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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
Predictability of ecological and evolutionary dynamics in a changing world
Claudio Bozzuto1 *, Anthony R. Ives 2
1 Wildlife Analysis GmbH, Oetlisbergstrasse 38, 8053 Zurich, Switzerland.
bozzuto@wildlifeanalysis.ch. ORCID 0000-0003-0355-8379.
2 Department of Integrative Biology, University of Wisconsin-Madison, Madison, WI 53706, USA.
arives@wisc.edu. ORCID 0000-0001-9375-9523.
* Corresponding Author: Claudio Bozzuto, Wildlife Analysis GmbH, 8053 Zurich, Switzerland.
bozzuto@wildlifeanalysis.ch.
Abstract
Ecological and evolutionary predictions are being increasingly employed to inform decision-makers
confronted with intensifying pressures on biodiversity. For these efforts to effectively guide conservation
actions, knowing the limit of predictability is pivotal. In this study, we provide realistic expectations for
the enterprise of predicting changes in ecological and evolutionary observations through time. We begin
with an intuitive explanation of predictability (the extent to which predictions are possible) employing an
easy-to-use metric, predictive power PP(t). To illustrate the challenge of forecasting, we then show that
among insects, birds, fishes, and mammals (i) 50% of the populations are predictable at most one year in
advance, and (ii) the median one-year-ahead predictive power corresponds to a prediction R2 of only 20%.
Predictability is not an immutable property of ecological systems. For example, different harvesting
strategies can impact the predictability of exploited populations to varying degrees. Moreover,
incorporating explanatory variables, accounting for time trends, and considering multivariate time series
can enhance predictability. To effectively address the challenge of biodiversity loss, researchers and
practitioners must be aware of the information within the available data that can be used for prediction
and explore efficient ways to leverage this knowledge for environmental stewardship.
Keywords: biodiversity loss; conservation; forecasting; phenotypic trait; predictive power; time series analysis.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
1. Introduction
Three decades ago, Jared Diamond [1] and Edward O. Wilson [2] summarized the main human-caused
stressors to biodiversity as habitat degradation, fragmentation and loss, overexploitation, introduction of
non-native species and diseases, and pollution. Not only do these threats continue to impact life on Earth,
they are joined by global warming accelerating the decline of biodiversity worldwide [3,4]. Worryingly,
these stressors combine synergistically to form a perfect storm [5], and their impacts can percolate via
cascading effects through entire ecosystems: biodiversity loss may be regarded as a harbinger of
ecosystem collapse [6]. In addition to threatening population persistence and species richness, human
stressors are potent agents of rapid evolutionary change in the wild, with exploitation leading the other
stressors in driving phenotypic changes in populations [7,8]. Thus, we are experiencing not only loss of
species but also loss of variation within species, with both of these types of losses changing life on Earth
(e.g. [9–11]).
Our understanding of the mechanisms underlying stressor effects and the global extent of these effects on
biodiversity has improved (e.g. [12–14]), and collections of long-term data allow testing hypotheses about
changes in wildlife populations (e.g. [15] and references therein). Global synoptic indicators of
biodiversity change include the Red List Index [16] and the Living Planet Index [17]. These indicators are
invaluable to inform – and ideally prompt actions by – governments and the general public. Several
authors have nonetheless bemoaned conservation biology and restoration ecology as generally lacking
substantial efforts to model and predict future changes to guide mitigation actions (e.g. [18–20]). The
reasons are multifaceted, and mechanistically predicting biodiversity responses to current and predicted
future stressor levels continues to be a formidable task (e.g. [21]; but see [22]). Yet, Bodner et al. ([23],
p.10) aptly note that “as researchers, we are often acutely aware of how much we do not know and
therefore get stuck at ‘more research is required’. However, environmental changes are increasingly
affecting our world, and decisions are made whether or not we are involved.”
Compared to the ecological dimensions of biodiversity loss, our understanding of genetic diversity loss is
less developed [12,24]. This is unfortunate because genetic variation is the raw material enabling species
to robustly adapt to a changing world [24]. Despite the growing recognition of this dimension of
biodiversity, predictions in evolutionary science face many challenges similar to ecology [25–27].
Evolutionary changes depend on changes in the genetic make-up of populations through time, but long-
term genetic datasets continue to be rare. Therefore, analyses are often performed on phenotypic changes
in the wild, which are routinely measured [13]. Phenotypic changes, however, do not necessarily imply
genetic changes; instead, they could be the result of phenotypic plasticity. Nonetheless, phenotypic
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
changes in larger populations, for species with shorter generation times, or in response to environmental
conditions are more likely to represent genetic changes [13].
How well can future population changes in abundance or phenotype be predicted? And is it
fundamentally possible to predict future states of biodiversity [28]? Answers to the first question often
center around forecast accuracy (e.g. [29]), that is, how good a model has to be to make good predictions.
Unfortunately, even a very good model can fail to make useful predictions of inherently unpredictable
events. For example, assuming heads and tails have equal probability is an excellent statistical model for
flipping a coin, but this is not much help in predicting the outcome of the next flip. Therefore, instead of
focusing on how well models fit data, we address the second question, asking how much information is
contained in a dataset to make predictions, and how a lack of information creates a barrier to prediction
dictated by a predictability limit of the system [28–32]. Here, we follow the definition that "predictability
is the study of the extent to which predictions are possible" [31, p. 2425]. While the issue of an inherent
limit of predictability has been studied for several decades in different fields, chief among them
climatology [33], ecologists and evolutionary biologists have only recently started tackling it
[25,26,28,29,34].
Our goal is to argue that biodiversity conservation needs to adopt the concept of predictability to better
design strategies and policies to constrain human-caused threats to life on Earth. After an intuitive
presentation of the concept of predictability, we apply it to approximately 1,600 population abundance
and phenotype time series of invertebrate and vertebrate species to ask about the limit to predictability of
ecological and evolutionary systems. We then discuss predictability addressing two central topics in
biodiversity conservation: managed (exploited) populations and environmental forcing (explanatory
variables) that affects the dynamics of population abundance and phenotypic traits. Our examples show
that predictability is not an immutable property of species but instead can be influenced by management
strategies and environmental changes. Furthermore, we offer statistical evidence and theoretical proof for
several previously stated but unresolved hypotheses ([29,35], and references therein). We conclude by
summarizing why adopting the concept of predictability in biodiversity conservation will improve our
strategies to counteract human-caused stressors of wildlife populations.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
2. Methods
Instead of an exhaustive technical description of predictability (see e.g. [29–31]), we introduce the
concept using an illustrative ecological example, the population of wolves on Isle Royale (Michigan,
USA). We focus on the period 1959 – 2011 when the population experienced large fluctuations, but
before it started a precipitous decline [36] (Fig. 1, ESM Appendix 2). We use this dataset for two related
analyses. We start by focusing on the wolf data alone to introduce the main ideas and elements of
predictability (Fig. 1a). Second, we include covariate data to demonstrate how it may increase
predictability (Fig. 1b). For the second analysis we consider (i) the wolves’ main prey (moose; given the
available data, we analyze senescent moose) and (ii) the short- and long-term effects of a canine
parvovirus outbreak (Fig. 1b); additional methodological details and information on the covariates can be
found in the ESM Appendix 2.
Fig. 1. The main elements characterizing
intrinsic predictability. (a) depicts the
abundance time series of the wolf
population on Isle Royale (black line, 1959-
2011) with the stationary and transition
distribution. For illustrative purposes, we
take the year 2006 as the ‘current’ year to
make predictions. (b) depicts the analogous
situation as in (a), except that now two
explanatory variables – moose population
(old, i.e. senescent, moose; top of panel)
and effects of canine parvovirus (CPV;
asterisk) – are used. The stationary and
transition distributions are therefore
conditional on the covariates. (c) gives
intrinsic predictive power, () (Box 1),
for the analyses (a-b) and forced predictive
power, () (Box 3), for the analysis with
explanatory variables (b). All distribution
variances in (a-b) are visualized as 66%
confidence intervals. Drawing: Pearson
Scott Foresman, public domain.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
Many textbook deterministic models show how density-regulated populations eventually reach a
‘carrying capacity’. The stochastic analogue is the stationary distribution, with a mean (the ‘carrying
capacity’) and a variance capturing, for example, yearly fluctuations in abundance around the mean (Fig.
1a); a density-regulated population in the long-term will settle on an average size (the mean of the
stationary distribution) and fluctuate around this mean according to the variance and higher statistical
moments of the stationary distribution. If the properties of a stochastic process do not change over time,
the stationary distribution offers the simplest way of forecasting future abundances: just predict that the
mean and uncertainty in the abundance of a species in any future year is given by the stationary
distribution, regardless of the current abundance. This approach is easy to implement because the
estimated sample mean, variance, and other moments of the time series suffice to characterize the
stationary distribution. A different but equally simple approach is to assume that next year’s abundance
will be equal to the current one; this assumption implies there is no density dependence regulating the
population, but instead the population fluctuates by a random walk. Both of these approaches, however,
are unlikely to be very accurate. Predictability, in a nutshell, takes a position between these two extremes:
predictability is based on the amount of information in a time series that can be used to project a
population's transition from the current state to the stationary distribution.
When conditioning forecasts on current abundance, the forecast dynamics are characterized by the
transition distribution, with mean and variance (also called forecast error variance: [30]) that change with
the forecast horizon (Fig. 1a). Given the stochastic nature of population fluctuations, it is not possible to
predict exactly how the population will change in the future. Nonetheless, for a density-regulated
population it is possible to predict the overall tendency – the mean of the transition distribution
approaches the stationary mean – and the uncertainty around this tendency, captured by the variance of
the transition distribution around its mean. Thus, starting at the current abundance, the transition
distribution will change through time and eventually converge to the stationary distribution. These two
distributions – transition and stationary – are the main elements characterizing predictability: the less the
transition distribution overlaps with the stationary one, the more predictive information is available for
reliable forecasts based on the present state of the population, and predictability is high [31].
The predictability measure we use is the metric predictive power, PP(t), rooted in information theory and
developed in climatology to measure the uncertainty in predictions [30] (Box 1). Previous work on
predictability in ecology has used model-free methods based on information theory [29,32] or
computational irreducibility [28], methods that typically cannot be integrated into further modeling and
forecasting for biodiversity conservation. Predictive power, ()< 1, depends on the forecast
horizon – in this study, time in years – and () will decrease to zero as increases to infinity and the
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
transition distribution converges to the stationary distribution (Fig. 1c). In practice, it makes sense to
define a predictability barrier that gives a threshold below which the predictability of the system is
negligible. To cast predictability in familiar statistical terms, () can be shown to be related to the
theoretically maximum possible prediction R2 reflecting forecast accuracy (Box 1). Using this link, to
determine a predictability barrier we will use ()= 0.05 as a threshold, corresponding to a maximum
possible prediction R2 of ~10% for univariate time series (Fig 1c). The selection of a threshold should
change depending on a researcher's needs. Here, we have selected a low threshold because, even with this
low threshold, many of the time series we analyzed reach this threshold in only a few years; if we selected
a higher threshold, the resolution would not be high enough to separate predictive power for many time
series. Finally, where necessary to distinguish it from other predictability metrics, we will refer to ()
as the intrinsic predictive power, because it only depends on the variation of the observed data.
We implement PP(t) using time-series models from the ARMA family [37] (Box 1). ARMA models are
familiar to many ecologists and evolutionary biologists (e.g. [38]), and are used extensively due to their
parsimonious and flexible nature. ARMA models make it possible to address many central topics in
wildlife data, ranging from measurement error to population decline due to human stressors. ARMA
models are often accurate approximations to nonlinear systems [37], although for highly nonlinear
processes like systems with multiple stable states (however elusive these are: [39]), ARMA models will
fail to capture patterns caused by the nonlinearities and therefore may underestimate the true
predictability of the data. This might argue for an approach that computes the information content of a
dataset without the need to specify a model [29,32]. The cost of taking a model-free approach, however,
is that there is no way to directly translate these information-based metrics into terms that assess model fit
and predictive ability.
Box 1: Intrinsic predictive power
Predictive power, PP(t), can be easily formulated for Gaussian processes like widely used time-series
models. For a univariate time series (ESM Appendix 1.1, eq. S1),
(
)
=
1
−
(
)
⁄
,
(1.1)
where () and are the variances of the transition and stationary distributions, and is the forecast
horizon. Calculation of () and is outlined in the ESM Appendix 1.1. The formula can be extended
to include estimation uncertainty in model parameters used to calculate () and [30]. Generally,
(
)
declines over time (i.e. forecast horizon), eventually approaching zero: in this limiting case
(
)
=
and predictability is lost (Fig. 1c). For univariate processes,
(
)
is related to the theoretical limit of
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
forecast accuracy,
(
)
=
1
−
(
)
[40], as
(
)
=
1
−
1
−
(
)
. The link
between
(
)
and forecast accuracy as measured by
(
)
, and the decrease of predictability over
time theoretically confirms previously proposed but unresolved hypotheses [35].
Because eq. 1.1 is suitable for Gaussian processes, time-series models from the ARMA(p,q) family are
apt candidates for implementing (). For the purpose of assessing predictability, it is convenient to
state an ARMA(p,q) process as a dynamic regression model,
(
)
=
(
)
+
(
)
,
(
)
=
(
−
)
+
(
−
)
.
(1.2a)
(1.2b)
Here, () is the potentially time-varying process governing the mean of () (Box 3), and () are
temporally autocorrelated errors; without a dependence of the mean on time, () is simply a constant
(e.g. ‘carrying capacity’). For a univariate ARMA model, () depends on the ARMA
coefficients (eq. 1.2b), whereas the residual variance of the process (
) cancels out (ESM Appendix
1.1). From a simulation study presented in the ESM Appendix 1.2, we recommend setting q = 0 for
unbiased estimates of predictive power.
(a) Broad comparisons of predictability among species
To gain a bird's eye view on predictability of animal population sizes and phenotypes, we analyzed 320
invertebrate and 963 vertebrate ecological time series from the four taxonomic groups: insects, birds,
mammals and fish. Invertebrate data were collected either at the species level or at the plot level (total
abundance or biomass, but not species richness or similar), and data of the other three groups were all
collected at the species level [41,42] (ESM Fig. S3, Table S1, Appendix 4). We also analyzed 307
phenotypic time series of bird, fish, and mammal populations [43] (ESM Fig. S3, Table S3, Appendix 4).
For selecting datasets, we used the following criteria (ESM Appendix 4): (i) only yearly data; (ii) no
presence-absence data; (iii) a minimum length of 20 years for ecological data and 15 years for phenotypic
data due to the smaller amount of time series available; (iv) a maximum proportion of missing values of
25%; and (v) a maximum of five consecutive missing values. In the ESM Appendix 4 we address time
series length and predictability in more detail. As for potential time trends, we ran analyses using no,
linear, or quadratic time trends (ESM Appendix 4). All analyses were performed using Python [44].
Although for this broad comparison we analyzed population-level data without explicitly considering
additional abiotic and biotic variables, for simplicity we will broadly refer to these population-level
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
dynamics as ecological dynamics and phenotypic dynamics. Later on (section §2d), we will additionally
consider covariate data, where the term ecological dynamics more correctly applies (instead of population
dynamics).
(b) Multivariate time series
Sometimes, multiple correlated time series are available, for example, for an ecological community or for
different phenotypic measures from the same organism. Predictability () is defined for the
multivariate case and can be applied to multivariate ARMA(p,q) models (ESM Appendix 1.1). The
univariate formula (Box 1) is just a special case [30].
When multiple time series are available, they could each be analyzed separately, they could be analyzed
as a multivariate set, or a dimensionality reduction method like Principle Components Analysis could be
applied and a univariate analysis on a single dimension conducted. Which approach is best? There is no
simple answer, and the best approach will depend on the time series and objectives. To illustrate the three
approaches, we use a phenotypic dataset of beak morphology of three populations of Darwin's finch
species on Daphne Major Island for the period 1973-2012 [45] (ESM Fig. S2, Appendix 4).
(c) Predictability of exploited populations
People worldwide rely on and benefit from the use of about 50,000 wild species for food, medicine, and
recreation [46]. Unfortunately, exploitation has become a main cause of elevated extinction risk, affecting
species from many faunal and floral taxonomic groups [4,47,48]. Wildlife management has increasingly
sought harvesting strategies to minimize overexploitation, and nowadays many approaches include
forecasting population dynamics to abide by sustainable management [23,49,50].
We expect predictability of exploited populations to reflect life history characteristics: because larger
animals tend to be exploited relatively more often [51] and their life history characteristics correlate with
a decreased degree of density dependence [52], this will lead to higher predictability (Box 1). But
different harvesting strategies can also affect predictability (Box 2). To illustrate how management affects
predictability, we analyzed data of 13 Swiss cantonal populations of the northern (Alpine) chamois
(Rupicapra rupicapra) for the years 1980-2020, for which hunting returns and abundance time series are
available [53] (ESM Appendix 5.2).
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
It is important to note that management strategies that lead to greater predictability are not necessarily
indicators of good management. High predictability implies that more information about future
abundances is available from information about the present state of the population, and if a good
management strategy compensates rapidly when there is overexploitation, the desired rapid return of the
population to its stationary distribution would associate low predictability with good management.
Box 2: Harvesting strategies and predictive power
A convenient way to categorize harvesting strategies is by expressing annual harvest,
(
)
, as the
product of the proportion of the population harvested, (), and population size, (): (
)
=
()(), where (∙) is a function of (). We consider the case in which () can be
approximately expressed as a linear function of ()= ln(), so that ()=ℎ+ ℎ(
)
+
()() and () are the residuals of the fitted proportion. If ℎ= 0, we recover the abundance-
independent proportional strategy. For ℎ> 0, the proportion harvested increases with increasing
(
)
,
and the intercept ℎ influences the threshold below which no animal is harvested. Finally, ℎ< 0
gives
a linear approximation to a constant yield strategy at equilibrium, ()()(), where
() is the constant yield. This model for a harvested population generates the autoregressive parameter
that depends on the harvesting parameter ℎ, =1 − ℎ(), where is the birth rate and
() is the autoregressive parameter of the unharvested population (ESM Appendix 5.1). Because
parameter directly affects predictability (Box 1), harvesting strategies with ℎ≠ 0 will do the same.
As detailed in the ESM Appendix 5.1, while this approach gives a linear approximation of widespread
harvesting strategies, the results might not apply to highly nonlinear systems.
(d) Predictability of forced systems
So far, we have not considered the possibility of external or environmental variables that affect either the
mean or variation of the variable of interest. If, for example, the mean abundance of a population depends
on mean annual temperature, then how would knowledge of this dependence affect our ability to predict
the future mean population abundance under different scenarios of global warming? Similarly, if
temperature affected the year-to-year variation in the abundance of a population, then how does
knowledge of temperature in a given year help us predict the population abundance in that year? Here, we
address predictability for forced systems – systems that depend on external variables – which have been
largely ignored in the ecological predictability literature [29,35].
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
To avoid confusion about the meaning of ‘prediction’, we distinguish between explanatory and
anticipatory predictions [54]. Explanatory predictions are tested with observed data, whereas anticipatory
predictions forecast future states of the system; the concept of predictability involves anticipatory
predictions. If knowledge can be obtained about external variables affecting observations (explanatory
predictions), then these might be used for making anticipatory predictions. Therefore, we have
incorporated external variables into a measure of forced predictive power, () (Box 3; Fig. 1b).
If an external variable () explains in part patterns in the observations of a time series (), to make
multi-year forecasts of () the future values of the mean of () would ideally be known. While in many
situations these values will be unknown, values of () might be assigned to make forecasts under
different scenarios for the future, such as different projections of future global warming or different
assumptions about habitat degradation. Furthermore, it might be possible to predict future values of ()
from the existing data (Fig. 1b). Finally, time can be treated as an external variable, making it possible to
ask how predictability changes when there is a time trend in the population’s mean. In all of these
situations, the goal is to determine the maximum anticipatory predictability when the mean value of ()
changes.
Predictive power can be expanded to account for year-to-year variation in (). The effects of this
variation on () are often studied using observed data for explanatory prediction, although conceptual
issues must be addressed to use the results in anticipatory prediction. Predictive power () (Box 1)
addresses the information provided by the observed values of () on future values of () that depend
on the variation in the future values around the mean of (); in other words, () concerns
predictability given by the transition distribution of the residual variation in () that is not explained by
(). When variation in () drives variation in () and future values of () are unknown, then the
explanatory predictability provided by () does not aid in anticipatory predictability, and therefore the
explanatory predictability of () should be ignored. However, this seems to throw away information that
() provides about (). Our measure of forced predictive power, () (Box 3), addresses this by
using observations to compute the conditional predictability of a future value of () given that () is
predicted once the value of () for that year is known. This formulation of () guarantees that as
() provides greater explanatory predictability of the observed data, () increases.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
Box 3: Forced predictive power
The definition of predictive power can be extended for processes with a mean that changes with
external, explanatory variables; a
possible explanatory variable is time, allowing incorporation of time
trends. The simple case of a single explanatory variable can be expressed as
(
)
=
(
)
+
(
)
, (3.1)
where ()()() is broken down into deterministic and stochastic components. Th
e regression
coefficient for () on () can be estimated from time-series data or treated as a known value for
projecting different forecasting scenarios. Writing the process mean () as a linear function of
(
)
causes little loss of generality, because an explanatory variable can be transformed to generate a linear
relationship with (). To be consistent with the ARMA approach we have taken (Box 1)
, we assume
that () has a Gaussian distribution with variance . Multiple explanatory variables can be
incorporated similarly.
Changes in the mean () and variation () in () have different consequences for predictability.
() addresses the rate at with the transition distribution approaches the stationary distribution, where
both distributions correspond to the variation in () around () given by (). Therefore, () does
not include variation in () created by variation in (). When predictions are made conditional on a
specific future value of (), and when a researcher wants to incorporate the information this provides
into predictive power, then the appropriate reference stationary distribution becomes the distribution of
() around () () rather than () around (). The stationary distribution of () around
(
)
has variance , which is always greater than for . When there are changes in
(), the ‘forced predictability’ is
(
)
=
(
)
−
(
0
)
(
2
)
, (3.2)
which is derived for Gaussian processes from the Kullback–Leibler Divergence (or relative entropy)
between the current distribution of (0) and the distribution of () [55,56]. Incorporating both the
mean and variation in () gives a measure of predictive power for explanatory variables (see also [55]),
(
)
=
1
−
exp
−
(
)
(
)
⁄
.
(3.3)
If there is no variation in (), then and eq. 3.3 gives the effects of pure forcing of the process
through changes in (). If there is no change in (), then exp()= 1 and () is similar to
(), but with replacing . Finally, if only the effect of forcing through changes in () is of
interest, then
(
)
can be set equal to
; this case we define as
(
)
.
To illustrate (), we performed univariate analyses of the three beak measures of the medium ground
finch (G. fortis) we described in section §2b. Moreover, for a broader view we analyzed the ecological
12
Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
and phenotypic datasets time series with statistically detectable time trends; we selected these 948 time
series from the 1,590 time series described in section §2a.
3. Results
(a) Broad comparisons of predictability among species
For the 1,283 ecological and 307 phenotypic analyzed time series, we found that 50% of all time series
are predictable at most one year ahead, with interquartile ranges of 0-3 years and 0-2 years, respectively
(ESM Fig. S4). Because values of the predictability barrier and one-year-ahead predictive power, PP(1),
are strongly associated (Spearman’s = 0.95, P < 0.0001), in Fig. 2 we present PP(1) values separated by
taxonomic groups, and for ecological time series also by illustrative sub-groups/families. The results
indicate that mammal populations have the highest predictability, followed by fishes, birds, and insects
(ecological data only). All results combined (grey lines in Fig. 2), 50% of ecological and phenotypic time
series have a PP(1) value of at most 0.13 and 0.06, respectively: PP(1) = 0.1 translates to a maximum
possible one-year-ahead prediction R2 of ~20% (Box 1), for the modest endeavor of predicting next year’s
ecological or phenotypic population value.
For ecological time series, the dependence of (1) on ecological characteristics (Fig. 3) gives some
interesting lessons for conservation. Time series from protected areas had on average the same predictive
power as those outside protected areas (Fig. 3b), and primary threats (human stressors) and the total
number of threats ([4]; ESM Table S2) were not associated with predictive power (ESM Fig. S5).
Therefore, as a group, species of conservation concern are neither more nor less inherently predictable
than other species, leading to two corollaries. First, making predictions for species of concern presents no
special challenges, at least from the perspective of population dynamics. Second, the threats themselves
do not seem to create unpredictability in population dynamics. Comparing across habitat realms,
freshwater time series of all taxonomic groups have the lowest predictive power (Fig. 3a), implying that
these species are more susceptible to annual environmental fluctuations. This is noteworthy because
freshwater populations have been suffering the most pronounced human-driven declines [17]. An
additional useful result in practice relates to data aggregation (Fig. 3c): for insects the degree of data
aggregation (single-species vs. sample plot abundance or biomass) seems not to influence predictive
power. This result contrasts with previous studies arguing that aggregated data may be expected to be
more predictable ([35], and references therein).
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
Fig. 2. Predictability across animal
populations. The distributions of one-year-
ahead predictive power values, PP(1), are
shown for (a) ecological and (b) phenotypic
population time series, sorted into the
taxonomic groups insects (only panel a),
birds, mammals and fish, and in panel (a)
additionally sorted into 10 illustrative
taxonomic subgroups. For each taxonomic
group a vertical line gives the median value
over all respective populations; smaller
vertical lines give the median and min/max
values. In both panels, the overall median
value is marked with a vertical grey line.
Drawings: Andrea Klaiber, © Wildlife
Analysis GmbH, Switzerland.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
Fig. 3.
Global predictive power of ecological dynamics
. (
a
) summarizes the one-year-ahead predictive power values, PP(1),
sorted by realm (median and interquartile range). (b-c) depict the distribution of PP(1) values of insects, sorted by protection
status and data level (species or plot time series), respectively (median and interquartile range), in both cases further classified by
realm. (d) depicts, as a hexagonal grid, a summary of geographically sorted PP(1) values (see color bar).
Finally, across the globe (Fig. 3d) populations along the Pacific coast in the northern hemisphere show
relatively low predictive power, a pattern driven mainly by fish populations (ESM Fig. S6). To better
understand this pattern, in the ESM Appendix 3 we present preliminary statistical evidence showing that
predictability may in part be driven by a populations' food web embedding, supporting a previously
published hypothesis ([35], and references therein).
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
(b) Multivariate time series
Analyses of three beak measures of the medium ground finch (Geospiza fortis) and small cactus finch (G.
scandens) gave similar results for all three approaches employed to investigate the potential advantage of
a multivariate analysis (Fig. 4): analyzing traits separately, first performing a PCA and using the first
component, and analyzing all three traits together. For the large ground finch (G. magnirostris), however,
the multivariate assessment leads to greater predictive power; this implies that there is information in the
co-variances between the traits that are not incorporated into univariate analyses.
Fig. 4. Predictability of univariate vs. multivariate time series.
The first row gives predictability results for each finch species
(column title) with all three measures analyzed separately, and the second row gives analogous results based on the first
component of a PCA (PC1) and the multivariate assessment. Drawings: John Gould, public domain.
This example illustrates that, while there is no guarantee that a multivariate approach provides higher
predictive power, it would be prudent to try a multivariate analysis. A multivariate approach will be
especially worth trying within conservation projects involving management areas, as opposed to projects
focusing on single species. If a project's success for a management area depends on its ability to protect
multiple species – or aggregate measures of ecosystem health such as carbon sequestration and nutrient
run-off prevention – then multivariate predictive power gives an assessment of how well the outcome of
management goals can be predicted. In sum, multivariate predictive power provides a potential holistic
assessment of the ability to make predictions.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
(c) Predictability of exploited populations
The distribution of one-year-ahead predictive power values, PP(1), of 13 Swiss chamois populations
reveals no intuitive geographic pattern (Fig. 5a). The harvesting strategy captured by the harvesting
parameter ℎ (Box 2), however, shows a clear effect on predictive power (Fig. 5b).
Fig. 5. Effect of harvesting strategy on predictability.
(a)
geographically presents estimated one-year-ahead predictive power
values, PP(1), of 13 Swiss cantonal chamois populations. (b) illustrates the change in the one-year-ahead predictive power,
∆PP(1) (harvested minus non-harvested), as a function of the harvesting strategy characterized by the slope value (ℎ) on the x-
axis (i.e., how the proportion harvested depends on abundance; Box 2). The dot colors are the same as in panel (a). Drawing:
Andrea Klaiber, © Wildlife Analysis GmbH, Switzerland.
For populations subject to a harvesting strategy in which the proportion harvested is adaptively adjusted
according to the current abundance (ℎ> 0), predictive power is lower, while populations subject to
constant yield-like harvesting (ℎ< 0) have higher predictive power. This result occurs because the state-
dependent management approach tracks the yearly population dynamics to decrease the variation in
population fluctuations which thus decreases predictability (Box 1). In contrast, a constant yield-like
harvesting strategy does not dampen population fluctuations but instead increases them relative to the
mean population by depressing the population, thereby increasing predictability. Other species and
management strategies might give different patterns; we cannot claim generality from this single example.
Nonetheless, the example shows that predictive power is not an immutable property of a population or
phenotypic trait but instead depends on the forces that affect populations, including management
strategies.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
For conservation, the chamois example shows that low predictability is associated with better
management. High predictive power implies that there is a lot of information in the residual, unexplained
variation in a population. Thus, the residual information revealed by high predictive power is information
that has not been used in the management strategy. A state-dependent harvesting strategy is more likely to
lead to sustainability and reduced extinction risk [49], and the lower predictive power caused by such
strategy is an indication that management is effective.
(d) Predictability of forced systems
For time series of the three beak measures of the medium ground finch, beak depth and width decrease
over time, while beak length showed no time trend (ESM Fig. S2). Therefore, we compared the intrinsic
and forced predictive power for these time series (Fig. 6). For both beak depth and width, the intrinsic
predictive power () remains above the threshold 0.05 (section §2) for 4 years. As () decreases,
the forced predictive power caused by the time trend, () (Box 3), increases, crossing the threshold at
2 and 4 years for beak depth and length, respectively. The increase in () occurs because, as the
predicted means of beak depth and length increase above the observed values, the predicted means
provide more information. The differing patterns of () and () show a switch in information
useful for making predictions. For the first 4 years () > (), implying that information about the
last observations provide better predictions, while afterwards () < (), implying that using
information about the time trends will provide better predictions.
For all ecological and phenotypic time series with time trends, we computed the number of years before
() > 0.05, the time at which information about the time trend provides substantial predictive power.
For approximately half of the time series, the time trend remains unpredictable for at least one year, and
potentially many more years (Fig. 6c,d; ESM Fig. S8).
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
Fig. 6. Predictability of forced
systems. For beak depth and width of the
medium ground finch (G. fortis), (a-b)
depict the z-transformed data (2004-
2012; ESM Fig. S2), the current
stationary distribution (in grey), the
forced stationary distribution (dark blue),
and the transition distribution (light blue).
All distribution variances are visualized
as 66% confidence interval. (c-d) give the
predictive power, (), and explanatory
predictive power, () (Box 3), with
the colored areas corresponding to the
unpredictable period of the forced (dark
blue) and intrinsic (light blue) predictive
power, respectively. In panels (a-d), the
current year is assumed 2004, from
which forecasts would be made. (e-f)
show the duration of the initial
unpredictable period of the forced
predictability (see the dark blue areas in
panels c-d) for all (e) ecological and (f)
phenotypic time series with a time trend.
Drawing: John Gould, public domain.
4. Discussion
For any biodiversity change assessment, the first step should be to evaluate predictability. Predictive
power, (), gives an objective summary of the information content in the data and a first cut at more-
detailed modeling to make predictions. () focuses on the ‘unexplained’ variation in a time series that
is not associated with any explanatory variable, asking whether this variation can be used for predictions
(Box 1). We also present () that incorporates explanatory variables () (Box 3). If () accounts
for some of the observed variation in a time series, then () will be greater than (). () can
also be used to assess predictions made from time trends in data. Thus, () and () parse out where
informative patterns exist in time-series data, directing attention to how reliable and credible predictions
can best be made.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
The predictability measure () is generic, in the sense that it can be applied to any time series or
simultaneously to multiple time series. Our finding that the predictability of the ecological and phenotypic
data collections was similar in magnitude surprised us (Fig. 2), because we expected population-level
phenotypic changes to be too slow to be detectable in most time-series datasets. However, if the fitness
consequences of traits are large, then traits should change on similar time scales as population abundances
[26]. Less surprising were the generally low () values (Fig. 2), confirming previous findings in
ecology [28]: in ecological systems, low predictability is assumed to be pervasive due to the complexity
of ecological systems involving many species with different behaviors, physiologies, etc. Differences in
() among time series may also reflect heterogeneity in quality among time series and their short
lengths. This underscores the need to compute () in biodiversity conservation where time series of
interest are often short and of suboptimal quality [15]. Low () values also caution against further
statistical analyses and argue for future concerted efforts to obtain information to increase predictive
power; a ‘better model’ will be of little help for data that are inherently unpredictable.
Is high predictive power ‘good’ for conservation management? High () values imply that a
mathematical/statistical model can be used to predict future values, because information about the present
state of the system is ‘remembered’ in the future. While the ability to make good predictions is valuable,
the same memory will make a system slow to respond to conservation management strategies. Thus, high
predictability measured by () might make it difficult to detect responses of populations to
conservation actions, hindering adaptive management. To obtain a more complete and broadly applicable
assessment of predictability, we defined () that explicitly incorporates explanatory variables (Fig.
1b) and time trends (Fig. 6).
Biotic data for explanatory variables might not be widespread in conservation studies except for well-
studied and monitored systems (e.g. [36,45]). Abiotic data for explanatory variables, however, may be
readily available or even be routinely collected (e.g. data on temperature and precipitation trends, or land-
use change [42]). Furthermore, time trends can and should be investigated for a predictability assessment,
as illustrated by our detection of time trends in 948 of 1,590 ecological and phenotypic time series. We
cannot give general recommendations for how much effort should be devoted in a study towards
obtaining information about explanatory variables () versus obtaining better time-series data on the
focal variable (); this will depend on the costs and benefits for a given study. Nonetheless, () and
() give the tools to assess the possible benefits of obtaining explanatory variables; both () and
() can be calculated with available data, and values of () substantially greater than () would
argue for devoting more effort to obtaining information about ().
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
For conservation management, our analyses of 1,590 ecological and phenotypic time series give a striking
contrast between intrinsic predictive power, (), and predictive power generated by a time trend,
(). The intrinsic predictability barrier was greater than one year for only 50% of the time series,
implying that current observations provide information for at most short-term predictions. However, for
50% of the 948 time series with time trends, () exceeded the same threshold in 2 or fewer years. In
other words, for many time series the information provided by an estimate of the time trend exceeded that
provided by the last observations after only a few years. Of course, the predictive power of recent
observations versus time trends will depend on the time series under study, and time trends may be
transitory and changeable. Nonetheless, our results argue that, if there is a time trend, it may provide more
information for making even short-term forecasts than current observations.
Many authors have encouraged conservation scientists and practitioners to incorporate predictions into the
management cycle (e.g. [19,54], and references therein). A predictability assessment should help to
interpret these predictions in a realistic and useful way for decision-making. For example, conservation
prioritizations should include forecasting to identify populations of greatest need of interventions.
Predictive power can help to assess which population-level predictions are more 'trustworthy' (i.e., have
high predictive power). The distribution of () values among taxonomic groups (Fig. 2) suggests it is
possible to adopt group-specific predictive approaches. For example, many insect populations show low
intrinsic predictive power (Fig. 2), making forecasts for insect populations more challenging: in these
cases, directing the focus to forced predictability may be sensible.
Predictions can also benefit the selection and implementation of conservation actions [19]. For example,
the expected effect of suitable conservation actions on future population dynamics could be compared,
and the resulting ranking be further weighted by the respective resources needed. The assessment of
predictability at the implementation stage will be especially valuable when thinking about monitoring the
expected action-specific outcomes. A declining population represents a forced system (Box 3), and many
conservation actions aim at halting or reversing declines. Forced predictive power, (), gives a metric
to ask how quickly changes in a forced stationary distribution – forced by conservation actions – will be
predictable. In the 948 analyzed time series with time trends (Fig. 6), the predictability from a changing
mean can be detected in only a few years. Such an analysis for a specific conservation program will give a
time frame for when a conservation action might be deemed a success or failure, or help ranking different
conservation actions. This approach differs conceptually from an impact assessment based on a
counterfactual analysis, increasingly employed in a conservation context ([19], and references therein).
To compute forced predictability (Box 3), the current (observed) stationary distribution is compared to the
time-varying (predicted) forced stationary distribution, the latter being forced by conservation actions. For
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
an impact assessment, in contrast, the forced stationary mean is compared to a counterfactual population
trajectory, i.e. the time-varying predicted dynamics if no actions were implemented. We feel that
comparing one forced (action-driven) trajectory to the current (observed) population state is more prudent
than comparing it to a second, non-measurable counterfactual trajectory.
As scientists worried about biodiversity loss, we often call for more and higher-quality data; what we
really want are data that contain more predictive information. Given the urgency for action dictated by a
rapidly changing world, we need to use the hard-won data we already have and explore ways of extracting
predictive information from these data: we cannot afford to wait for ‘better’ data to become available
while human-caused stressors are downgrading wildlife populations and habitats. We believe that
embracing the concept of predictability will benefit the urgent and global endeavor of protecting life on
Earth.
Acknowledgments: We thank Thorsten Hens and two anonymous reviewers for valuable suggestions on improving the
manuscript.
Author contributions: C.B. Conceptualization, Methodology, Formal analysis, Data Curation, Writing - Original Draft, Writing
- Review & Editing, Visualization. A.R.I. Methodology, Formal analysis, Writing - Review & Editing.
Data availability: Raw results are available from the Zenodo repository [57]. Time series data analyzed in this study have all
been previously published and are publicly available online. The sources are referenced in the main text and also collected in the
ESM Appendix 6.
Code availability: Code to compute predictability metrics (in Python) is available from the Zenodo repository [57].
Conflict of interest declaration: We declare we have no competing interests.
Declaration of AI use: We have not used AI-assisted technologies in creating this article.
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Bozzuto C, Ives AR (2024): Predictability of ecological and evolutionary dynamics in a changing world.
Proceedings of the Royal Society B. DOI: 10.1098/rspb.2024.0980.
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