Grimm & Wissel
The intrinsic mean time to extinction: a unifying
approach to analysing persistence and viability
of populations
Volker Grimm1*, Christian Wissel2
UFZ Umweltforschungszentrum Leipzig-Halle
Sektion Ökosystemanalyse
Postfach 500 136
D-04301 Leipzig
Germany
Fax: ++ 341 235 3500
1
phone: ++ 341 235 2903, e-mail: [email protected]
2
phone: ++ 341 235 3245, e-mail: [email protected]
*
Correspondence author
To appear in OIKOS (accepted in October 2003)
1
Grimm & Wissel
2
ABSTRACT
Analysing the persistence and viability of small populations is a key issue in extinction theory
and population viability analysis. However, there is still no consensus on how to quantify
persistence and viability. We present an approach to evaluate any simulation model concerned
with extinction. The approach is devised from general Markov models of stochastic
population dynamics. From these models, we distil insights into the general mathematical
structure of the risk of extinction by time t, P0(t). From this mathematical structure, we devise
a simple but effective protocol – the ln(1-P0)-plot – which is applicable for situations
including environmental noise or catastrophes. This plot delivers two quantities which are
fundamental to the assessment of persistence and viability: the intrinsic mean time to
extinction, Tm, and the probability c1 of the population reaching the established phase. The
established phase is characterized by typical fluctuations of the population’s state variable
which can be described by quasi-stationary probability distributions. The risk of extinction in
the established phase is constant and given by 1/Tm. We show that Tm is the basic currency for
the assessment of persistence and viability because Tm is independent of initial conditions and
allows the risk of extinction to be calculated for any time horizon. For situations where initial
conditions are important, additionally c1 has to be considered.
Grimm & Wissel
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INTRODUCTION
Persistence and viability are the fundamental properties of populations in extinction theory
and population viability analysis (PVA). Surprisingly, however, there is no consensus on how
to quantify persistence and viability, especially in simulation models. Extinction theory
usually focuses on the arithmetic mean time to extinction, T (Leigh 1981, Wissel 1983,
Goodman 1987, Lande 1987, 1993, Wissel and Stöcker 1991, Grasman 1996; but see Sæther
et al. 2000, Engen et al. 2001). However, the usage of T has been criticized because the
distribution of extinction times is skewed and therefore the qualifier ‘mean’ would be
misleading. Therefore, it has been argued that the median time to extinction is a more
significant measure of persistence than the mean (e.g. Goodmann 1987, Groom and Pascual
1998). The significance of T has also been questioned because T may be strongly biased by
the initial state of the population and therefore may not really capture the intrinsic ability of a
population to persist (Ludwig 1996).
This disagreement on how to quantify persistence is even more pronounced in PVA.
Here, T is often considered as inappropriate because mean times to extinction of thousands of
years or more seem to project too far in the future (Beissinger and Westphal 1998). PVA
therefore mostly refers to viability, a concept introduced by Shaffer (1981): a population is
considered as viable if the risk of extinction, P0(t), over a certain time horizon t does not
exceed a certain threshold. However, the choice of both the time horizon and the threshold is
arbitrary. This makes it difficult to compare viability analyses which are based on different
time horizons and thresholds. For example, Marshall and Edwards-Jones (1998) use a
threshold of 5% in 50 years to assess the minimum viable population (MVP; Shaffer 1981) of
Tetrao urogallus (capercaillie) in Scotland, whereas Grimm and Storch (2000), modelling
T. urogallus in the Bavarian alps, use 1% in 100 years. The MVPs resulting from these two
studies cannot be directly compared.
Some argue that the time horizon t for assessing viability has to be large if long
generation times are involved (e.g. elephants, Armbruster and Lande 1993, Armbruster et al.
1999). Others argue that only time horizons of a few decades are relevant because model
parameters cannot be assumed to be constant for longer times, and because uncertainties in
model parameters would accumulate for longer time horizons and thus render any model
prediction worthless (Beissinger and Westphal 1998). And as with T , initial conditions, i.e.
Grimm & Wissel
4
the initial state of the model population, may bias the assessment of P0(t). There is no clear
methodology to distinguish between the intrinsic ability of a population to persist in a certain
landscape and the effect of its initial state. But the usage of extinction risk, P0(t), has even
been questioned in the first place, because populations may become quasi-extinct, i.e. no
longer detectable or no longer capable of recovering, at non-zero population sizes. So-called
quasi-extinction curves specifiy the risk that the population drops below a certain threshold
population size during a certain time horizon (Ginzburg et al. 1982, Burgman et al. 1993,
Fiedler and Kareiva 1998, Goldwasser et al. 2000, Engen and Sæther 2000, Coulson et al.
2001), but it is not clear how quasi-extinction risk is related to the risk of real extinction.
Some even argue that rather than extinction or quasiextinction risk the distribution of
population sizes at the end of the simulated time horizon is more appropriate for the purposes
of PVA (e.g. Beissinger and Westphal 1998).
But even if there were no debate about when and how to use T and P0(t), what is the
relationship between these two quantities? There is, indeed, a theoretical relationship (e.g.
Renshaw 1991, Verboom et al. 1991, Stephan 1993, Lande 1993, Foley 1994, Mangel and
Tier1994, Wissel et al. 1994, Vucetich and Waite 1998, Gosselin and Lebreton 2000) which
we present below. But this relationship has not yet been fully acknowledged, let alone utilized
in extinction theory and PVA. Moreover, it is not clear how general this relationship is and
how it can be verified for specific simulation models. The problems with assessing
persistence and viability summarized above are due to the fact that all models in extinction
theory and most models in PVA are stochastic and therefore produce probabilistic results.
What is needed is a unifying currency to evaluate stochastic population models.
This lack of a unifying currency is linked to the lack of a general theoretical
framework for the models used in extinction theory and PVA. On the one hand, there are
abstract mathematical models of hypothetical species which use differential or difference
equations. These models usually address general problems of population extinction and are
designed to gain general insights. On the other hand, there are very detailed simulation
models (individual-based, spatially explicit) which are designed to give a rather realistic
description of a specific population in a specific landscape. There are all kinds of intermediate
sorts of models between these two extremes, but still the question is: what is the relationship
between the insights gained from these different modelling approaches, which are seemingly
fundamentally different? Do the results of mathematical models which ignore virtually all
Grimm & Wissel
5
biological details help analyse the detailed models? Is it possible to obtain general insights
from the very specific detailed models?
As a possible solution to all these problems, here we introduce a general protocol for
dealing with all kinds of models addressing population extinction. Our approach to develop
this protocol is: First, from the theory of Markov processes we distil general insights into the
structure of the extinction process. Second, we translate these insights into a protocol of how
to analyse stochastic simulation models in general. Third, we test the protocol using complex
simulation models. Finally, the insights from the protocol are used to formulate three general
concepts: the ‘intrinsic mean time to extinction’, which provides a unifying currency for
extinction theory and PVA, the ‘established phase’ in which populations have ‘forgotten’
specific initial conditions and therefore show typical fluctuations of the state variables, and a
more general notion of viability. The protocol and concepts we present will not solve the
problems of extinction theory and, in particular, PVA per se (e.g. the lack of appropriate data
and the problem of model validation). However, our approach could contribute to focusing on
these problems instead of arguing about problems which, as we show, do not really exist.
THE GENERAL STRUCTURE OF THE EXTINCTION PROCESS
In the following, we present the theoretical background of our approach to evaluate stochastic
simulation models of population dynamics. Readers who are not interested in mathematical
details (which are well-established in the theory of Markov processes; e.g. Verboom et al.
1991 and references therein) may skip the following considerations and proceed directly to
Eqn. (6). We use a simple Markov population model to demonstrate the general mathematical
structure of the probability P0(t) that a population is extinct by time t. This general structure is
then used as a theoretical prediction, i.e. we expect to find the same structure in more realistic
simulation models, too. The protocol developed in the following section will allow us to
check whether this prediction holds for more complex simulation models. ‘Markov model’
means that we consider the stochastic population dynamics as a Markov process, i.e. a process
without memory: the current change of the population does not depend on earlier states but
only on the current state of the population. We will discuss later why this assumption does not
limit the generality of the following considerations.
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Grimm & Wissel
We consider a population of n individuals. The probability Pn(t) of having n
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individuals at time t is determined by the so-called master equation (Markov process of birth
and death type):
dPn (t )
= bn −1 Pn −1 (t ) + d n +1 Pn +1 (t ) − bn Pn (t ) − d n Pn (t ) ,
dt
(1)
which is a simple bookkeeping of the probabilities that the population had the same size n one
time step dt earlier or that it had one individual more (or less). dndt and bndt are the
B
B
B
B
probabilities of the population size n decreasing (death) or increasing (birth), respectively, by
one individual in the infinitesimal time interval dt. For the birth and death rates bn and dn,
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B
B
B
B
several submodels can be chosen, such as the logistic equation (Goodman 1987a,b, Wissel
B
and Schmitt 1987, Wissel and Stöcker 1991, Wissel et al. 1994), although the particular
choice of submodels for bn and dn is not relevant for the following considerations. Eqn. (1)
B
B
B
B
can be rewritten with the help of a tridiagonal matrix A where the elements of this so-called
transition matrix, An,m, are defined by bn and dn :
B
B
B
dPn (t )
=
dt
(2)
B
B
B
∑ An, m Pm (t ) ,
m
with m running from 1 to some ceiling population size. In the following it is sufficient to
consider only n>0 because P0(t) follows from the normalization of Pn(t), i.e.
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B
B
B
P0 (t ) = 1 − ∑ Pn (t ) . Eqn. (2) is a linear matrix equation and therefore its solution Pn(t) can be
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B
n >0
written with the help of the eigenvalues and eigenvectors of the matrix A:
Pn (t ) = ∑ u n, i ci e −ωi t
(3)
i
It can be shown that all eigenvalues –ω i of the matrix A are real and negative (Wilkinson
B
B
1965). un,i is the n-th component of the normalized right-hand eigenvector of A
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B
corresponding to the i-th eigenvalue –ω i , and ci is the inner product of the corresponding
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left-hand eigenvector with the initial condition, Pn(t=0). We applied a very robust method to
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Grimm & Wissel
solve numerically the eigenvalue problem for the matrix A for different submodels of bn and
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dn, for example the stochastic logistic equation (Wissel et. al. 1994). In virtually all cases we
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found that the absolute first eigenvalue is much smaller than the absolute value of higher
eigenvalues:
ω 1 << ω i
(4)
for i > 1
Note that Eqn. (4) refers to properties of the matrix A and does therefore not consider effects
of initial conditions, Pn(t=0). Since all eigenvalues –ω i of A are negative, Eqn.(4) means that
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B
B
B
in Eqn. (3) the contributions of the exponential functions with the higher eigenvalues (i>1)
quickly fall to zero and the dynamics of the probability Pn(t) of the population having n
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individuals at time t is dominated by the so-called dominant eigenvalue –ω1. Thus, after a
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short transient time (which has a duration of the order of magnitude Tt =ω 2 -1<< ω 1 -1), the
B
B
B
PB
P
B
PB
P
solution Pn(t) may be approximated as:
B
(5)
B
Pn (t ) = u n,1 c1 e
−ω 1 t
,
for n > 0 and t > Tt
Since the eigenvector un,1 is normalized, i.e.
B
B
∑ u n,1 = 1,
n >0
we obtain
∑ Pn (t ) = c1e
−ω1t
n >0
which is the probability that the population size n is larger than zero. Hence, the probability
P0(t) that n=0 is:
B
B
(6)
P0 (t ) = 1 − c1 e
−ω 1 t
.
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Grimm & Wissel
The state n=0, however, is simply extinction and thus Eqn. (6) provides the mathematical
structure of the risk of extinction by time t. This mathematical structure is determined by a
negative exponential with two constants, c1 and ω 1, which have a clear ecological meaning
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explained below. Eqn. (6) provides the basis for a general protocol which allows us both to
determine the two constants from simulations and to check whether the general structure
described in Eqn. (6) holds for specific models.
A GENERAL PROTOCOL FOR EVALUATING SIMULATION MODELS
The first steps of the protocol to evaluate any simulation model concerned with extinction are
as follows: Run the stochastic simulation model, which uses discrete time steps, say, 1,000
times. Start each simulation with the same initial state of the population. Run the simulations
for a maximum time of, say, 1,000 years. For each run, record the time when the population
went extinct. Then, assemble a histogram of extinction times. All simulations where the
population still exists after 1,000 years are ignored in the histogram. Fig. 1A shows a typical
histogram of extinction times gained from a metapopulation model of a grasshopper species
(Bryodema tuberculata, Stelter et al. 1997). The model includes demographic and
environmental noise at the patch level and a dynamic landscape composed of suitable habitat
patches with varying carrying capacity. The typical shape of this histogram, i.e. the
exponential decay, can be found in virtually all stochastic population models (Burgman et al.
1993) and corresponds to the mathematical structure of P0(t) (Eqn. (6)).
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The next steps of the protocol are: Divide all histogram bars by the total number of
simulation runs. This normalizes the histogram and the bars give an estimate of the
probability p(t’) of the population going extinct in the time interval t’. From this we obtain the
cumulative probability P0(t) of the population being extinct by time t by cumulation (i.e. the
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B
bars of the normalized histogram are successively added up to time t):
t
(7)
P0 (t ) = ∑ p (t ' )
t '= 1
Note that this probability may be influenced by the initial conditions of the simulation and
contains different sampling errors for different values of t because of the finite number of runs
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Grimm & Wissel
used to estimate this probability. A typical example resulting from Fig. 1A is shown in
Fig. 1B.
Because of our considerations above we expect this probability determined from a
number of repeated simulation runs to show the structure of Eqn. (6). In order to check this
prediction, we rewrite Eqn. (6) as:
(8)
− ln(1 − P0 (t )) = − ln(c1 ) +
t
Tm
where we have defined the time Tm as:
B
(9)
Tm =
B
1
ω1
Thus, the theoretical prediction behind Eqn. (6) is that a plot of –ln(1-P0(t)) versus time t
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should yield a straight line (Stephan 1993, see also Fig. 1B in Verboom et. al. 1991). We refer
to this plot as the ‘ln(1-P0)-plot’ hereafter. The predicted linearity can indeed be seen in our
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example in Fig. 1C. It is very easy to check whether a linear relationship is obtained in a
particular case. Thus, the ln(1-P0)-plot is a simple test if, in the model under consideration,
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P0(t) has the general mathematical structure as described in Eqn. (6). If this is so, the
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ecological interpretation of the two parameters of Eqn. (6) developed below should also hold.
Of course, due to the finite number of simulation runs some deviation from a straight line has
to be expected. But the higher the number of runs, the better the fit to a straight line.
Since the ln(1-P0)-plot is a linear plot, the parameters c1 and Tm (or ω1, respectively)
B
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B
B
B
B
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are easily obtained: c1 can be determined from the intercept with the y-axis, which is –ln(c1),
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and the characteristic time Tm is given by the inverse of the slope. But what is the ecological
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meaning of these two parameters? We obtain an initial idea of their meaning if we repeat the
above protocol using different initial states, for example extremely good, extremely bad, or
intermediate ones. ‘Good’ and ‘bad’ means that the population in that state can be qualified
intuitively as having a very low or very high short-term risk of extinction. In our example
(Fig. 2), ‘good’ corresponds to a state where all patches have maximum capacity and are
occupied by a subpopulation; ‘bad’ means that almost all patches have minimum capacity and
Grimm & Wissel
10
only a few of them are occupied. Note that a bad state may turn to better state after some time
because catastrophic disturbance events create new patches of maximum capacity (Stelter et
al. 1997). Fig. 2 shows that the slope of the straight line and thus the parameter Tm does not
seem to depend on the initial state, whereas the intercept and, in turn, c1 depends on the initial
state. Thus, Tm is an intrinsic property of the population whereas c1 is an indicator of the
‘quality’ of the initial state. What Fig. 2 also shows is that the simple arithmetic mean of the
length of all simulation runs (i.e. until extinction), T , reflects both initial conditions and the
ability of the population to persist. T does not, therefore, allow one to isolate these two
factors that influence population persistence.
ESTABLISHED PHASE AND THE INTRINSIC MEAN TIME TO EXTINCTION
We mentioned above that Eqn. (6) only holds after a transient phase. This phase is predicted
by Eqn. (4) to be very short if no ‘extreme’ initial conditions of the simulation are chosen (we
discuss the general case, including extreme initial conditions, in the next section below). In
the ln(1-P0)-plot this is reflected by the fact that not all points of the plot form a straight line
right from the beginning, but will approach a straight line after a time Tt which is the duration
of the transient phase (see also Fig. 1B in Verboom et. al. 1991). This transient phase,
however, may be so short that it may not be detected in the ln(1-P0)-plot. In this case the
regression line of the ln(1-P0)-plot will virtually start at the origin.
But how is a population characterized once the transient phase is finished? Intuitively,
one would call such a population ‘established’ because it has reached, for all its state
variables, its characteristic range of fluctuations. In particular, the state variables – e.g.
number of individuals, age, size or spatial distribution – no longer reflect an echo of the initial
state. Now, for any real population which is known to have persisted for some generations
and which did not experience extreme disturbance events recently, we can reasonable assume
– even if we don’t know its exact state – that it is established. We therefore refer to the phase
following the transient phase as the ‘established phase’ (for a nice visualization of the
transient and established phase, see Fig. 1 in Halley and Dempster 1996). Note that
‘established phase’ does not refer to a certain state of a population, which usually is not
known exactly anyway, but to a probability distribution which predicts the state variables. A
population has reached the established phase if the probability distributions of the state
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Grimm & Wissel
variables describing the population are quasi-stationary. If this is so, the risk of extinction per
short time interval is constant leading to the exponential decrease in Pn(t) (Appendix A). It
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can also be shown analytically (Appendix B) that Tm is in fact equal to the arithmetic mean
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time to extinction if initially, at time t=0, the population is already in the established phase.
This theoretical prediction is easy to test with simulation models (Fig. 2B).
Tm can, as we have shown, easily be determined from the slope of the regression line
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in the ln(1-P0)-plot. Since Tm does not depend on the initial state of the population but is an
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intrinsic property of a population in a given environment (e.g. Fig. 2), we propose referring to
Tm as the ‘intrinsic mean time to extinction’. For mathematical models based on a transition
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matrix, Tm is defined as the absolute of the inverse of the first eigenvalue of the transition
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matrix. For simulation models, Tm is determined as the inverse of the slope of the regression
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line of the ln(1-P0)-plot. We chose the term ‘intrinsic’ because of the analogy to the intrinsic
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rate of increase of populations, which also can be determined as an eigenvalue of a transition
matrix (the Leslie matrix).
THE PROBABILITY OF REACHING THE ESTABLISHED PHASE
Fig. 2 indicates that the coefficient c1 depends on the initial state of the population. First of
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all, two cases have to be differentiated: a positive or a negative intercept of the regression line
of the ln(1-P0)-plot with the y-axis. If the intercept is positive, i.e. –ln(c1) > 0, c1 is between 0
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and 1 and can thus be interpreted as a probability. From our example, i.e. the mathematical
model of Eqn. (1), we can deduce an appropriate ecological interpretation of the probability
c1: consider Eqn. (5)
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B
Pn (t ) = u n,1 c1 e
− t
Tm
,
where ω1 has been replaced by 1/Tm. The eigenvector un,1 describes the probability
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B
B
B
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B
distribution of finding, in the established phase, the population size n (Wissel et. al. 1994). On
the other hand we know, by definition of the transient phase, that at the end of this phase, i.e.
after the time Tt, the population has reached the established phase or has gone extinct. The
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probability of finding the population size n at time Tt is thus:
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B
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Grimm & Wissel
Pn (Tt ) = u n,1 c1 e
T
− t
Tm
.
It follows that the factor c1exp(-Tt/Tm) describes the probability of reaching the established
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phase. For Tt<<Tm this factor may be well approximated by c1. For populations which are
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intuitively considered viable, Tm will be of the order of magnitude of 10,000 years (see
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section on viability below) and therefore the condition Tt<<Tm should be fulfilled. In such
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cases c1 as determined by the ln(1-P0)-plot is a good approximation for the probability of
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reaching the established phase. Note, however, that in Fig. 2 we deliberately chose the
extreme cases of very low intrinsic mean times to extinction. Therefore in Fig. 2C, the
transient time Tt until the ln(1-P0)-plot approaches the straight line is of the same order of
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magnitude as Tm. In such extreme cases, c1 is not a good approximation of the probability of
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reaching the established phase but has to be corrected by the factor exp(-Tt/Tm).
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If initially the state variables of the population are within the range of fluctuations
described by the quasi-stationary probability distribution of the established phase, then c1≈1.
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The theoretical prediction is thus that for populations which are initally within the range of
fluctuations of the established phase, the risk of extinction is largely independent on the
peculiarities of the initial size and structure. This may be hard to understand intuitively but
can be checked with simulations. Fig. 3 shows results of an age-structured stochastic
simulation model of Tetrao urogallus (capercaillie; Grimm and Storch 2000). The risk of
extinction after 100 years, P0(100), depends on initial population size N0 up to about 250
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individuals, but for N0>250, the remaining dependence of P0(100) on N0 can, although still
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existing, largely be ignored and therefore the initial population considered as belonging to the
range of the fluctuations of the established phase.
However, populations may initially, at time t=0, not only be in a ‘bad’ state but may
also be in a ‘good’ state with practically no short-term probability of extinction. In the
ln(1-P0)-plot, a ‘good’ initial state leads to an intercept with the x-axis that is positive. This is
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the second case of the case differentiation regarding the intercept. Now we can write c1 > 0 as
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c1 = eTr / Tm which can be taken as an equation for defining a time Tr. With this definition
B
Eq. 6 can be rewritten as
P0 (t ) = 1 − e −(t −Tr ) / Tm
.
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Grimm & Wissel
This shows that Tr is a time which shifts the regression line in Fig. 2 to the right, yielding an
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intercept with the x-axis of Tr. In this ‘good’ case, the population starts with large individual
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numbers above the typical range of fluctuations of the established state. It takes some time
until the population reaches the established state because a net number of individuals has to
die, and this time is exactly given by the positive intercept with the x-axis in the ln(1-P0)-plot
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(Fig. 2a). In conservation biology, this time has been referred to as ‘relaxation time’ (Brooks
et al. 1999; see also Ovaskainen and Hanski 2002).
There may be several reasons why a population at a certain time t=0 is not established.
First, the population may be ‘over-established’, i.e. the short-term extinction risk may be
smaller than for populations which are established. This is, for example, the case if in a
metapopulation a certain fraction, e.g. 50%, of the habitat patches is lost due to sudden natural
or anthropogenic changes. Since the fraction of occupied patches decreases with the total
number of patches, it will take some time until the population has reached the established
phase (this is exactly the time delay responsible for the ‘extinction debt’ discussed in simple
models of metapopulation communities; e.g. Tilman 1994). Second, the initial short term
extinction risk may be larger than for the established phase due to an extreme disturbance
event (catastrophe) which may have diminished population size to an unusually small value.
In this case c1 describes the probability of recovery to establishment. Another reason may be
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that a population is in a colonizing phase. For instance, this appears during a recolonization of
a habitat patch of a metapopulation (Hanski 1999). In this case c1 gives the probability of
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colonization being successful which means that the population becomes established.
However, if a population exists for some time under certain environmental conditions
it can reasonably be assumed that it is in the established phase (see also Day and Possingham
1995). This situation will be investigated in more detail in the following section.
THE RELATIONSHIP BETWEEN Tm
B
AND P0(t)
B
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B
If we consider a population which is established at t=0, i.e. c1≈1, Eqn. (6) becomes:
B
(10)
P0 (t ) ≈ 1 − e
− t
Tm
.
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Grimm & Wissel
It has been argued that the (intrinsic) mean time to extinction Tm is not the quantity of interest
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in PVA, but rather that the probability of extinction P0(t) is the decisive quantity. Although
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we agree, it is evident from our above considerations that Tm is a signifier of an established
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population which is of central importance. Knowing Tm, P0(t) of the established population
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can easily be calculated for any time horizon t of interest using Eqn. (10). Thus the discussion
about a proper time horizon for P0(t) is superfluous.
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Tm can be considered as a general currency for PVA since the extinction risk P0(t) can
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be determined from it for any desired time horizon t. This is especially easy if P0(t)<<1,
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which is of special importance for PVA. Using this and a Taylor expansion, Eqn. (10)
simplifies to:
P0 (t ) ≈
(11)
t
Tm
This means that for times t much smaller than Tm, the risk of extinction P0(t) increases
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approximately linearly. This simple relationship shows that the consideration of the intrinsic
mean time to extinction Tm per se may be misleading. A value of Tm =1,000 years may give
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the impression that there is no significant risk of extinction over time horizons of, say,
decades or 100 years. But this value of Tm is equivalent to a risk of extinction of P0(100)≈0.1
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for a time horizon of 100 years. Thus, Tm (or its inverse) should be considered as: (1) in
B
B
extinction theory the currency for comparing the ability of populations to persist, and (2) in
PVA as the currency for determining viability, i.e. the risk of extinction for certain time
horizons.
Note, however, that when initially the population is not in the established phase the
risk of extinction P0(t) must be determined from the full Eqn. (6) which includes c1. In such
B
B
B
B
cases, once c1 is known, Tm and Eqn. (6) can be used to calculate P0(t) for any time horizon of
B
B
B
B
B
B
interest.
GENERALIZING VIABILITY
For established populations the general currency Tm also provides the basis for overcoming
B
B
the arbitrariness in the definition of a minimum viable population (MVP; Shaffer 1981) or of
15
Grimm & Wissel
viability in general. Let us suppose that the demand in viability is that P0(t) does not exceed a
B
B
risk limit Pmax for times smaller than a chosen time horizon, tmax:
B
B
B
Po (t ) ≤ Pmax
(12)
B
for t ≤ t max .
If the risk limit is small, i.e. Pmax<<1, we obtain from Eqs. (11) and (12) that the demand of
B
B
Eqn. (12) is equivalent to the demand:
Tm ≥ Tmin ≈
(13)
t max
Pmax
For instance, the choice of a risk limit Pmax = 0.01 or 1% for a time horizon of tmax = 100 years
B
B
B
B
is equivalent to the demand that the intrinsic mean time to extinction Tm must be at least
B
B
100/0.01=10,000 years. Now let us suppose that we have used the ln(1-P0)-plot to determine
B
B
Tm for different values of a model parameter, for example the carrying capacity K. That means
B
B
that we have produced a plot of Tm versus K as shown, for a model of Tetrao urogallus in the
B
B
Bavarian Alps (Grimm and Storch 2000), in Fig. 4. The demand of a risk limit Pmax for a time
B
B
horizon tmax, which is equivalent to a demand for Tm in Eqn. (13), can now be transformed to a
B
B
B
B
demand for the parameter K: K has to exceed Kmin (Fig. 4).
B
B
This provides a simple procedure for determining whether an established population is
viable without having to predefine the risk limit Pmax and the time horizon tmax. It is sufficient
B
B
B
B
to provide a Tm-plot as shown in Fig. 4. Using the simple relationship of Eqn. (13), everyone
B
B
can choose their own Pmax and tmax and can determine the corresponding demand for the
B
B
B
B
parameter K.
MARKOV PROCESSES AND COMPLEX SIMULATION MODELS
We found the ln(1-P0)-plot to be very robust. It has been applied to both simple and complex
B
B
models of wide ranging complexity, including, for example, age structure (Wiegand et. al.
1998, Grimm and Storch 2000), territorial behaviour (Bender et al. 1996), social behaviour
(Neuert et al. 1995, Grimm et al. 2003), and metapopulations in dynamic landscapes (Stelter
et al. 1997). In all these applications, the ln(1-P0)-plot could be fitted by a straight line. Given
B
B
Grimm & Wissel
16
the assumption of a Markov process, i.e. of a stochastic process without memory, it is perhaps
surprising how well the plot works even for complex models which seem to violate the
Markov assumption, e.g. models with age structure or with succession within habitats. There
is, however, a theoretical explanation as to why this violation of the Markov assumption does
not occur more often: possible memory effects in population dynamics can be described by a
Markov process if enough state variables are introduced which carry this memory. Typical
examples include age, weight, size or other covariates of fitness. If these additional state
variables carry the memory of earlier states of the population, the basic assumption of Markov
processes that the current change only depends on the current state still can be used.
DISCUSSION
The purpose of this paper was to unify the numerous different approaches of extinction theory
and PVA to quantifying the persistence and viability of populations. To this end, we distilled
the general mathematical structure of the extinction process, i.e. Eqn. (6), from the theory of
Markov processes. This mathematical structure, which is characterized by a negative
exponential with two parameters, c1 and Tm , suggested a simple but effective protocol – the
ln(1-P0)-plot – to evaluate stochastic simulation models of population dynamics. The ln(1-P0)plot delivers the two parameters c1 and Tm, which turned out to be fundamental to describe the
extinction process or, respectively, the persistence and viability of populations.
An important feature of the ln(1-P0)-plot is that it automatically provides a test as to
whether it can be applied in the first place: if the plot is linear, the mathematical structure of
Eqn. (6) is given and therefore the ecological interpretation of the parameters c1 and Tm is
adequate. We applied the ln(1-P0)-plot to models of both hypothetical (Johst and Wissel 1997)
and real species, and the model structure varied from very simple (e.g. a simulation of the
model of Eqn. (1)) to very complex. In all these cases, the ln(1-P0)-plot was linear. This
finding unifies the theoretical models of extinction theory with the more realistic models of
PVA: the Markov property of stochastic processes, i.e. a process without ‘memory’, which is
used in theoretical models, still exists inherently in realistic models because here the memory
is taken into account by additional state variables (e.g. age, size, habitat quality, etc.).
c1 and Tm are basic quantities in extinction theory and PVA. Tm is not affected by
initial conditions of the simulations and therefore describes the intrinsic ability of a
population to persist in a certain landscape. We therefore referred to Tm as the ‘intrinsic mean
Grimm & Wissel
17
time to extinction’. The qualifier ‘intrinsic’ is appropriate because Tm is as fundamental to
stochastic population dynamics as the intrinsic rate of increase, rm, to deterministic population
dynamics (Cooper 1984). The median time to extinction is by no means more fundamental or
appropriate than Tm, because it can easily be shown that the median time is equal to Tm ln(2)
(Appendix C).
In conservation biology, Tm is the general currency to define viability. Soulé (1987)
defines viability as the capacity to „maintain itself ... for the foreseeable ecological future
(usually centuries) with a certain, agreed on, degree of certitude, say 95%“ (p.2). This
definition refers to a principle, inherent, and hence ‘intrinsic’ ability of a population which is
established. However, referring to Tm as being intrinsic and the general currency does not
mean that initial conditions may always be neglected. There are cases where initial conditions
are important, e.g. after catastrophes and during colonization or after reintroductions. In such
cases, the population cannot realise its intrinsic ability to persist because it is too small (the
same is true for Leslie matrices and initial age structures which are ‘worse’ than the stable
exponential distribution). Therefore, the full Eqn. (6), including the probability of reaching
the established phase, which is approximated by c1, has to be considered to determine the risk
of extinction P0(t). The real achievement of the ln(1-P0)-plot is that it allows to distinguish
clearly and for any kind of stochastic population model between intrinsic and initial aspects.
For many problems in extinction theory and PVA, the focus will be on the intrinsic property,
but for certain classes of problems, the focus will be on effects of initial conditions.
In the definition of viability given by Soulé (see above), both the threshold risk one is
willing to accept and the time interval of consideration cannot be deduced scientifically, but
are more or less arbitrary. However, the currency Tm is an intrinsic property, independent of
initial conditions, thresholds and time horizons and therefore allows different assessments of
viability to be compared. The purpose of Tm is not to project thousands of years into the
future, but to provide a currency to quantify the risk of extinction P0(t) for any, usually short,
time horizon of interest, e.g. t=20, 50, or 100 years.
c1 describes the probability of a population to reach the established phase, i.e. to
achieve quasi-stationary probability distributions of the populations’ state variables (including
population size) which are linked to a risk of extinction per short time interval which is
constant (and equal to 1/Tm). We would like to emphasize that ‘establishment’ by no means
implies a quantitative assessment of persistence or viability. Established populations may
Grimm & Wissel
18
have a very short or a very long intrinsic mean time to extinction. The question with c1 is
whether a population, which starts from a certain initial state, will realise its intrinsic ability to
persist in the given landscape, or whether it will go extinct beforehand.
Certainly, there may be problems in extinction theory and PVA where the approach
presented here might not be appropriate. For example, Vucetich and Waite (1998) mention
alternatives to the exponential distribution of extinction times; Halley and Kunin (1999)
discuss the influence of the distribution of environmental noise on persistence, e.g. red or pink
instead of white noise. However, the ln(1-P0)-plot by itself provides a test of its applicability
and there is therefore no risk of it being applied erroneously. Moreover, it is easy to check
whether Tm is a good predictor of the arithmetic mean of extinction times of established
populations, i.e. if all simulations are run until their ‘bitter end’ (extinction). Likewise, it can
easily be checked whether P0(t), as calculated from Tm using Eqn. (10) or (11) (provided
simulations are started with established populations), matches P0(t) as determined directly
from simulations over a fixed time horizon (Burgman et al. 1993).
Our general approach here was first to distil general properties of a general process
which may be described by simple mathematical relationships (Eqn. (6)) and then to use this
relationship to evaluate more detailed and complex simulation models. This, precisely, is
what Grimm (1999) called for in order to unify population level and individual level
ecological models. Abstract population models deliver an integrated view of how to analyse
individual-based models. Individual-based models, on the other hand, allow the significance
of biological details, e.g. individual and environmental variation and behaviour, to be
analysed, so that the (mathematical) limitations of abstract population models can be
overcome. Thus, what we tried to show in this paper is that the unification of different
modelling approaches leads to synergistic effects, i.e. it enables theoretical, conceptual and
methodological problems to be solved that could not be elucidated with either abstract or
detailed models alone.
ACKNOWLEDGEMENTS
We would like to thank: T. Stephan, the ‘pioneer’ of the ln(1-P0)-plot, for discussions and
advice; H. Hildenbrandt for implementing the software that was used to produce Figs. 2-4;
and L. Fahse, S. Higgins, H. Hildenbrandt, R. Lande, K. Johst, G. Nachmann, E. Revilla,
Grimm & Wissel
B.-E. Sæther, T. Stephan, K. Wiegand for their valuable comments on earlier drafts of this
manuscript.
19
20
Grimm & Wissel
APPENDIX A
The constant probability of extinction per short time interval is given by the first eigenvalue
ω1 = Tm-1 of the transition matrix A.
B
B
B
PB
P
To show that a constant risk of extinction per short time interval dt and the exponential
structure of Eqn. (5) are equivalent, consider the probability Ps(t) that the population is not
B
B
extinct but still survives at time t:
Ps (t ) = 1 − P0 (t ) = ∑ Pn (t ) .
n>0
If there is a constant probability ω1 of extinction per time interval dt we get, with (1-ω1dt)
B
B
B
B
being the probability of survival in dt:
Ps (t + dt ) = Ps (t )(1 − ω1dt )
which for infinitesimally small dt corresponds to:
dPs (t )
= −ω1 Ps (t )
dt
with the solution:
Ps (t ) ∝ e −ω1 t ,
which corresponds to the mathematical structure of Eqn. (5).
APPENDIX B
To prove the inverse relationship between ω1 and the intrinsic mean time to extinction, Tm,
B
B
consider the probability that extinction occurs within the time interval [t, t+dt], which is
Ps(t)ω1 dt. Therefore the mean time to extinction is :
B
B
B
B
B
B
Grimm & Wissel
∞
Tm = ∫ t Ps (t )ω1dt
0
∞
= ∫ e −ω1 t tω1dt
0
=
1
ω1
.
APPENDIX C
The median time to extinction, TM, is defined by
B
B
1
⇔
2
1
=
⇔
2
= ln(2)
P0 (TM ) =
e −TM / Tm
TM
Tm
which follows, for the established phase (c1≈1), from Eqn. (6).
B
B
21
Grimm & Wissel
22
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Grimm & Wissel
26
FIGURE LEGENDS
Figure 1:
The general protocol to evaluate stochastic simulation models of small populations. A,
Histogram of extinction times. B, Cumulative probability of extinction by time t, P0(t),
obtained from A by normalization and cumulation (see text). C, ln(1-P0)-plot, i.e. the plot of
-ln(1-P0(t)) versus time t (points) and a linear regression line with a slope of 1/220 years-1
(modified after Stelter et al. 1997).
Figure 2:
ln(1-P0)-plot applied to the same metapopulation model as in Fig. 1 (Stelter et al. 1997;
parameters changed to obtain shorter times to extinction), but for different initial conditions.
The left panels show the initial state of the population and of the landscape (patches). A, All
patches occupied and with maximum capacity. B, a state of intermediate quality. C, an
extremely poor initial state. The right panels show the corresponding frequency distribution of
times to extinction from 1,000 runs of the model, the arithmetic mean times to extinction,
T (in years), the ln(1-P0)-plot, and the intrinsic mean times to extinction, Tm, which is largely
independent of the initial conditions. Note that in the model includes environmental variations
and catastrophes and that patch capacity is not static but may become maximal after
catastrophes and then decrease linearly due to succession.
Figure 3:
Probability of extinction after 100 years versus initial population size at time t=0 for a model
population of Tetrao urogallus (capercaillie), assuming a ceiling capacity of the habitat of
K=500 (after Grimm and Storch 2000).
Figure 4:
Intrinsic mean time to extinction, Tm, (in years) versus capacity of the habitat for a model
population of Tetrao urogallus (capercaillie). The Tm=10,000–line indicates the threshold
mean time to extinction needed to fulfil our criterion of viability (extinction risk not larger
than 1% in 100 years); the Tm=1,000–line indicates the threshold for the criterion used by
Marshall and Edwards-Jones (1998) (5% in 50 years). The minimum capacity needed for the
Grimm & Wissel
viability is Kmin=470 for Tm =10,000 criterion, and Kmin=215 for Tm =1,000 criterion (after
Grimm and Storch 2000).
27
28
Grimm & Wissel
FIGURES
Fig. 1
A
Number of runs
150
100
50
0
0
250
500
750
1000
1.0
B
P0(t)
0.8
0.6
0.4
0.2
0.0
0
250
500
750
1000
4
C
-ln(1-P0(t))
3
2
m=
1
1
yrs-1
220
0
0
250
500
750
Time to extinction t [yrs]
1000
29
Grimm & Wissel
Fig. 2
A
400
5
Tm= 42
T = 71
300
4
3
200
2
100
1
B
C
0
400
50
100
150
Tm= 40
T = 40
300
0
200
5
4
3
200
2
100
1
0
0
200
5
0
400
50
100
150
Tm= 41
T = 28
300
4
3
200
2
100
1
0
occupied patch
empty patch
_
5
_
capacity
30
0
0
50
100
150
Time to extinction
200
-ln(1-P0(t))
Number of simulation runs
0
30
Grimm & Wissel
Fig. 3
Probability of extinction
after 100 years
1.0
0.8
0.6
0.4
0.2
0.0
0
100
200
300
400
500
Initial population size
Fig. 4
Mean time to extinction Tm
10000
8000
6000
4000
2000
0
0
100
200
300
400
Capacity of the habitat
500