AUSTRALASIAN JOURNAL OF ECOTOXICOLOGY
Vol. 14, pp. 31-35, 2008
Chemical concentration on population size
S H O R T
Hayashi et al
C O M M U N I C A T I O N
EXAMINING THE RELATIONSHIP BETWEEN CHEMICAL CONCENTRATION AND
EQUILIBRIUM POPULATION SIZE
Takehiko I Hayashi*1, Masashi Kamo2 and Yoshinari Tanaka1
1
Research Center for Environmental Risk, National Institute for Environmental Studies, 16-2 Onogawa, Tsukuba, Ibaraki
306-8506, Japan.
2
Research Institute of Science for Safety and Sustainability, National Institute of Advanced Industrial Science and Technology,
16-1 Onogawa, Tsukuba, Ibaraki 305-8569, Japan.
Manuscript received: 3/2/2009; accepted: 1/5/2009.
ABSTRACT
We present a model and method for examining the potential effect of toxic chemicals on the equilibrium size of a wildlife
population. As a case study, we applied the model and method to the effect of zinc on fathead minnow (Pimephales promelas)
populations. Our analysis suggested a simple linear relationship between zinc concentration and equilibrium population size
of fathead minnow, namely that an increase of 1 µg/L in zinc concentration can cause a decrease of 0.04 individuals/m3 in the
equilibrium population size. The major limitations of our model are ignorance of the bioavailability of zinc, the assumption
that the density effect and toxic effect act on the same life-history traits (e.g., fertility), and the assumption that the density
effect acts only via total population size.
Key words: ecological risk assessment; population level; population size; dose-response; zinc.
INTRODUCTION
Revealing the relationships between chemical concentration
and biological responses has been a major goal of
ecotoxicological studies (e.g., Hendriks and Enserink 1996;
Penttinen and Kukkonen 1998; Tanaka and Nakanishi
2001; Hendriks et al. 2005; Jonker et al. 2005) because the
relationships predict toxic effects at given concentrations and
thus provide a basis for quantitative ecological risk assessment
and management. In general, mathematical description of the
ecotoxicological concentration-response relationships at the
population-level can be complex because of nonlinearities
inherent in toxicological effects and population dynamics.
Quantitative relationships between chemical concentration
and toxic effects at the individual level (e.g., the effects on
individual survivability) are typically nonlinear and often
described by probit or logistic curves. In addition, quantitative
relationships between the toxic effects at the individual level
and the effects at the population level (e.g., the effect on
population growth rate) are also typically nonlinear (Caswell
2001). The two levels of nonlinearity make it difficult to
develop models to derive clear quantitative concentrationresponse relationships between chemicals and toxic effects
at the population level.
In this paper, we present a model linking chemical
concentration and equilibrium population size. We also
perform a case study that analyses the effect of zinc on
fathead minnow (Pimephales promelas) populations. We use
equilibrium population size as an index of the population-level
effect because this parameter has several advantages over other
indices of population-level effects, such as intrinsic growth
rate or the extinction probability of a population (Hayashi
et al. 2009). First, the concept of equilibrium population
size is easy to understand, even for a nonspecialist, which
provides a great benefit in risk communication and consensus*Author for correspondence, email: [email protected]
building among stakeholders (Landis and Kaminski 2007).
Second, equilibrium population size is relatively easy to
discuss based on a comparison with field data, because
field data generally include information about the number
of individuals. Third, population size is often used as a
practical target of conservation management. Thus, the use
of equilibrium population size as an index can be helpful in
the development of an integrated ecological risk management
approach that considers various factors in a common
framework. Fourth, population size has clear ecological
importance because decreases in equilibrium population
size can affect community structure and ecological services.
Thus, the analysis of equilibrium population size allows us to
provide intuitive, quantitative, and testable predictions about
population-level effects of toxic substances.
MODEL
We adopted the Leslie matrix model (Caswell 2001) to link
chemical concentration and equilibrium population size for
the following reasons. First, age-structured models such as the
Leslie matrix model are more suitable to integrate age-related
toxic effects (e.g., juvenile mortality) into population models
(e.g., Lin et al. 2005; Kamo and Naito 2008) than are non-agestructured models such as non-age-structured logistic growth
models. Second, the Leslie matrix model is a standard model
in population ecology, and analytical tools for the model are
available (Caswell 2001). Third, the simplicity of the Leslie
matrix model allows us to analyse the relationship between
chemical concentration and equilibrium population size. More
detailed models such as individual-based models are more
complex than necessary for the calculation of equilibrium
population size and both the model-building and analysis
of population dynamics are time-consuming. We did not
use models that consider environmental and demographic
stochasticities in this study. Although it is important to
31
AUSTRALASIAN JOURNAL OF ECOTOXICOLOGY
Vol. 14, pp. 31-35, 2008
Chemical concentration on population size
Hayashi et al
consider these stochasticities when estimating the extinction
probability of populations (Beissinger and McCullough 2002;
Morris and Doak 2002), our goal in this study was to estimate
equilibrium population size. Including stochastic effects does
not add essential information to the analysis of equilibrium
population size, because equilibrium population size itself
is independent of stochastic effects, whereas stochastic
effects cause the size of populations to fluctuate around the
equilibrium population size.
To present our model, we first use the following simple
two-stage matrix model (Neubert and Caswell 2000;
Caswell 2001) as an example. The changes in the number of
individuals in a time step are described as:
(1),
where n1(t) and n2(t) are the number of first-life-stage
individuals (juveniles) and second-life-stage individuals
(adults) at time t, respectively. The total population size is
N(t) = n1(t) + n2(t). The projection matrix, B, is a life-history
matrix given by:
(2),
where J, A, M and F represent juvenile survivability, adult
survivability, maturation probability and fertility, respectively.
We assume that both chemical toxicity and density dependence
affect only fertility. The regulation of fertility by density is
appropriate if there is a limit on food availability and the
number of offspring produced by a female is dependent on
her food intake; it is also appropriate for some types of strong
density-dependent mortality of young (Grant 1998).
Let us denote the rate of reduction in fertility by θ (0 ≤ θ ≤
1), such that Eq. (2) becomes
(see Appendix for the derivation). Although obtaining the
algebraic solution of θ * will become increasingly difficult
as more age classes are considered, the θ * value can always
be computed numerically.
If the two effects of density dependence and chemical
exposure are independent, θ can be given in the explicit
form as
(5),
Where e-bN describes the Ricker-type density dependence and
b (b>0) indicates the intensity of the dependence. Rickertype density dependence is assumed throughout this analysis
because it is one of the most standard and general models
of density effects in population ecology (Caswell 2001).
A Ricker-type density model was also used in previous
population-level ecotoxicological models for fish populations
(Grant 1998; Brown et al. 2003; Miller and Ankley 2004).
φ(x) describes a rate of reduction in fertility by chemical
exposure at a given concentration x, and it is assumed to be
a monotonically decreasing function of x with asymptotic
values of 1 at the limit of x → 0 and 0 at the limit of x → ∞.
The form of function φ is determined by the concentration–
response relationship derived by toxicity tests.
The total population size at equilibrium (Neq) can be derived
by combining Eqs. (4) and (5):
(6).
Eq. (6) gives Neq as
(7).
Eq. (7) gives the equilibrium population size as a function
of chemical concentration. Note that in the absence of the
chemical (i.e., x = 0), we have
[
(3).
In general, the population growth rate (λ) is defined by the ratio
of population sizes between two subsequent generations after
the population has reached a stable age distribution (Caswell
2001, p. 87), and λ is given by the dominant eigenvalue of the
life-history matrix (Caswell 2001, p. 72). In Eq. (3), there is
a unique θ (say θ *) that results in the dominant eigenvalue
λ = 1, in which case the decrease in fertility does not cause
population size to change in subsequent generations. The θ
* must satisfy the following relationship:
(4)
32
= 1 and
].
This gives a total population size in the absence of the
adverse effects of chemicals. Eq. (7) shows that equilibrium
population size can be predicted by specifying only the
toxicity function φ(x) from ecotoxicological test data if θ *
and b can be calculated from previous ecological studies on
focal species (see the next section). Note that Eq. (7) holds
even when more age classes are considered as long as the
density dependence and adverse effect of chemicals work
only on fertility.
APPLICATION OF THE MODEL TO THE
FATHEAD MINNOW
Here, we present an analysis of the population-level effect of
zinc on the fathead minnow (P. promelas). The life-history
matrix of the fish based on age class was composed by Miller
and Ankley (2004). Kamo and Naito (2008) calculated the
AUSTRALASIAN JOURNAL OF ECOTOXICOLOGY
Vol. 14, pp. 31-35, 2008
Chemical concentration on population size
Hayashi et al
(8),
where ni is the number of individuals at age i and the total
population size is N(t) = n1(t) + n2(t) + n3(t). Using birth
pulse fertility values and a prebreeding census with an annual
time step, fertility (Fi) values are defined as the product of
survival from eggs to age 1 year and the reproductive output
(i.e., the number of eggs) of an individual upon reaching
age i (Caswell 2001; Miller and Ankley 2004). Si is the
survivability of an individual (i.e., the probability that an
individual does not die) from age i to i + 1. We assume that
the rate of reduction in fertility [denoted by e-bN φ(x)] is the
same for all age classes.
The value of θ = [e-bN φ(x)] satisfying the population
growth rate λ = 1 (i.e., θ of the equilibrium population) can
be numerically calculated as θ * = 0.56 from Eq. (8). The
relationship between zinc concentration and equilibrium
population size is then:
(9).
Eq. (9) suggests that equilibrium population size decreases
linearly with increasing zinc concentration when x ≥ 30
(Figure 1), and a zinc concentration of x = 172 µg/L results
in population extinction [i.e., Neq(172) = 0]. Note that the
concentration leading to population extinction (172 µg/L)
���
���
relationship between zinc concentration and egg production
rate using the toxicity tests conducted by Brungs (1969),
which is φ(x) = exp[5.647 – 0.00409x]/exp[5.5243], where x
is the concentration of zinc (µg/L). The function is normalised
to be 1 at a zinc concentration of 30 µg/L, which was the
control concentration in the test by Brungs (1969). In the
following analysis, we assume that there is no adverse effect
on egg production at concentrations ≤ 30 µg/L (i.e., φ(x) = 1
when x ≤ 30). The acute half-lethal concentration (LC50) of
the fathead minnow is high (e.g., 96-h zinc LC50 of juvenile
fathead minnow was reported as 2.61 mg/L by Broderius and
Smith 1979), and therefore we assume that the death of the
fish by zinc exposure does not occur within the concentration
ranges we examine here (less than about 200 µg/L). We
assume Ricker-type density-dependent fertility (e.g., Levin
and Goodyear 1980; Grant 1998; Brown et al. 2003), as in
Eq. (5). By combining the density-dependent terms and the
density-independent life-history matrix that considers the
toxic effect by zinc (Kamo and Naito 2008), the changes in
the number of individuals in a year are described as:
�
�
��
���
��������������������������
Figure 1. Relationships between zinc concentration and equilibrium
population size. The values of equilibrium population size are
relative to the Neq in the absence of zinc.
is the same as that predicted from a density-independent
model (Kamo and Naito 2008). When N = 0, the densitydependent term (e-bN) equals 1 and Eq. (8) is identical to
the life-history matrix of the density-independent model
(Kamo and Naito 2008). Because the concentration leading
to population extinction is defined as the concentration that
satisfies the condition of population growth rate = 1 (e.g.,
Lin et al. 2005; Kamo and Naito 2008), this concentration is
also identical to the concentration for Neq = 0 in our densitydependent model. Reported density estimates of fathead
minnow field populations range from 5.19 to 6.14 fish/m3,
and the midpoint of the range was 5.665 fish/m3 (Miller and
Ankley 2004; Miller et al. 2007). We use 5.665 fish/m3 as a
density estimate and assume that Neq(0) = 5.665 fish/m3. The
intensity of the density effect can then be calculated as b =
0.10235 by Eq. (9). This means that an increase of 1 µg/L in
zinc concentration causes a decrease of 0.040 individuals/m3
in the equilibrium population size, providing intuitive insights
into the population-level effect of zinc on fathead minnow
populations.
DISCUSSION
We presented a model to link chemical concentration and
toxic effect at the population level. As a case study, our
model was applied to the effect of zinc on the fathead
minnow population and predicted a simple linear relationship
between zinc concentration and equilibrium population size.
Our model uses equilibrium population size as an index of
population-level effects, which enables our model to provide
quantitative and testable predictions about population-level
effects of toxic substances. For example, Eq. (9) predicts that
an increase of 1 µg/L in zinc concentration causes a decrease
of 0.04 individuals/m3 in the equilibrium population size of
fathead minnow. The quantitative presentation of ecological
risk based on the decreased number of individuals would
also be useful in the analysis of the economic impact on
fisheries. Moreover, the validity of Eq. (9) can be tested by
examining the relationship between the zinc concentration
and equilibrium population density of fish in the field. These
benefits of our model will substantially enhance the feedback
between field, laboratory, and theoretical studies of ecological
risk assessment and management.
33
AUSTRALASIAN JOURNAL OF ECOTOXICOLOGY
Chemical concentration on population size
Another feature of our model is the explicit consideration
of density effects. In general, most natural populations are
subject to density effects. However, most previous studies on
population-level effects of toxic chemicals did not consider
density effects (e.g., Forbes et al. 2001; Stark et al. 2004;
Lin et al. 2005). The ignorance of density effects carries the
risk of providing incorrect predictions about population-level
effects of toxic chemicals (Moe 2007; Hayashi et al. 2009).
Therefore, it is important to develop computational methods
that consider both density dependence and the adverse effects
of chemicals, as our model does. Although we assumed only
Ricker-type density dependence, our approach can be applied
to other types of density dependence as well.
Our model and analysis have several limitations. First, the
model is only applicable to cases in which the density effect
and toxic effect act on the same matrix element in life tables
(so that these effects can be expressed in a single variable,
θ). Second, we assumed that density-dependent effects act
only via total population size, and this may not be valid in
many populations. Third, our analysis of the effect of zinc
did not consider bioavailability of zinc and water hardness,
and the toxicity of zinc is known to depend on these
factors. Fourth, we ignored interspecific interactions and
environmental variability, although these factors also tend
to regulate population size in the field. Finally, our model
did not consider the degree of uncertainty involved in the
estimation of equilibrium population size. Removing these
limitations is necessary for improving the model and analysis
presented in this paper.
ACKNOWLEDGEMENT
This study was supported by the Global COE Program E03
(Eco-risk Asia) of the Ministry of Education, Culture, Sports,
Science and Technology of Japan.
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Then, the log transformation of Eq. (4) leads to Eq. (7):
and
(7).
APPENDIX
This appendix shows the derivation of Eqs. (4) and (5). From
Eqs. (1) and (2), a straightforward matrix calculation reveals
that n1(t + 1) and n2(t + 1) are:
(A1)
and
(A2).
At the equilibrium population in which population size is
unchanged, the number of individuals in each life stage
satisfies the following conditions:
(A3)
and
(A4),
where n1* and n2* are the equilibrium number of individuals
in the first and second life stages, respectively. From Eqs.
(A2) and (A4), n2* is derived as:
(A5).
Substituting Eq. (A5) into (A1) leads to Eq. (4), as follows:
35