EXAMINING THE RELATIONSHIP BETWEEN CHEMICAL CONCENTRATION AND EQUILIBRIUM POPULATION SIZE

Transcription

1 S H O R T 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 Kamo 2 Yoshinari Tanaka 1 1 Research Center for Environmental Risk, National Institute for Environmental Studies, 16-2 Onogawa, Tsukuba, Ibaraki , Japan. 2 Research Institute of Science for Safety Sustainability, National Institute of Advanced Industrial Science Technology, 16-1 Onogawa, Tsukuba, Ibaraki , Japan. Manuscript received: 3/2/2009; accepted: 1/5/2009. ABSTRACT We present a model 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 method to the effect of zinc on fathead minnow (Pimephales promelas) populations. Our analysis suggested a simple linear relationship between zinc concentration 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/m 3 in the equilibrium population size. The major limitations of our model are ignorance of the bioavailability of zinc, the assumption that the density effect toxic effect act on the same life-history traits (e.g., fertility), 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 biological responses has been a major goal of ecotoxicological studies (e.g., Hendriks Enserink 1996; Penttinen Kukkonen 1998; Tanaka Nakanishi 2001; Hendriks et al. 2005; Jonker et al. 2005) because the relationships predict toxic effects at given concentrations thus provide a basis for quantitative ecological risk assessment 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 population dynamics. Quantitative relationships between chemical concentration toxic effects at the individual level (e.g., the effects on individual survivability) are typically nonlinear often described by probit or logistic curves. In addition, quantitative relationships between the toxic effects at the individual level 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 toxic effects at the population level. In this paper, we present a model linking chemical concentration 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 underst, even for a nonspecialist, which provides a great benefit in risk communication consensus- building among stakeholders (Lis 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 ecological services. Thus, the analysis of equilibrium population size allows us to provide intuitive, quantitative, testable predictions about population-level effects of toxic substances. MODEL We adopted the Leslie matrix model (Caswell 2001) to link chemical concentration 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 Naito 2008) than are non-agestructured models such as non-age-structured logistic growth models. Second, the Leslie matrix model is a stard model in population ecology, 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 equilibrium population size. More detailed models such as individual-based models are more complex than necessary for the calculation of equilibrium population size both the model-building analysis of population dynamics are time-consuming. We did not use models that consider environmental demographic stochasticities in this study. Although it is important to *Author for correspondence, hayashi.takehiko@nies.go.jp 31

2 consider these stochasticities when estimating the extinction probability of populations (Beissinger McCullough 2002; Morris 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 Caswell 2000; Caswell 2001) as an example. The changes in the number of individuals in a time step are described as: (1), where n 1 (t) n 2 (t) are the number of first-life-stage individuals (juveniles) second-life-stage individuals (adults) at time t, respectively. The total population size is N(t) = n 1 (t) + n 2 (t). The projection matrix, B, is a life-history matrix given by: (2), where J, A, M F represent juvenile survivability, adult survivability, maturation probability fertility, respectively. We assume that both chemical toxicity density dependence affect only fertility. The regulation of fertility by density is appropriate if there is a limit on food availability 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 (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), 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) (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 chemical exposure are independent, θ can be given in the explicit form as (5), Where e -bn describes the Ricker-type density dependence b (b>0) indicates the intensity of the dependence. Rickertype density dependence is assumed throughout this analysis because it is one of the most stard 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 Ankley 2004). φ(x) describes a rate of reduction in fertility by chemical exposure at a given concentration x, it is assumed to be a monotonically decreasing function of x with asymptotic values of 1 at the limit of x 0 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 (N eq ) can be derived by combining Eqs. (4) (5): Eq. (6) gives N eq as (6). (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 [ = 1 ]. 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 * 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 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 Ankley (2004). Kamo Naito (2008) calculated the 32

3 relationship between zinc concentration egg production rate using the toxicity tests conducted by Brungs (1969), which is φ(x) = exp[ x]/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 Smith 1979), 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 Goodyear 1980; Grant 1998; Brown et al. 2003), as in Eq. (5). By combining the density-dependent terms the density-independent life-history matrix that considers the toxic effect by zinc (Kamo Naito 2008), the changes in the number of individuals in a year are described as: (8), where n i is the number of individuals at age i the total population size is N(t) = n 1 (t) + n 2 (t) + n 3 (t). Using birth pulse fertility values a prebreeding census with an annual time step, fertility (F i ) values are defined as the product of survival from eggs to age 1 year the reproductive output (i.e., the number of eggs) of an individual upon reaching age i (Caswell 2001; Miller Ankley 2004). S i 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 equilibrium population size is then: (9). Eq. (9) suggests that equilibrium population size decreases linearly with increasing zinc concentration when x 30 (Figure 1), a zinc concentration of x = 172 µg/l results in population extinction [i.e., N eq (172) = 0]. Note that the concentration leading to population extinction (172 µg/l) Neq Concentration of Zn (!g/l) Figure 1. Relationships between zinc concentration equilibrium population size. The values of equilibrium population size are relative to the N eq in the absence of zinc. is the same as that predicted from a density-independent model (Kamo Naito 2008). When N = 0, the densitydependent term (e -bn ) equals 1 Eq. (8) is identical to the life-history matrix of the density-independent model (Kamo 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 Naito 2008), this concentration is also identical to the concentration for N eq = 0 in our densitydependent model. Reported density estimates of fathead minnow field populations range from 5.19 to 6.14 fish/m 3, the midpoint of the range was fish/m 3 (Miller Ankley 2004; Miller et al. 2007). We use fish/m 3 as a density estimate assume that N eq (0) = fish/m 3. The intensity of the density effect can then be calculated as b = by Eq. (9). This means that an increase of 1 µg/l in zinc concentration causes a decrease of individuals/m 3 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 toxic effect at the population level. As a case study, our model was applied to the effect of zinc on the fathead minnow population predicted a simple linear relationship between zinc concentration equilibrium population size. Our model uses equilibrium population size as an index of population-level effects, which enables our model to provide quantitative 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/m 3 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 equilibrium population density of fish in the field. These benefits of our model will substantially enhance the feedback between field, laboratory, theoretical studies of ecological risk assessment management. 33

4 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;. 2009). Therefore, it is important to develop computational methods that consider both density dependence 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 analysis have several limitations. First, the model is only applicable to cases in which the density effect 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, this may not be valid in many populations. Third, our analysis of the effect of zinc did not consider bioavailability of zinc water hardness, the toxicity of zinc is known to depend on these factors. Fourth, we ignored interspecific interactions 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 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 Technology of Japan. REFERENCES Beissinger SR McCullough DR (Eds) Population Viability Analysis. University of Chicago Press, Chicago, USA. 593 pp. Broderius S Smith LL Lethal sublethal effects of binary mixtures of cyanide hexavalent chromium, zinc, or ammonia to the fathead minnow (Pimephales promelas) rainbow trout (Salmo gairdneri). Journal of Fisheries Research Board of Canada 36, Brown AR, Riddle AM, Cunningham NL, Kedwards WJ, Shillabeer N Hutchinson TH Predicting the effects of endocrine-disrupting chemicals on fish populations. Human Ecological Risk Assessment 9, Brungs WA Chronic toxicity of zinc to the fathead minnow, Pimephales promelas Rafinesque. Transactions of the American Fisheries Society 98, Caswell H Matrix Population Models: Construction, Analysis, Interpretation. 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5 Morris WF Doak FD Quantitative Conservation Biology: Theory Practice of Population Viability Analysis. Sinauer Associates, Sunderl, USA. 480 pp. Neubert MG Caswell H Density-dependent vital rates their population dynamic consequences. Journal of Mathematical Biology 41, Penttinen OP Kukkonen J Chemical stress metabolic rate in aquatic invertebrates: threshold, dose-response relationships, mode of toxic action. Environmental Toxicology Chemistry 17, Stark JD, Banks JE Vargas R How risky is risk assessment: the role that life history strategies play in susceptibility of species to stress. Proceedings of the National Academy of Sciences USA 101, Tanaka Y Nakanishi J Model selection parameterization of the concentration-response functions for population-level effects. Environmental Toxicology Chemistry 20, (4). Then, the log transformation of Eq. (4) leads to Eq. (7): (7). APPENDIX This appendix shows the derivation of Eqs. (4) (5). From Eqs. (1) (2), a straightforward matrix calculation reveals that n 1 (t + 1) n 2 (t + 1) are: (A1) (A2). At the equilibrium population in which population size is unchanged, the number of individuals in each life stage satisfies the following conditions: (A3) (A4), where n 1 * n 2 * are the equilibrium number of individuals in the first second life stages, respectively. From Eqs. (A2) (A4), n 2 * is derived as: (A5). Substituting Eq. (A5) into (A1) leads to Eq. (4), as follows: 35