reduction in
average rainfall through its negative effect on all vital rates,
but also by increased variability in rainfall. The latter may be
important also in long-lived species if variability is high and
many vital rates are affected.
Acknowledgements
We are very grateful to K. Al-Khafaji, C.V. Haridas, S. Tuljapurkar,
S. Beissinger and two anonymous reviewers who provided comments
and suggestions that significantly improved this study. Carol Horvitz
kindly provided the MATLAB code used in Tuljapurkar et al.
(2003). N.J. and J.K. were financially supported by the Swedish
Research Council.
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Supporting Information
Additional supporting information may be found in the online version of this article.
Appendix S1. Probability of reproduction within the three age classes
as a function of annual rainfall.
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2009 The Authors. Journal compilation 2009 British Ecological Society, Journal of Animal Ecology, 79, 109–116