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# Biodiversity and ecosystem productivity in a fluctuating environment: The insurance hypothesis

^{*}To whom reprint requests should be addressed. e-mail: rf.sne@uaeroL.

## Abstract

Although the effect of biodiversity on ecosystem functioning has
become a major focus in ecology, its significance in a fluctuating
environment is still poorly understood. According to the insurance
hypothesis, biodiversity insures ecosystems against declines in their
functioning because many species provide greater guarantees that some
will maintain functioning even if others fail. Here we examine this
hypothesis theoretically. We develop a general stochastic dynamic model
to assess the effects of species richness on the expected temporal mean
and variance of ecosystem processes such as productivity, based on
individual species’ productivity responses to environmental
fluctuations. Our model shows two major insurance effects of species
richness on ecosystem productivity: (*i*) a buffering
effect, i.e., a reduction in the temporal variance of productivity, and
(*ii*) a performance-enhancing effect, i.e., an increase
in the temporal mean of productivity. The strength of these insurance
effects is determined by three factors: (*i*) the way
ecosystem productivity is determined by individual species responses to
environmental fluctuations, (*ii*) the degree of
asynchronicity of these responses, and (*iii*) the
detailed form of these responses. In particular, the greater the
variance of the species responses, the lower the species richness at
which the temporal mean of the ecosystem process saturates and the
ecosystem becomes redundant. These results provide a strong theoretical
foundation for the insurance hypothesis, which proves to be a
fundamental principle for understanding the long-term effects of
biodiversity on ecosystem processes.

**Keywords:**stochastic dynamic model, species richness, ecosystem processes, temporal variability, ecosystem stability

Recently the effects of biodiversity on ecosystem processes have received much attention because of the growing concern that loss of biodiversity may impair ecosystem functioning (1–5). A number of experiments have been performed or are in progress (see e.g. ref. 6) and theoretical studies are emerging in this area (7, 8). However, most of these studies are restricted to situations where environmental fluctuation is negligible or excluded (but see refs. 9–12). In the long term, all ecosystems are bound to experience environmental changes (13). Therefore a critical question is: how are ecosystem processes affected by biodiversity or by a loss of biodiversity in a fluctuating environment?

The insurance hypothesis so far has been an intuitive idea that increasing biodiversity insures ecosystems against declines in their functioning caused by environmental fluctuations (12, 14–16). Such an effect is expected because different species respond differently to environmental changes, hence the contribution of some species to ecosystem processes may decrease while that of others may increase when the environment changes. Thus greater species richness should lead to a decreased variability in ecosystem processes because of compensation among species. Here we define insurance effects of biodiversity more generally as any long-term effects of biodiversity that contribute to maintain or enhance ecosystem functioning in the face of environmental fluctuations. These effects may differ depending on the type of fluctuations experienced and the ecosystem properties regarded as desirable, such as long-term average performance, reduced variability, long-term probability of persistence, or resilience to pulse perturbations.

New theoretical studies have started to emerge on these issues
(17–20). However, the generality and implications of the insurance
hypothesis in real ecosystems are still unclear because of the specific
assumptions and analyses made in these studies. Here we present a
stochastic dynamic model to show that (*i*) this hypothesis is
expected to be true under very general conditions within a trophic
level or functional group; (*ii*) species richness may not
only decrease the temporal variance of ecosystem processes but also
increase their temporal mean; and (*iii*) the species richness
beyond which an ecosystem becomes redundant depends on the way the
various species respond to environmental fluctuations. We take
ecosystem productivity as an example of an important ecosystem process,
but our results can be easily generalized to other processes.

## THE MODEL

We develop a stochastic dynamic model to assess the effects of species richness within a trophic level or functional group on the expected temporal mean and variance of ecosystem productivity based on individual species’ productivity responses to environmental fluctuations. The model consists of the following three elements.

#### Replicate Ecosystems.

A set of replicate ecosystems is constructed at each level of species richness by random sampling from a species pool. This procedure corresponds to recent experimental protocols, which is necessary to separate the effects of diversity on ecosystem functioning from combinatorial effects because of species identity (e.g., ref. 21).

#### Productivity Response of Each Species.

Each species in an
ecosystem is characterized by a specific productivity response to
environmental fluctuations. We assume that the productivity of species
*i* at time *t* obeys an unspecified stochastic
process and hence is a random variable of time,
*X*_{i}(*t*). Discrete time is chosen for mathematical
convenience. Each species’ productivity is assumed to take on values
between 0 and 1 (0 ≤ *X*_{i}(*t*) ≤ 1) without
loss of generality.

#### Total Ecosystem Productivity.

The total productivity of a
replicate ecosystem with species richness *n* at time *t*,
*X*(*t*:*n*), is a function of the individual species’ productivities
at that time and is also a stochastic process:

In this model, we measure the magnitude of an ecosystem process by
the temporal mean of that process, *X*_{n} = (1/*T*)
Σ_{t}*X*(*t*:*n*), and the temporal variation of the process
by its temporal variance, *V*_{n} = (1/*T*)
Σ_{t}(*X*(*t*:*n*) − *X*_{n})^{2}. When an
ecosystem is subjected to environmental fluctuations, its processes do
not have a stable equilibrium value. Using other measures of temporal
variation, such as the coefficient of variation, *CV*_{n}
= /*X*_{n}, does not change our
results qualitatively. The expected values of the temporal mean and
variance of total productivity,
*E*_{e}[*X*_{n}] and
*E*_{e}[*V*_{n}], respectively, are then
calculated as functions of species richness by averaging
*X*_{n} and *V*_{n} over all
replicates (Appendix):

where *Var*_{e}[.] and
*Cov*_{e}[.] denote expected variance and
covariance, respectively, and *Var*_{T}[.]
denotes temporal variance.

## RESULTS

The results depend on the way total ecosystem productivity is
determined by the individual species’ productivities. We analyze two
limiting cases here: (*i*) determination by dominance, i.e.,
total productivity at any time is approximated by the productivity of
the most productive species because of interspecific competition:

(*ii*) determination by equivalence, i.e.,
total productivity is simply the average of the individual species’
productivities because interspecific interactions are negligible:

Equivalence was implicitly assumed in some previous works (17, 19), but determination by dominance is likely in grassland ecosystems where a single resource limits plant growth (7). Plant competition experiments showed that the yield of two-species mixtures often was close to the yield of the most productive monoculture (e.g., refs. 22–24). Thus, the real determination of total productivity is expected to generally lie between these two limiting cases.

#### Determination by Dominance.

The effects of species richness on
the expected values of the temporal mean,
*E*_{e}[*X*_{n}], and variance,
*E*_{e}[*V*_{n}], of productivity depend on
the degree of asynchronicity of the species responses as follows (see
proof in the Appendix). (*i*) If the responses of
all species are perfectly positively correlated, i.e., the coefficient
of correlation, *r*_{ij}, between
*X*_{i}(*t*) and *X*_{j}(*t*) is equal
to 1 for all pairs of species *i* and *j*, then, for
any *n* ≥ 2,

(*ii*) Otherwise,

and, for sufficiently large *n*,

Thus, in the highly unlikely case when there is no asynchronicity
at all in the species responses, the ecosystem behaves exactly as a
single species. In all other cases,
*E*_{e}[*X*_{n}] is greater than
*E*_{e}[*X*_{1}] and
*E*_{e}[*V*_{n}] is smaller than
*E*_{e}[*V*_{1}] for a large enough species
richness *n*. Furthermore, unless all pairs of species
responses are perfectly correlated (either positively or negatively,
i.e., |*r*_{ij}| = 1),
*E*_{e}[*X*_{n}] increases to its maximum value
(in this case, 1) and *E*_{e}[*V*_{n}]
decreases to zero as species richness *n* increases (Figs. (Figs.1
1
and
and22*A*).
These effects occur irrespective of any autocorrelation in individual
species responses. The autocorrelation of total productivity can be
shown to vanish as species richness increases (Appendix); in
other words, the total ecosystem response can be regarded as an
independent process when species richness is large.

#### Determination by Equivalence.

In contrast to the previous
case, the expected temporal mean of productivity is now constant
irrespective of species richness *n* (Appendix):

If the responses of all species are perfectly positively
correlated (*r*_{ij} = 1 for all pairs of species
*i* and *j*), then, for any *n*,

Otherwise, *E*_{e}[*V*_{n}] can be
either smaller or greater than
*E*_{e}[*V*_{1}], and thus the behavior of
*E*_{e}[*V*_{n}] as a function of species
richness *n* can be complex and idiosyncratic (16) depending
on the details of the system because of the correlations between
species responses at different times (Appendix). In the
special case when these responses are independent,
*E*_{e}[*V*_{n}] becomes smaller than
*E*_{e}[*V*_{1}] for a sufficiently large
*n* and converges to the minimum value
*Var*_{T}[*E*_{e}[*X*(*t*:1)]] (Fig.
(Fig.22*B*). Note that
*Var*_{T}[*E*_{e}[*X*(*t*:1)]] is the
temporal variance of the expected productivity at each time, and thus
vanishes if the species responses have no directional trend in time
(i.e., *E*_{e}[*X*(*s*:1)] = *E*_{e}[*X*(*t*:1)]
for any *s*, *t*).

#### Productivity-Diversity Pattern.

The pattern of average
ecosystem productivity as a function of species richness and the degree
of ecosystem redundancy can be greatly affected by both the species
responses and the degree of asynchronicity of these responses in the
ecosystem. Here, we call an ecosystem redundant (16, 19, 25) for some
functional process if this process has attained a plateau for a lower
value of species richness and is not enhanced by the addition of
further species in the system. We examine in particular the effect of
the variance of each species response,
*Var*[*X*_{i}(*t*)], on the species richness
required for an ecosystem to become redundant.

Assume that total ecosystem productivity is determined by dominance of
the most productive species, and all species responses are independent
of each other and obey the same stochastic process. Further assume that
the mean of the response is 1/2, as in the β-distribution (Fig.
(Fig.33*A*), in which case the
variance of the species response may be viewed as a measure of a
species’ contribution to maximum ecosystem productivity (in this case,
1). This is because a greater variance means a greater probability of a
species taking on the maximum productivity at each time. The results
are summarized in Fig. Fig.33*B*. For a given species richness, as
the variance of the species response increases, the temporal mean of
ecosystem productivity is elevated, that is, the effect of species
richness is enhanced. But as a result, the species richness beyond
which the ecosystem is redundant also decreases.

## DISCUSSION

This work shows two major effects of species richness on ecosystem
productivity in a fluctuating environment: (*i*) a reduction
in the temporal variance of productivity, in short a buffering effect,
and (*ii*) an increase in the temporal mean of productivity,
in short a performance-enhancing effect. We call both these effects
insurance effects in agreement with our definition (see
*Introduction*) because they both contribute to maintain or
enhance ecosystem functioning in the face of environmental
fluctuations.

Under what conditions can we expect such insurance effects to occur in an ecosystem? Our results show that asynchronicity of the species responses to environmental fluctuations is the basis for the buffering effect. In the extreme case of an ecosystem with only one species in a functional group, a low productivity of this species at a time results directly in a low ecosystem productivity at that time. On the other hand, in an ecosystem with a high species richness, species with high as well as low productivities can be expected to occur because of asynchronicity of their responses, so that a low productivity in some species does not necessarily affect ecosystem productivity. The origin of asynchronicity of species responses does not matter; it can be generated in many ways, whether by competitive release, physiologically determined differences in the response to environmental fluctuations or purely stochastic effects. On the other hand, the performance-enhancing effect needs, in addition to asynchronicity, some adaptive mechanism that gives greater weight to those species that perform better in each environmental condition, for example, selection by interspecific competition in the case of determination by dominance. In this case, the variation in productivity increases with species richness at any time and the selection process through dominance picks up the most productive species within this variation; as a result, productivity increases with species richness. In contrast, in the case of determination by equivalence, the mean ecosystem productivity remains constant irrespective of species richness because high and low productivities cancel each other out around the mean.

Regarding the buffering effect in the case of determination by equivalence, we obtain qualitatively similar results as those obtained by previous authors who used a different approach and more restrictive assumptions (17, 18). In these studies, total community productivity (biomass) was kept constant, and the variance of individual species’ productivity responses were assumed to be related to the mean by a power-law relation. Our results show that the buffering effect does not depend on these particular assumptions. In our modeling framework, these assumptions are not necessary but can be derived as consequences for a special case. Further, our model shows that correlations between species responses can cause more complicated behaviors.

For the present analysis, we assumed that all species have the same
response range, 0 ≤ *X*_{i}(*t*) ≤ 1
(Appendix). This assumption, however, can be easily relaxed,
which leads to slightly different predictions (unpublished results).
For instance, in the case of determination by dominance, if there are
two types of species A and B, such that species A always have a
productivity that is higher than that of species B, there is no
difference as regards ecosystem productivity between a monoculture of
one species A and a mixture of one A plus any number of B. That is,
species with a consistently low productivity have no contribution to
insurance effects, whatever their species richness. If species response
ranges overlap each other, however, all contribute to insurance
effects.

The importance of insurance effects is thus determined by three
factors: (*i*) the way ecosystem productivity is determined by
individual species responses to environmental fluctuations,
(*ii*) the degree of asynchronicity of these responses, and
(*iii*) their detailed characteristics including their range
of variation. To what extent are insurance effects expected to be
present in real ecosystems? On a long enough time scale for significant
environmental fluctuations to take place, species response ranges are
likely to usually overlap, and some degree of asynchronicity of
responses seems inevitable if we remember the high dimensionality of
the physiological niche space to which each species responds. As
already mentioned earlier, the real determination of total productivity
also should lie between the two extremes considered here, i.e.,
determination by dominance and determination by equivalence. Only a
slight selection mechanism favoring species that are better adapted to
current environmental conditions and have a higher than average
productivity under these conditions would suffice to deviate from the
perfect determination by equivalence and lead to a
performance-enhancing effect. Thus, it seems highly probable that
species richness has both a buffering and a performance-enhancing
effect in real ecosystems in the long term. Recent experimental
evidence on ecosystem predictability in aquatic microcosms (11, 12)
does support the buffering effect. The performance-enhancing effect
awaits experimental tests; it is likely to be more commonly found in
communities governed by strong competition for a limiting factor, such
as terrestrial plant communities. Food-web configuration may play a
critical role in more complex ecosystems, which we have
ignored deliberately here, the productivity of each trophic level being
likely to depend on features such as the number of trophic levels (26),
etc.

Our model also shows that the variance of the species responses has a
significant effect on the productivity-diversity pattern and on the
species richness required to reach ecosystem redundancy. Consider the
two limiting cases when the variance of the species responses is
maximum (α = 0) and zero (α = ∞) in Fig. Fig.33*B*. When
variance is maximum, because each species has a high probability of
achieving the maximum productivity at any time, the ecosystem becomes
redundant at a relatively small number of species (typically, of the
order of 10). All the additional species above the first 10 species
have little effect on total ecosystem productivity. On the contrary,
when variance is minimum, each species has a very small probability of
achieving the maximum productivity, and thus the species richness at
which the ecosystem becomes redundant is infinite. Thus, the higher the
probability each component species has to contribute to an ecosystem
process, the lower the species richness at which this process saturates
and the ecosystem becomes redundant.

This result can have profound consequences from a conservation point of view (2). Suppose that the species richness of an ecosystem is being reduced from, say, 100 species. If the ecosystem can be regarded as randomly constructed from a pool of species with little temporal variability, as might be the case for some tropical ecosystems, ecosystem productivity is expected to decrease gradually and roughly linearly. On the other hand, if it is composed of species with highly variable responses, as might be the case in temperate ecosystems, the change in ecosystem productivity is expected to be nonlinear and sudden. Ecosystem productivity would be maintained close to its maximum value as long as species richness is high enough for the ecosystem to be redundant, but would decline abruptly when species richness is further reduced beyond this point. Thus the characteristic responses of component species may greatly affect the response of ecosystem performance to changes in biodiversity.

## Acknowledgments

We thank A. Hector, M. Higashi, J. H. Lawton, P. J. Morin, S. Naeem, and N. Yamamura for their comments on the manuscript. This work was supported by the Environment and Climate program of the European Union and the Environnement, Vie et Sociétés program of the Centre National de la Recherche Scientifique (France).

## Appendix

We define the response range of species *i*, [*a*_{i},
*b*_{i}], as the range of productivity values that species
*i* can take on with a positive probability, where the
inequality 0 ≤ *a*_{i} ≤ *b*_{i} ≤ 1
holds. For simplicity, we assume in the following that the response
range of any species at any time is [0, 1], but this restriction can
be easily relaxed.

#### Determination by Dominance.

Let *q*(*x*_{1},
*x*_{2},…, *x*_{n}, *t*) be the joint
probability density distribution of *X*_{i}(*t*) (*i* =
1, 2,…, *n*), i.e., the joint probability density with
which each species *i* has productivity
*x*_{i} at time *t*, and
*q*(*x*_{1}, *x*_{2},…, *x*_{n},
*t*|*y*_{1}, *y*_{2},…, *y*_{n}, *s*)
the joint conditional probability density with which each species
*i* has productivity *x*_{i} at time
*t* under the condition that it had productivity
*y*_{i} at time *s* (*s* < *t*).
Similarly, let *p*_{n}(*x*, *t*) and
*p*_{n}(*x*, *t*|*y*, *s*) be the probability density
distribution and conditional probability density distribution of
*X*(*t*:*n*), respectively.

*E*_{e}[*X*_{n}] and
*E*_{e}[*V*_{n}] are expressed in terms of
*E*_{e}[*X*(*t*:*n*)],
*Var*_{e}[*X*(*t*:*n*)] and
*Cov*[*X*(*s*:*n*), *X*(*t*:*n*)] as in Eqs. 2 and
3, and thus rewritten in terms of
*E*_{e}[*X*^{k}(*t*:*n*)] (*k* is a positive
integer) and *E*_{e}[*X*(*s*:*n*)*X*(*t*:*n*)]. These in turn
can be expressed in terms of the probability density distributions as
follows:

where

Eqs. A1 and A2 are obtained by partial integration after substituting the following two equations (which are derived on the basis of elementary probability calculus) into them:

The following propositions are obtained from Eqs.
**A1–A6** except in the special case where all pairs of species
responses are perfectly correlated.

##### Proposition 1.

*p*_{n}(*x*, *t*) converges to the
delta function δ(*x* − 1) as species richness
*n* increases.

##### Proposition 2.

*E*_{e}[*X*(*t*:*n*)] increases
monotonically with species richness *n* toward the maximum
possible productivity in the species pool (in this case, 1). As a
result, the expected value of the temporal mean of productivity,
*E*_{e}[*X*_{n}], follows the same behavior.

##### Proposition 3.

*Var*_{e}[*X*(*t*:*n*)]
decreases toward zero for a sufficiently large species richness.

##### Proposition 4.

*Cov*_{e}[*X*(*s*, *n*), *X*(*t*,
*n*)] converges to zero as species richness *n* increases.
This means that the autocorrelation of the stochastic process
*X*(*t*:*n*) vanishes for a sufficiently large species richness.

##### Proposition 5.

*E*_{e}[*V*_{n}] vanishes
for a sufficiently large species richness.

The proofs of *Propositions 2–5* follow directly from
proposition 1 and can be obtained on request.

#### Proof of *Proposition 1.*

Because *Q*_{n}(*x*,
*t*) in Eq. A1 is a cumulative probability distribution
of *X*(*t*:*n*), 0 ≤ *Q*_{n}(*x*, *t*) ≤ 1 holds, and
*Q*_{n}(*x*, *t*) is a nonincreasing function of species
richness *n*:

When the inequality holds in Eq. A7,
*Q*_{n}(*x*, *t*) converges monotonically to zero because
the hypervolume *x*^{n} of the integration range of
variables *x*_{1}, *x*_{2},…,
*x*_{n} is embedded in the *n*-dimensional
hypercube [0, 1]^{n} and converges monotonically
to zero as *n* increases as long as *x* < 1. It
follows that the last term in the right-hand side of Eq. A1
decreases monotonically to 0. Thus,
*E*_{e}[*X*^{k}(*t*:*n*)] increases
monotonically to 1, which means that *p*_{n}(*x*, *t*)
converges to δ(*x* − 1) because the characteristic
functions of these two functions coincide at the limit *n* →
∞.

When all pairs of species responses are perfectly correlated, i.e.,
|*r*_{ij}| = 1 for all *i* and
*j*, either *X*(*t*:*n*) = *X*_{1}(*t*) for any
*n* ≥ 1 (*r*_{ij} = 1) or *X*(*t*:*n*) =
*X*_{1}(*t*) for *n* < *k* and *X*(*t*:*n*) =
*max* {*X*_{1}(*t*), *X*_{k}(*t*)} for
*n* ≥ *k* (*r*_{ij} = −1 for at least one pair)
holds, where *k* is the smallest integer such that
*X*_{1}(*t*) ≠ *X*_{k}(*t*). This means that
either *Q*_{n}(*x*, *t*) = *Q*_{1}(*x*, *t*) for any
*n* ≥ 1 or *Q*_{n}(*x*, *t*) =
*Q*_{1}(*x*, *t*) for *n* < *k* and
*Q*_{n}(*x*, *t*) = *Q*_{k}(*x*, *t*) for *n*
≥ *k*.

#### Determination by Equivalence.

It follows directly from Eq. 4b that:

The right-hand side of Eq. A8 is the
expected productivity of a monoculture,
*E*_{e}[*X*(*t*:1)]. Therefore, using Eqs. **2,
**Eqs. 7 and A9 follow.

The variance and covariance terms in Eq. 3 then can be further expressed as follows:

Because
Σ_{i<j}*Cov*_{e}[*X*_{i}(*s*),
*X*_{j}(*t*)] can take on both positive and negative values
depending on the combination of species responses in the ecosystem,
from Eq. 3, *E*_{e}[*V*_{n}] can
be greater than *E*_{e}[*V*_{1}] and behave
idiosyncratically as a function of species richness. Here, for further
analysis, we focus on three limiting cases, where
*Var*_{e}[*X*_{i}(*t*)]
*Var*(*t*) for all *i* is assumed for simplicity.

##### Independent responses:

*r*_{ij} = 0 for all
*i, j*.

Thus, *E*_{e}[*V*_{n}] converges to
*Var*_{T}[*E*_{e}[*X*(*t*:1)]] at a
speed of *O*(*n*).

##### Perfect positive correlation in responses:

##### Perfect correlation in responses:

*|r _{ij}| =
*1 for all

*i, j*.

Species can be classified into two groups, say A and B, such that
any two species in the same group have a perfect positive correlation
in their responses (*r*_{ij} = 1), while any two
species in different groups have a perfect negative correlation in
their responses (*r*_{ij} = −1). Assume that the two
groups have the same number of species, *m* (hence,
*n* = 2*m*). In this case,

If *i* and *j* belong to the same
group, *Cov*_{e}[*X*_{i}(*s*),
*X*_{j}(*t*)] = *Cov*_{e}[*X*_{i}(*s*),
*X*_{i}(*t*)]. Otherwise,
*Cov*_{e}[*X*_{i}(*s*), *X*_{j}(*t*)] =
−*Cov*_{e}[*X*_{i}(*s*),
*X*_{i}(*t*)]. It follows that:

If the two groups have unequal numbers of species, e.g.
*m* and *m* + 1, it can be shown similarly that
*E*_{e}[*V*_{n}] converges to
*Var*_{T}[*E*_{e}[*X*(*t*:1)]] but with
oscillations.

## References

*Theor. Popul. Biol*., in press.

**National Academy of Sciences**

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