Download Carrying Capacity and Demographic Stochasticity: Scaling Behavior

Theoretical Population Biology 57, 5965 (2000)
doi:10.1006tpbi.1999.1434, available online at http:www.idealibrary.com on
Carrying Capacity and Demographic
Stochasticity: Scaling Behavior of the
Stochastic Logistic Model
Jonathan Dushoff
Institute of Physics, Academia Sinica, Nankang 105, Taipei, Taiwan
Received March 18, 1999
The stochastic logistic model is the simplest model that combines individual-level
demography with density dependence. It explicitly or implicitly underlies many models of
biodiversity of competing species, as well as non-spatial or metapopulation models of
persistence of individual species. The model has also been used to study persistence in simple
disease models. The stochastic logistic model has direct relevance for questions of limiting
similarity in ecological systems. This paper uses a biased random walk heuristic to derive a
scaling relationship for the persistence of a population under this model, and discusses its
implications for models of biodiversity and persistence. Time to extinction of a species under
the stochastic logistic model is approximated by the exponential of the scaling quantity
U=(R&1) 2 NR(R+1), where N is the habitat size and R is the basic reproductive number.
] 2000 Academic Press
1. INTRODUCTION
biodiversity among competing species is maintained by a
trade-off between competitive ability on one hand and
reproductive and dispersal ability on the other. The same
model was discussed, from the perspective of competing
disease strains, by May and Nowak (1994). In this sort
of hierarchical competition, the ``spacing'' between species
at equilibrium may be determined by demographic
stochasticity (Kinzig et al., 1999).
Finally, the stochastic logistic model is equivalent to
the stochastic SIS model in epidemiology, which was
investigated in some detail by Na# sell (1995, 1996).
Na# sell's results are similar in many ways to those presented here, but he did not focus on scaling based on
habitat size (in the case of epidemiology, the habitat size
is the population of available hosts), nor did he use the
biased random walk heuristic.
The stochastic logistic model investigated in this paper
is the ``patch dynamic'' version, where the habitat is given
an explicit size, and the only effect of density is that newborns cannot establish if they land on an occupied site.
Historically, many authors (e.g., Pielou (1991)) have
The stochastic logistic model is the simplest population model with discrete individuals and density
dependence. Thus, it is the starting point for investigating
the effects of demographic stochasticity in finite habitats.
Many models of population viability are based on the
stochastic logistic model, with added features such as
population structure, Allee effects (Dennis, 1989; Gabriel
and Burger, 1992; Stephan and Wissel, 1994) and
environmental stochasticity (Leigh, 1981; Goodman,
1987). Furthermore, the stochastic logistic represents a
good starting point for within-patch behavior of
metapopulation persistence models, if it is thought that
demographic stochasticity is an important component of
persistence at the level of individual patches.
The stochastic logistic model is also relevant to some
models of biodiversity of competing species, and by
extension models of speciesarea relationships. Tilman
(1990) proposed a model, based on earlier work of
Levins (Levins, 1969; Levins and Culver, 1971), where
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60
Jonathan Dushoff
used ``stochastic logistic'' to refer to a somewhat broader
model, where birth rates as well as death rates depend
linearly on the number of adults present. The behavior of
the two models should be similar.
This paper presents an argument based on a biased
random walk suggesting a scaling relationship between
population size and reproductive number for the
stochastic logistic model, together with evidence of the
scaling relationship based on analysis and numerical
simulation of the system. The scaling ideas presented
here can be applied to a discrete-time version of this
model as well, with some complications (Dushoff,
submitted).
Systems of this kind have a great deal in common with
percolation models and other phase-transition systems
that have been extensively studied by statistical
physicists. The approach presented here is strongly
analogous to what physicists call ``finite-size scaling'' of
such models (see Stauffer and Aharony (1991, Chap. 4)
for a review). Further use of techniques from statistical
physics may have the potential to greatly increase our
understanding of stochastic population models.
2. THE MODEL
In the stochastic logistic model we consider a habitat
with a finite number of sites N. Individuals die at rate m
and give birth at rate f randomly into one of the N sites.
If the site is already occupied, the newborn disappears.
This is precisely the behavior of the dominant species
in Tilman's competitioncolonization model (Tilman,
1990).
The number of parameters in the model can be
reduced by counting time in ``generations'' (units of 1m).
Thus the density-independent birth rate becomes R=
f m. Counting in terms of generations, we have the
following instantaneous birth and death rates, b j and d j ,
for a population of size j :
b j =Rj(1& jN )
d j = j.
(1)
To investigate persistence over long time periods, I will
consider only populations whose basic reproductive
number R is greater than 1. There are, however, interesting questions about time to extinction for populations
with R1 as well. For R>1, define the carrying
capacity K as the point at which birth rates are equal to
death rates. The carrying capacity is given by
K=(1&1R) N.
(2)
In the deterministic version of this model, the population
would reach an equilibrium of K individuals.
3. A HEURISTIC SCALING ARGUMENT
The key question is whether a population with birth
and death rates given by (1) will persist. Of course,
defining persistence is itself a tricky question. For the
infinite-habitat version of (1), where b j becomes Rj, it is
possible to ask simply whether a finite population will
persist forever or eventually go extinct. If N is finite,
however, the population will always eventually go
extinct, although the mean time to extinction may be
considerably longer than the expected lifetime of the
universe.
For a population that starts from a small number of
individuals, we can heuristically break the question of
persistence into two components. First, will the population establishthat is, reach or approach its carrying
capacity? Second, how long can we expect the population to survive if it does succeed in establishing?
For the question of establishment, we refer to the
infinite system, which can be solved exactly (Karlin and
Taylor, 1975). The probability of (infinite) persistence
starting from j 0 individuals in an infinite habitat is
P =1&1R j0.
(3)
If we think of the establishment question as representing
the density-independent component, of demographic
stochasticity, P makes sense as a heuristic estimate of
establishment.
To evaluate the probability of extinction of an established population, consider the population as performing
a biased random walk. Whenever the population is less
than the carrying capacity, the birth rate will be greater
than the death rate; thus the random walk will be biased
towards increasing population. In order to go extinct, the
population must wander against this bias from the
carrying capacity down to zero.
The question is then how far a biased random walk is
likely to stray against the direction of the bias. If the
number of steps S is reasonably large we expect the distribution of positions of a biased random walk to be
approximated by a normal distribution with mean &;S
and variance S(1&; 2 ), where ; measures the amount of
bias, defined here as the mean distance traveled per step.
To approximate the stochastic logistic model as a
biased random walk, we will look at the dynamics in
terms of the number of births or deaths that have
occurred (steps to the right and to the left), rather than
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Scaling Behavior of the Stochastic Logistic Model
in terms of time. It is straightforward to show that if the
population size is j, the probability that the next step will
be to the right is given by p j =b j (b j +d j ), and thus that
the expected value of the next step is given by
2p j &1=(b j &d j )(b j +d j ). When j is zero, and there is
no density dependence, and the mean distance traveled in
one step is given by R&1
R+1 . Since the population will have
to traverse the distance from the carrying capacity
(where there is no bias) to zero in order to go extinct, we
can assume, as a rough approximation, that the effective
R&1
.
mean bias is half of the bias at zero, so that ;r 2(R+1)
2
Assume that ; is small compared to 1, and imagine
that we can estimate a quantity, :, that tells us how many
standard deviations away from the mean our walk is
likely to achieve in some long period of time. The
standard deviation after S steps is approximately - S
and the distance that the population can walk against the
grain is given by : - S&;S. The maximum value of this
quantity occurs when S=: 24; 2, giving a distance
traveled of : 24;. At this point, we are not claiming to
have any ideas about the value of :. Instead, the point of
the heuristic argument is that we expect the characteristic
wandering distance of the biased random walk to be on
the order of 1;.
The scaling argument of this paper is based on the
hypothesis that the persistence of the population can be
predicted by K;, which is the distance that the population needs to wander in order to go extinct, given in units
of the characteristic wandering distance. If K is on the
order of the characteristic wandering distance 1;, then
we would expect the population to wander to extinction
within some reasonable period of time, while if K is much
larger than 1;, we would expect wandering to extinction
to be quite unlikely and thus to take quite a long time.
Hence, the hypothesis is that persistence of established
populations will be predicted by
U=K;=
(R&1) 2
N.
R(R+1)
(4)
In fact, since : 2 is proportional to U, the extinction probability or mean time to extinction might be expected to
be exponential in U, if U is not too small.
4. METHODS
It is straightforward to use a recursive algorithm to
calculate the mean time to extinction for the system (1),
starting from a single individual (Karlin and Taylor,
1975). The mean time to extinction is given by
:
j
1
k
.
` 1&
j k< j
N
\
+
(5)
Probability of persistence was calculated by directly
integrating differential equations for the probability of
the system (1) being in a given state, given that the population was not extinct. From the birth and death rates
(1) we can write differential equations for the probability
that there are exactly j individuals at a given time, f j :
f4 j =b j&1 f j&1 +d j+1 f j+1 &(b j +d j ) f j .
(6)
The probability of the population being j increases with
births from populations of j&1 and deaths from populations of j+1, and decreases with births and deaths out of
populations of size j.
The probability that the population has gone extinct
by a given time is given by f 0 , and the probability that it
has persisted, P, is given by 1& f 0 . We can therefore
model the distribution of individuals, given that the population has survived, using p j = f j P. Modeling the p j
instead of the f j improves numerical stability, and also
allows integration to a steady state, after which it is
possible to extrapolate exponentially to find the survival
probability to any desired time. Results were checked
by calculating the mean time to extinction from the
integration results, and comparing with the analytic
solution (5).
5. RESULTS
Figure 1 (top) shows the mean time to extinction for
populations in different-sized ``habitats'' (different values
of N) plotted against the rescaled birth rate, R. As would
be expected, if two populations have the same value of R,
the one with a larger value of N will survive longer. In
Fig. 1 (middle), the same values are plotted against
carrying capacity K, rather than against the birth rate. It
is useful to consider these two panels in terms of R and
K. Figure 1 (top) shows that if we hold R constant,
populations with a higher K survive longer, and Fig. 1
(middle) shows that if we hold K constant, populations
with a higher R survive longer.
In Fig. 1 (bottom) the same values are plotted yet
again, this time against the quantity U derived from the
random walk argument. U works surprisingly well as a
predictor of mean time to extinction. Away from U=0,
the line is also straight on a semilog plot, indicating that
MTE is roughly exponential in U, as predicted. In fact,
the scaling works too well to be explained by the heuristic
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FIG. 1. Mean time to extinction starting from a single individual,
plotted against reproductive number R, carrying capacity K, and the
scaling parameter U.
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Jonathan Dushoff
FIG. 2. Mean time to extinction starting from the steady-state distribution, plotted against reproductive number R, carrying capacity K,
and the scaling parameter U.
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Scaling Behavior of the Stochastic Logistic Model
argument alone; there must be some more detailed
reason why this scaling works, which may be worth
looking for.
Figure 2 shows the mean time to extinction starting
from the steady-state distribution. Once again the results
are plotted first against R, then against K, and finally
against U. Results are qualitatively similar to those in
Fig. 1. Surprisingly, the scaling does not work as well as
when we started from a single individual; this reinforces
the suspicion that we do not understand everything
about why the scaling works.
Figures 1 and 2 show that the quantity U is useful in
predicting the mean time to extinction of a model population with demographic stochasticity and density
dependence. This is useful for understanding models of
biodiversity. For models of single endangered species,
however, we may be more interested in the probability of
a population surviving for a fixed long time.
Since extinctions from the steady-state distribution are
distributed exponentially in time, we can calculate the
probability of survival from the steady state to a fixed
time T generations in terms of the mean time to extinction from the steady state M. The probability is
exp(&TM ). Since Mrexp(U ), we can approximate the
probability of survival to time T as
FIG. 4. Probability of survival for 1000 generations, starting from
a single individual, plotted against reproductive number R, together
with the theoretical curve for a population in an infinite habitat to
survive to time infinity.
This is an S-shaped curve which is steep in the range
where U is on the order of ln T. Figure 3 shows extinction
probabilities from the steady state, together with this
rule-of-thumb approximation.
The question of survival from a small (non-steadystate) population is more difficult. As discussed above,
the survival probability to time infinity for an infinite
habitat (Eq. (3)) should provide an upper bound if T is
large. Figure 4 shows the probability of survival for 1000
generations, starting from a single individual, together
with the theoretical curve for an infinite habitat and
infinite time. It is clear that the finite-habitat curves
approach the theoretical curves as R becomes large.
Based on the results for mean time to extinction, we
might expect that the scaling quantity U would be a good
predictor of the ratio between the observed survival
probabilities and the theoretical value for an infinite
habitat. Figure 5 shows the ratio PP plotted against U.
Although U does a much better job of predicting the
additional extinctions caused by a finite habitat than
either R or K alone, it is clear that there is still room for
FIG. 3. Probability of survival for 1000 generations, starting from
the steady-state distribution, plotted against U, together with the rule
of thumb (7).
FIG. 5. Relative probability of survival for 1000 generations,
PP , plotted against U, together with the approximation (7).
Srexp(&T exp(&U )).
(7)
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improvement. Part of the problem is probably the fact
that the expected relation between S and U (Eq. (7)) is
rather sensitive to small errors in approximation.
6. DISCUSSION
Mean time to extinction in a population following the
stochastic logistic model (1) can be predicted by a single
parameter, U, derived from a biased random walk
heuristic argument. This parameter provides a way of
linking spatial scale with the reproductive number of
species whose persistence is governed by demographic
stochasticity. Comparisons of predictions based on U
with data for subpopulation persistence can provide
evidence for the importance of demographic stochasticity
in comparison with that of other factors such as
trophic interactions, Allee effects, and environmental
stochasticity.
Although estimates of survival probability to a fixed
time based on the scaling parameter U are not as precise
as estimates of mean time to extinction, it is still possible
to develop a rule of thumb for when a population can be
expected to survive. For a population to be confidently
expected to survive, starting from a steady-state distribution, we want to be beyond the steep part of the rule-ofthumb curve in Fig. 3, meaning that U should be greater
than ln(T ). For the chosen time of 1000 generations,
ln(T )r7.
Estimates of survival probability from a small population are even less precise, but we can extend the rule of
thumb above. For a population to survive, it should be
reasonably protected from both wandering to extinction
from the steady state, and from failure to establish at all.
Thus we want U to be large compared to ln T, as before,
and we also want R j0 to be small, meaning that j 0 should
be large compared to 1(R&1). Note that these estimates
apply only to effects due to stochasticity in birth and
death rates. For many species, environmental stochasticity
may also be important, which could sharply reduce persistence estimates. For monoecious, sexual species, it is
known that effects such as failure to find mates and
unfavorable sex ratios are important for small populations (Dennis, 1989; Gabriel and Burger, 1992). This
problem can be partially corrected in this model by
counting only females (and thus halving the habitat size),
but this is only a rough approximation to a full sexual
model.
The scaling result also has some implications for
models of biodiversity of competing species. If the longterm survival probability is closely related to the quantity
UrN$ 2, then as the number of sites, N, increases, the
Jonathan Dushoff
amount of ``breathing space'' by which R must exceed
one for the species to persist decreases at the rate 1- N.
Hence in species diversity models based on the paradigm
of Tilman (1990) we would expect the number of species
found in the long run to increase roughly as - N.
This prediction provides a null model for speciesarea
relationships under Tilman's paradigm, under the
assumption that extinction rate is determined primarily
by environmental stochasticity (rather than disease or
environmental stochasticity, for example) and that
spatial structure may be ignored. Tilman's paradigm
postulates an absolute competitive hierarchy. It can
usefully be seen as one end of a spectrum of theories of
diversity of competing species, where Hubbell's (1997)
models of identical species are the other end of the
spectrum.
It should be pointed out that both Leigh (1981) and
Na# sell (1995) have already shown that N$ 2 is a useful
scaling quantity for this model, although neither one
found the further correction embodied in (4), which
seems to substantially improve results at this scale. The
work of Leigh also provides insight into why the
exponential approximation may work better for populations that start from a single individual than those that
start from carrying capacity, or from the equilibrium distribution. Future work may be able to incorporate his
ideas into the framework provided here to improve
results for populations starting from the equilibrium distribution. Both works contain more complicated
approximations which in fact work better than the
approximation given here.
It is hoped that the simple scaling relation here will
prove useful both in assessing the relative importance of
demographic stochasticity compared with other factors
contributing to local or global extinction in the field, and
in assisting with the analysis of more complex and
realistic models, which may have similar underlying
structure and scaling rules. It is further hoped that the
biased random walk heuristic will prove useful in understanding structurally more complex models, in particular
``SIR'' disease models, which consider the possibility of
acquired immunity to disease.
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