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Dec 10, 2001 - AN ILLUSTRATION WITH BROWN TROUT. ARNAUD .... brown trout due to the spatial segregation of age classes (Fig. 1). No spatial shift ...
Environmental Toxicology and Chemistry, Vol. 22, No. 5, pp. 958–969, 2003 q 2003 SETAC Printed in the USA 0730-7268/03 $12.00 1 .00

ECOTOXICOLOGY AND SPATIAL MODELING IN POPULATION DYNAMICS: AN ILLUSTRATION WITH BROWN TROUT ARNAUD CHAUMOT,*† SANDRINE CHARLES,† PATRICK FLAMMARION,‡ and PIERRE AUGER† †UMR CNRS 5558 Laboratoire de Biome´trie et de Biologie Evolutive, 43 Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France ‡Ministe`re de l’Ecologie—D4E Service de la Recherche et de la Prospective, 20 avenue de Se´gur, 75302 Paris 07 SP, France ( Received 10 December 2001; Accepted 18 September 2002) Abstract—We developed a multiregion matrix population model to explore how the demography of a hypothetical brown trout population living in a river network varies in response to different spatial scenarios of cadmium contamination. Age structure, spatial distribution, and demographic and migration processes are taken into account in the model. Chronic or acute cadmium concentrations affect the demographic parameters at the scale of the river range. The outputs of the model constitute populationlevel end points (the asymptotic population growth rate, the stable age structure, and the asymptotic spatial distribution) that allow comparing the different spatial scenarios of contamination regarding the demographic response at the scale of the whole river network. An analysis of the sensitivity of these end points to lower order parameters enables us to link the local effects of cadmium to the global demographic behavior of the brown trout population. Such a link is of broad interest in the point of view of ecotoxicological management. Keywords—Ecotoxicology

Population level

Spatial modeling

Brown trout

Ecotoxicology is currently addressing the issue of assessing the impact of environmental pollution on natural populations by using data extracted from bioassays (exposure of small groups of individuals to a toxicant in the laboratory). Introducing effects of pollutants on demographic parameters into population dynamics models (i.e., demographic methodology [1]) is one way to examine impacts at the population level. Such an extrapolation step often assumes that individuals exposed to the toxicant accomplish their life cycle at only one place. However, many species move between several habitats, e.g., due to the ecology of their different development stages. Ecotoxicological studies underline the relevant role played by spatial distribution of individuals in fragmented populations, for instance [2,3]. Some processes of refuge or certain ‘‘actions at distance’’ [3] emerge from such a metapopulation approach. The spatial dimension of toxicant exposure modulates population response to pollutant discharges. Caswell [1] thus stresses the importance of integrating the spatial dimension into ecotoxicological population models. Therefore, focus on a larger spatial scale is required in order to study toxic effects on population dynamics. Spatial modeling in a metapopulation approach is a way to solve this issue. As an example, we developed multiregion matrix population models to explore how the demography of a hypothetical brown trout population living in a river network varies in response to different spatial scenarios of cadmium contamination. The main objective of our work is to elucidate spatial population modeling methodologies in an ecotoxicological framework. We previously showed how the high dimension of spatial systems can be treated by using aggregation methods

[4]. However, this methodology is only suitable under certain assumptions concerning the migration processes occurring in the population [5,6]. In the case of a trout population, we had to suppose a complete mixing of individuals each year into the river network. Although some authors discuss the existence of a mobile fraction in trout populations [7–12], the literature mainly reports sedentary behaviors during the year and site fidelity during seasonal movements for resident brown trout in river networks [7,12–16]. Thus, we present here a model following these assumptions of spatial behaviors without population mixing. We refer to resident brown trout populations observed in western France characterized by two seasonal migrations [17–23]. Our multiregion Leslie model considers chronic or acute toxicity of cadmium in terms of survival and fecundity reduction through dose–response curves. We test different scenarios that correspond to different spatial patterns of exposure to cadmium. Age structure, spatial distribution, and demographic and migration processes are taken into account. The outputs of this model constitute population-level end points (the asymptotic population growth rate, the stable age structure, and the asymptotic spatial distribution) that allow comparing the different spatial scenarios of contamination. The analysis of the sensitivity of these end points to lower order parameters enables us to link the local effects of cadmium to the global impact on demography [1,24]. In other words, this allows us to examine how, where, and which vital rates or age classes are involved in explaining the effect of pollution on the population. Such a link is important in the context of ecotoxicological management. Indeed, it solves the issue of the loss of information due to the extrapolation process and change of scale.

* To whom correspondence may be addressed ([email protected]). Presented at the 22nd Annual Meeting, SETAC North America, Baltimore, Maryland, USA, November 11–15, 2001.

A hypothetical brown trout population (Salmo trutta) is considered in this study. The general framework is close to previous contributions [4,25,26].

INTRODUCTION

BIOLOGICAL DATA

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Table 1. Demographic parameters in the river network (YOY01 5 young of the year, juv11 5 one-year-old juveniles)a,b Level

YOY01

Juv11

Adults

1 2 3 4 1 2 3 4

— — — — s1 s1/a s1/a2 s1/a3

— — — — s2/a2 s2/a s2 s2/a

200 230 270 300 s3/a3 s3/a2 s3/a s3

Average length (mm)

Survival rates

a

s1, s2, and s3: maximal annual survival rates of each age-class, estimated with data from Maisse and Baglinie`re [23]; s1 5 0.07; s2 5 s3 5 0.4. b a 5 gradient parameter in annual survival rates fit by Charles et al. [25]; a 5 2. Average lengths are chosen according to field studies [17,21,23]. Fig. 1. Life cycle of resident brown trout, Salmo trutta fario. (YOY01 5 young of the year; juv1 5 one-year old juveniles.)

Demographic process According to field studies [23,27], the population is subdivided into three age classes, i.e., young of the year (YOY01), one-year-old juveniles (juv11), and adults. The reference life cycle presented in Figure 1 reflects features of resident trout populations living in western France [17–23]. Four symmetric levels comprise the theoretical river network (i.e., 15 patches standing for 15 stretches of river; see Fig. 2). The dynamic process takes place in each patch. Annual survival rates and fecundity are assumed to depend on the level in the river network (Table 1). A decreasing gradient from up- to downstream affects YOY01 annual survival rates (probability that a YOY01 survives to one-year-old juvenile stage), i.e., the habitat conditions in the first levels of the river networks (e.g., oxygen, bed [gravel], temperature) result in higher survival rates of YOY01. Inversely, adult survival rates (probability that an adult survives from year to year) increase from up- to downstream. Annual survival rates of juv11 are

maximal in the third level. Such patterns of heterogeneity in survival rates between river stretches are explained by the fact that feeding and growth areas for the older age classes are situated downstream in river networks. Thus, the heterogeneity of habitat from up- to downstream and the requirements of the different age classes lead to a spatial segregation of young and old age classes [23]. This segregation is evident in spatial distributions observed in field studies (Table 2). The fecundity (number of ova per female) depends on the size of the female [23]. In addition, adults resident downstream tend to be larger than adults resident upstream [23]. Therefore, mean size of adult females is specific to each level according to field studies [17,21,23]. Chosen values are reported in Table 1. The fecundity f (number of eggs produced by female during one spawning season) is calculated using the relationship between fecundity and fish length l as f 5 5.2719 3 l 2 689.12 [21]. The fertilization rate is assumed to be 0.9 [23] and the survival rate from fertilized egg stage to new-emerged fry stage to be 0.5 (J.L. Baglinie`re, Laboratoire d’e´cologie Aquatique—INRA, Rennes, France, personal communication [28]).

Migrations Two major spatial changes occur during the life cycle of brown trout due to the spatial segregation of age classes (Fig. 1). No spatial shift affects the distribution of individuals during the first year; YOY01 remain in the stretch of river where they were hatched [19,29–32]. A downstream movement is described in spring for the juvenile stage [18–20,23]. This corresponds to a spatial shift between nursery areas (upstream) and growth areas (downstream). Quantitative data about this migration are often unavailable. We thus describe it by a very simplistic pattern with only three parameters (Fig. 3). In order to fit these parameters, we use the spatial distributions of YOY01 and juv11 obTable 2. Distribution in autumn of each age class in the different levels of the river network; estimates from field data [23] (YOY01 5 young of the year, juv11 5 one-year-old juveniles) River network level

Fig. 2. Reference river network, 15 patches in four levels from upto downstream.

1 2 3 4

YOY01 (%)

Juv11 (%)

Adults (%)

51 33 12 4

2 18 35 45

0.3 16 40 43.7

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Fig. 3. Downstream migration of one-year-old juveniles (juv11) in spring. The migration rates a, b, and g are fit using the spatial distributions of YOY01 (young of the year) and juv11 observed in autumn in the field (Table 2). The fit assumes a survival of six months for YOY01, then a downstream migration event, and afterward, six months of survival for juv11 (survival process described in Table 1): a 5 20%, b 5 25%, g 5 50% after fitting.

served in autumn (reported in Table 2), taking into account survival and migration processes (see legend of Fig. 3). The fits of the parameter values are within the range reported through the rudimentary data of the literature [18–20,33]. We assume that juv11 (after this downstream migration) and adults (except for spawning) stay in the same stretch of river. During the end of the year (November–December), adults migrate upstream in order to spawn [17,22,34]. After spawning, the adults quickly return to their resident stretch of river just after the reproduction event [12,13,15]. The use of the qualitative pattern of this upstream migration results from a literature review [12,13,15,17,18,21], and the modeling approach is presented by Charles et al. [26]. This pattern is illustrated in Figure 4. Only one parameter, v, is introduced. We assume the adults living in level 4 mainly spawn in level 2 (respectively, level 3 in level 1, level 2 in level 1, level 1 in level 1). Nevertheless, some adults from level 4 spawn in the other levels. The parameter v stands for the ratio between the proportion of breeders going to the major level of reproduction and the proportion going to adjacent levels. For instance, if p is the proportion of adults from level 4 spawning in level 2, the proportion spawning in level 1 is p/v, the proportion spawning in level 3 is p/v, the proportion spawning in level 4 is p/v2. These proportions verify p 1 p/v 1 p/v 1 p/v2 5 1. Thus, it is that, for a given v, p is known (p 5 v2/

Fig. 4. Pattern for the upstream migration of adult breeders in late autumn. The migration rates are a function of v, the ratio between flows linking one level to two adjacent spawning levels: v 5 2 after fitting.

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Fig. 5. Chronic cadmium dose–response curves from logistic regressions applied to long-term bioassay results [35,36]. (1) Reduction in the fecundity of adults exposed during one year of exposure, (2) reduction in the annual survival rate of YOY01 (young of the year) exposed during one year of exposure, (3) reduction in the annual survival rate of 100-g-weight individuals exposed during one year of exposure, where reduction 5 (rate observed with exposed fishes)/(rate observed in control).

(1 1 2v 1 v2). The same process is applied for other starting levels, but with only three possible spawning levels for adults living in level 3, two for adults in level 2, and one for adults in level 1. Moreover, in order to calculate the migration rates between patches, we assume that the breeders starting from a given patch are symmetrically distributed into the different patches of the target spawning level. Different values for v are tested with the qualitative assumption v . 1.

Ecotoxicological aspects Cadmium effects on demographic parameters were expressed by means of dose–response curves (Figs. 5 and 6). The chronic curves are the same as in a previous contribution [4], corresponding to bioassay results from the literature [35,36]. First, they describe the insensitivity of young stages to chronic cadmium concentrations. Indeed, hatchability, survival to emergence, and annual survival rate during the first year (YOY01 survival rate) are not affected by chronic exposure. Fertilization rate is also assumed to be unaffected, according to conclusions of experiments in contaminated water with a heavy metal mixture [37]. Fecundity and annual survival

Fig. 6. Acute cadmium dose–response curves from logistic regressions applied to 96-h-exposure bioassay results [35,38,39]. (1) Survival rate of YOY01 (young of the year) exposed during 96 h of exposure, (2) survival rate of 100-g-weight individuals exposed during 96 h exposure.

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i 5 2 for juv11; i 5 3 for adults). Then we get 45 state variables gathered in the vector n(t) in order to describe the total population in spring of year t. This vector is a set of population vectors ni(t) describing the internal structure of each age class as follows: n(t) 5 [n1 (t)

n2 (t)

n3 (t)]T

(1)

where

ni (t) 5 [n(i,1) (t) · · · n(i,j) (t) · · · n(i,15) (t)]T

(2)

Here, a superior T denotes transposition.

Life cycle: Demographic and migration processes Let s(i,j) represent the annual survival rate of age class i on patch j and fj be the fecundity of adult females on patch j (number of eggs per female produced each year). On patch j, YOY01 in year t becomes juv11 in year t 1 1 with the survival rate s(1,j) and

n(2,j)(t 1 1) 5 s(1,j)n(1,j)(t) Fig. 7. Tested spatial scenarios of pollution. Pollution-free network. Scenario patch 1, pollution occurs in patch 1. Scenario patch 2, cadmium discharges take place in patch 2; the pollutant spreads downstream in patch 1. Scenario patch 3, patch 3 is the source of pollution. Scenario patch 5, discharges occur in patch 5. In each scenario, the cadmium concentration is divided by two between two successive patches from up- to downstream due to dilution (e.g., from patch 2 to patch 1 in scenario patch 2).

rate of 100-g-weight individuals are reduced by such a chronic exposure. These curves are applied to juv11 and adults. The acute curves (Fig. 6) correspond to reductions of survival after a 96-h exposure to cadmium [35,38,39]. Migrations are assumed to be invariant in the presence of cadmium. Indeed, brown trout seem to avoid heavy metals with a lower sensitivity than do other species, such as rainbow trout [40]. Different pollution scenarios were tested (Fig. 7). First, we consider a pollution-free network. Second, we consider chronic or acute pollution occurring in the network branch composed of patches 1, 2, 3, and 5. Chronic pollution corresponds to contamination with low concentrations of cadmium present all along the year. Acute pollution is taken to be an event of 96 h with high concentrations occurring every year between the downstream migration of juv11 and the upstream breeding movement of adults. The tested concentrations (approximately up to 100 mg/L for the chronic case and up to 100 mg/L for the acute one) are those employed in bioassays. These concentrations also correspond to the observed concentrations in real polluted rivers [40–44]. In the case of polluted scenarios, the toxicant is present in the river network from the patch of discharge to downstream (Fig. 7). The dilution effect divides the cadmium concentration by two between two successive patches. We test different concentrations of cadmium emission. Changes in the location of the cadmium discharge are also examined in order to take into account the spatial dimension of exposure, i.e., discharges occur successively in patches 1, 2, 3, and 5 (scenarios patch 1, patch 2, patch 3, and patch 5, respectively). MATHEMATICAL MODELING

Let ni,j(t) be the number of individuals of age class i on patch j in April of year t (j 5 1, . . . , 15; i 5 1 for YOY01;

(3)

So we can link n1(t) and n2(t 1 1) with following matrix equation

n2 (t 1 1) 5 S1n1 (t)

with

(4)

S1 5 diag{s(1,1) , . . . , s(1,j) , . . . , s(1,15) }

(5)

We could model the survival of juv11 and adults in a similar way, i.e., as

n3 (t 1 1) 5 S2n2 (t) 1 S3n3 (t)

with

Si 5 diag{s(i,1) , . . . , s(i,j) , . . . , s(i,15) }

(6) (7)

But we have to take into account migrations for these two age classes. Juv11 migrate downstream in spring according to the pattern discussed in the previous section (Fig. 3). This pattern of downstream migration is modeled by a square matrix MD (Scheme 1). The entry at row r and column c of MD is the proportion of juvenile trout on patch c going to patch r in spring; e.g., a proportion b of the juveniles on patch 5 migrate to patch 2. The structure of the juv11 age class just after the downstream migration is given by MDn2(t)

(9)

Then the first term in Equation 6, which stands for survival of these juv11, has to be replaced by S2MDn2(t)

(10)

That is not the case for the second term relative to adult survival. Indeed, a site-fidelity behavior is well described for brown trout whereby adults migrate upstream for breeding late in autumn and go back just after spawning to their starting stretch of river (in December). So we do not take into account any spatial change in the transition between adults in year t and adults in year t 1 1. To conclude, Equation 6 becomes

n3(t 1 1) 5 S2MDn2(t) 1 S3n3(t)

(11)

The last transition in the life cycle is the reproductive process. Adult females produce eggs with fecundity rates fj depending on their living stretch of river (Table 1). Moreover, we consider that adult survival on patch j during half a year (between spring and autumn) is given by (s(3,j))1/2. So, if we assume that the sex ratio is 0.5 for the adult class, the number of eggs laid in December of year t in patch j would be

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Scheme 1.

0.5 f jÏs(3,j) n(3,j) (t)

(12)

If e is the fertilization rate and g the survival rate between the embryo stage and emergence (i.e., survival under gravel), the number of new emerged trout on patch j in year t 1 1 would be calculated as

n(1,j) (t 1 1) 5 0.5eg f jÏs(3,j) n(3,j) (t)

(13)

So

n1 (t 1 1) 5 0.5egF ÏS3 n3 (t)

with

F 5 diag{ f 1 , . . . , f j , . . . , f 15 }

and

ÏS3 5 diag{Ïs(3,1) , . . . , Ïs(3,j) , . . . , Ïs(3,15) }

(14)

Effects of pollutant on demography

(15)

Chronic pollution scenarios. Chronic concentrations of cadmium affect demographic parameters, and so the matrix L is modified into a matrix Lchronic. Let c be the discharged cadmium concentration in the stretch of river of pollutant emission. We can then calculate for each stretch of the river network a concentration cj, corresponding to the chronic concentration of cadmium present in patch j all along the year (see Ecotoxicological aspects). The survival rate s(i,j) of individuals of the age class i on patch j is then reduced by a coefficient rs(i,j) calculated from cj using the chronic dose–response curve for the age class i (Fig. 5). Hence, the new survival rate in the presence of cadmium in patch j is the product s(i,j)rs(i,j). The reduction rfj in adult fecundity in stretch j is calculated in the same way. We then build the matrices

(16)

But adults do not breed in the stretch of river where they live all during the year. The migration of reproduction (Fig. 4) is represented in a way similar to the downstream migration of juv11 by the matrix (Scheme 2). If we take into account this spatial change before spawning, Equation 14 becomes n1 (t 1 1) 5 0.5egMRF ÏS3 n3 (t)

(18)

Finally, regarding Equations 1, 4, 11, and 18, we build the following matrix L in order to link the population vector in year t with the population vector in year t 1 1:

n (t 1 1) 5 Ln (t) O

 L 5 S1  O

with

O being a submatrix of 0 with 15 rows and 15 columns. L is a multiregional Leslie matrix (45 rows and 45 columns). It presents a first dominant eigenvalue l corresponding to the asymptotic population growth rate [24,45]. The right eigenvector w associated with this first eigenvalue gives the asymptotic structure of the population. So it yields either the asymptotic spatial distribution of each age class or the stable age structure of the population. We also use the first left eigenvector v for sensitivity processing.

(19)

O

0.5egMRF ÏS3 

O

O

S2MD

S3

   

s s s Rsi 5 diag{r (i,1) · · · r (i,j) · · · r (i,15) } f } R f 5 diag{r 1f · · · r jf · · · r 15

(20)

and

(21) (22)

Thus, the demography in the presence of chronic cadmium pollution is modeled by the matrix

Scheme 2.

Environ. Toxicol. Chem. 22, 2003

Ecotoxicology and spatial modeling in population dynamics 

 

O

0.5egMR (FR f )Ï(S3Rs3 ) 

O

O

(S2Rs2 )MD

(S3Rs3 )

O

Lchronic 5 (S1R1s )  O

  

(23)



Acute pollution scenarios. We model an event of acute cadmium pollution occurring every year during 96 h. This event happens between the downstream migration of juv11 and the upstream breeding movement of adults (i.e., late in spring, in summer, or in autumn). In the same way as we modeled chronic scenarios, c is the cadmium concentration in the stretch of discharge in the river network and cj, concentration in patch j, is calculated according to the assumptions formulated for the pollutant spreading (see Ecotoxicological aspects). Survival of age class i in patch j during this acute pollution, sa(i,j), is then estimated with acute dose–response curves in Figure 6. We gather these survival rates in three diagonal matrices as a a Sai 5 diag{sa(i,1) . . . s(i,j) . . . s(i,15) }

(24)

The population dynamics under acute pollution conditions are then modeled with the Leslie matrix as 

O

 Lacute 5 (S1Sa1 )   O

O

0.5egMRF(ÏS3 Sa3 )

O

O

(S2Sa2 )MD

(S3Sa3 )

  

(25)

963

Table 3. Distribution in autumn of each age class in the different levels of the river network; output of the model in the case of a freepollution network (YOY01 5 young of the year; juv11 5 one-yearold juveniles) River network level

YOY01 (%)

Juv11 (%)

Adults (%)

51.1 35.6 10.7 2.6

1.9 18.0 34.1 46.0

0.4 7.9 33.5 58.2

1 2 3 4

duction in the survival of age class i. Moreover, these global contributions are sums of local contributions due to our spatial modeling approach, xf 5

O xf j

j

and

xs i 5

O xs

(i,j)

(28)

j

In other words, xsj is the variation of l explained by a reduction in the fecundity of trout living on patch j and xs(i,j) the variation of l caused by a reduction in the survival of age class i on patch j. These local contributions are calculated with the same way of thinking as Equation 26 as



Both Lchronic and Lacute depend on the discharged cadmium concentration c and on the spatial pattern of pollutant distribution. So these Leslie matrices yield population end points (asymptotic growth rate, stable age structure, spatial distribution) that are functions of the tested pollution scenario.

Sensitivity analysis and decomposition of the output signal The entries of the Leslie matrix that leads the population dynamics under polluted conditions (Lchronic or Lacute) are functions of the natural logarithm of the discharged concentration log c (dose–response curves are expressed on a log scale). Caswell [1,24] showed how the sensitivity of the first eigenvalue l to changes in a lower order variable can be linked to the effects of this variable on each entry of the Leslie matrix. Let log c be that lower order variable. The decomposition can then be written as ]l 5 ](log c)

O ]]ll ](log]l c) xy

x,y

(26)

xy

where lxy are the entries of the Leslie matrix. The sensitivity of l to changes in one entry of the Leslie matrix (sxy 5 ]l/]lxy) can be calculated for a given c using the first right eigenvector w and the first left eigenvector v of the Leslie matrix [24] as

vx wy ]l 5 ]lxy ^w, v&

(27)

where ^ & denotes the scalar product and vx the xth coordinate of the vector v (respectively wy, the yth coordinate of w). In the chronic case, l can be affected by reduction in fecundity terms or in survival terms. Thus, using the methodology of Caswell [1], we identify four global contributions: one fecundity contribution, xf, corresponding to the variation of the asymptotic population growth rate due to reduction in fecundity terms and three survival contributions, xsi, one per age class, corresponding to the variation of l caused by re-

Fig. 8. Effect on the asymptotic population growth rate l of increasing concentrations of discharge cadmium in the patch source of pollution for each spatial scenario in the chronic case and in the acute one. The curve is normalized by l0 5 1.10 (pollution-free network).

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Fig. 9. Decomposition of the sensitivity of the asymptotic population growth rate l to the cadmium discharged concentration c according to the chronic scenario patch 1. All results are normalized by the reference value for the pollution-free river network, l0 5 1.10. This decomposition is performed with four global contributions, xs1, xs2 xs3, and xf, respectively, standing for the effect of cadmium due to reduction in the survival of each age class (YOY01 5 young of the year, juv11 5 one-year-old juveniles, adults) and in the adult fecundity. These global contributions are sums of local contributions relative to each patch. We only present here local contributions that differ from zero.

O s ]]lr ](log]r c) and ]l ]r 5Os ]r ](log c) xy

x fj 5

xy

x,y

xs(i,j)

f j

xy

xy

x,y

f j

s (i,j)

s (i,j)

(29)

Thus, we decompose the sensitivity of l into contributions as ]l 5 (x f 1 1 · · · 1 x f 15 ) 1 (xs(1,1) 1 · · · 1 xs(1,15) ) ](log c) 1 (xs(2,1) 1 · · · 1 xs(2,15) ) 1 (xs(3,1) 1 · · · 1 xs(3,15) ) 5 x f 1 xs1 1 xs2 1 xs3

(30)

We proceed in the same way for the acute scenarios. Instead of rs(i,j), we use sa(i,j) in the survival terms, and fecundity contributions are nil because we do not describe any effect of acute pollution on fecundity (see Ecotoxicological aspects). Such a decomposition (Eqn. 30) is very interesting from an ecotoxicological management point of view because, besides regarding how l is influenced by the pollutant concentration, it allows us to know how, where, and which vital rates

Fig. 10. Same decomposition as Figure 9 but according to the chronic scenario patch 2.

or age classes are involved in explaining the effect of pollution on the global population end point l for a given scenario. RESULTS

Population features without contamination First, we calculated a reference value of l0 5 1.10 for the asymptotic population growth rate in the case of a pollutionfree river network. This indicates a potential growth for the population (l . 1). Then the stable age structure in autumn is 83% of YOY01, 13% of juv11, and 4% of adults. Moreover, the asymptotic spatial distributions in autumn describe gradients in good coherence with field data (Tables 2 and 3): YOY01 are gathered upstream in the network, juv11 and adults are conversely spread in the downstream levels. These features are given with v 5 2 (parameter of the upstream migration pattern). We tested two other values (v 5 5 and v 5 10) in order to fit our model. That yielded l0 5 1.12 for v 5 5 and l0 5 1.13 for v 5 10. The stable age structure is the same in the three cases. Spatial distributions are quite unchanged for juv11 and adults. The distribution of YOY01 remains qualitatively the same (increasing gradient from upto downstream) but it presents a better fit with field data for v 5 2. We therefore chose this value. So, in unpolluted conditions, our model yields population features that constitute a likely description of a brown trout population living in a river network. Our work mainly aims at analyzing patterns of perturbation of these population features under different pollution scenarios.

Ecotoxicology and spatial modeling in population dynamics

Fig. 11. Same decomposition as Figure 9 but according to the chronic scenario patch 2, considering a fictional pollutant with no chronic effect on adult fecundity

Effects of cadmium discharges on the asymptotic population growth rate l Figure 8 shows a reduction in the asymptotic population growth rate for increasing cadmium discharges under the different spatial scenarios in the chronic or acute case. In the chronic case, the maximum reduction is about 20% for all scenarios. Nevertheless, the decrease occurs for lower concentrations when the location of the polluting source is more downstream. In the acute case, the reduction also reaches about 20%. Nevertheless, the effect of space seems to be less significant; i.e., for a given concentration, the maximal gap between two spatial scenarios is less than 5% of reduction in l, whereas it reaches more than 10% for chronic pollution. Using the decomposition of the sensitivity of l in contributions corresponding to the different demographic parameters (Eqn. 30 in the previous section) allows us to understand how these patterns emerge. Chronic case. Figure 9 reports results of this decomposition performed for chronic pollution with a discharge point situated downstream (scenario patch 1 in Fig. 7). Cadmium is only present in patch 1. First, the decrease in l is mainly caused by a reduction in the fecundity in patch 1. Second, a very weak contribution of survival of juv11 and adults in patch 1 appears for higher concentrations. The lag between these two contributions (first reduction in fecundity and second in juv11/adult survival) can be explained by the different median effective concentration (EC50) observed in dose–response curves (Fig. 5). Note that the YOY01 contribution is nil due to the assumed insensitivity of YOY01 survival rate to chronic exposure. Moreover, the sum of the different contributions equals the

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Fig. 12. Same decomposition as Figure 9 but according to the acute scenario patch 1.

sensitivity of l to cadmium concentration in agreement with Equation 30. Consider now that the cadmium discharge occurs in patch 2 (Fig. 10). Cadmium is then present in patches 2 and 1, with a dilution factor of two between the concentrations in the two patches (Fig. 7, scenario patch 2). The decrease in l is also mainly explained by a reduction in fecundity terms, with two local contributions here, i.e., reductions in the fecundity in patch 2 and patch 1. The negative spike of fecundity contribution in patch 1 appears for concentrations twice those for patch 2. In fact, this shift on the axis of concentrations has to be linked with the dilution process considered in this scenario. Moreover, the negative spike corresponding to patch 1 is larger in absolute value than the one relative to patch 2. This can be explained by the gradient of the spatial distribution of adults (more present downstream; Table 3). Thus, more breeders are exposed to cadmium in patch 1 than in patch 2, which leads to a more serious impact on the population dynamics for patch 1 than for patch 2. The weakness of juv11 and adult survival contributions is surprising when compared with results in our previous study [4]. In this article, with other migratory assumptions (population mixing), the impact of survival reduction is very significant. In order to understand the pattern in these results, Figure 11 shows the same analysis as for the chronic scenario patch 2, but we consider a fictional pollutant, which follows the same survival dose–response curves as cadmium but which does not affect fecundity. Strong negative peaks of juv11 and adult survival contributions occur in this case, one for survival reduction in patch 1 and one in patch 2. They appear at con-

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Fig. 13. Same decomposition as Figure 9 but according to the acute scenario patch 3.

centrations corresponding to the EC50 of dose–response curves. Thus, in the case of cadmium, the weakness of the decrease in the population growth rate l due to survival reductions of the two oldest age classes is explained by the effect of cadmium on fecundity. Indeed, the trout in the contaminated patches have fecundity totally reduced for the concentrations affecting their survival. Then these trout do not participate in the population renewal for these concentrations; as a consequence, the fact that they survive or not does not influence the population growth rate l. We do not present here the decomposition of sensitivity for other chronic scenarios (patch 3 and patch 5 in Fig. 7). Indeed, similar patterns as the two first scenarios can be described. Acute case. Figures 12 and 13 show the decomposition of sensitivity performed for scenario patch 1 and patch 3 in the case of acute pollution. No effect concerning fecundity is present because of the ecotoxicological assumptions. So the decrease in l is only explained by survival reductions for the three age classes in the contaminated patches. As in the chronic case, the extent of each local contribution notably depends on the number of trout present in the different contaminated patches. For instance, in Figure 13 (scenario patch 3), local contribution of adult survival is larger for downstream patches (e.g., patch 1) than for upstream ones (e.g., patch 3). That is the opposite for YOY01. This results from the different gradients in the spatial distribution of age classes in the river network. As in the chronic case, we can note shifts on the concentration axis due to dilution between the two scenarios (e.g., adult survival contribution in patch 1: xs(3,1)). Nevertheless,

Fig. 14. Changes in the stable age structure in autumn for increasing chronic or acute discharges of cadmium in patch 2 (scenario patch 2) (YOY01 5 young of the year, juv11 5 one-year-old juveniles).

they are not so marked because acute contribution curves are flatter than chronic ones. This has to be linked with the slopes of reference dose–response curves, which are slighter in the acute case (Figs. 5 and 6). As a consequence, the curves of l for the different acute scenarios remain more grouped than for chronic ones (Fig. 8); the effect of space is more significant in the chronic case than in the acute case.

Changes in the stable age structure Figure 14 presents the stable age structure in autumn for different cadmium concentrations for the scenario patch 2 in the chronic case and in the acute case. Other scenarios (Fig. 7) lead to similar patterns. First, chronic pollution gives rise to an aging effect in the

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Fig. 15. Asymptotic adult distributions into the river network in autumn under different chronic scenarios of contamination.

population. This can be understood by the reduction in adult fecundity. Second, this trend is reversed for higher concentrations. Here we see a significant effect of the reduction in juv11 and adult survival rates. Such an effect was not so clear regarding the impact on the asymptotic population growth rate. In contrast with the chronic case, no significant changes in the stable age structure appear for the acute scenario (Fig. 14).

Changes in the spatial distribution We present here the results in the chronic case for a high cadmium concentration (102 mg/L). Figure 15 shows the adult spatial distribution in autumn for different locations of cadmium discharges in the river network. In the case with no pollution, adults are distributed following an increasing gradient from up- to downstream. Moreover, the distribution is symmetrical. The scenario patch 1 gives rise to two disconnected subpopulations living on three uncontaminated levels. If cadmium is discharged in patch 2, the population is only present on a subarea on the right side of the river network. No trout live on the left side because breeders of these area are mainly situated in patches 1 and 2 (see Fig. 4), which are the contaminated patches for this scenario. Note that, for this

concentration (102 mg/L), the effect on the asymptotic growth rate l is the same for these two scenarios patch 1 and patch 2 (Fig. 8). So, despite the same growth rate, the consequences of pollution on the trout population are very different regarding the spatial distributions. If cadmium is discharged upstream in the river network (scenario patch 5), the resulting spatial pattern is close to the one for scenario patch 2 except for the presence of trout on patch 1 downstream. That is due to the dilution process considered for the pollutant spread. Thus, adults in patch 1 are less exposed to cadmium in scenario patch 5 than in the scenario patch 2. The acute case is not presented here because it yields conclusions similar to the chronic case. DISCUSSION

Our study shows how metapopulation methodologies (such as spatially explicit models) could be employed in an ecotoxicological framework in order to explore the effect of space in population response to contamination. Our multiregion Leslie model yields population end points allowing the comparison between different pollution scenarios. The analysis has

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to be processed with several end points. For instance, two different spatial scenarios can both give rise to the same reduction in the population growth rate and lead to totally different spatial distributions. Another example is the insensitivity of growth rate to chronic reduction of adult survival and the impact of this reduction on the age structure. The first basic conclusion (which can be made as spatial heterogeneity is considered) is that population consequences of pollution depend on the spatial pattern of exposure. That is true regarding either the population renewal (asymptotic population growth rate) or the age structure or most of all the spatial distribution of individuals. Furthermore, this effect of space is also modulated by the kind of exposure; chronic pollution consequences on renewal are more sensitive to space dimension than acute ones in the example of brown trout and cadmium. Thus, the toxic effects on population dynamics present complex patterns as the spatial dimension is considered. The population level intricately integrates levels of exposure, spatial repartition of pollutant, spatial distribution of individuals, weight of each age class in the demography, thresholds, and shapes of dose–response curves. That is why modeling approaches are well adapted to deal with such changes of scale. Nevertheless, one criticism could be that such a model constitutes a black box, which does not allow understanding the emergence of output patterns. That could be formulated for numerical models calling upon simulations. Here we show that using analytical models is a way to avoid this drawback. Indeed, the analysis of the sensitivity of population end points to lower order parameters in the model [1,24] enables us to link the local effects of pollutant to the global impact on demography. We can thus understand where and which vital rates or age classes are involved in explaining the effect of pollution on the population. Modeling methodologies can account for different spatial behaviors. In our previous contribution [4], we exposed the case of a population that is completely mixed at the beginning of each year. This assumption allowed aggregation of variables, an easier mathematical treatment because the dimension of the system is reduced. Then we noted differences between our models; in our example, the demographic effect of reduction in survival of older age classes is conditioned by the population mixing. So different patterns of migration and spatial behaviors give rise to different impacts of pollution, and spatial modeling techniques are diverse enough to take into account these differences. The freshwater system chosen as an illustration is absolutely not a constraint for the modeling methodology we developed. Any topology of habitat could be treated. Marine contexts, terrestrial systems (e.g., agroecosystems) could be modeled in the same way. The spatial scale also might be larger (e.g., regional, continental) or smaller. This methodology is interesting in any case where a spatial structure supporting heterogeneity of physical/chemical/biological conditions may be identified. Otherwise, ecotoxicology is not the only potential field for application. We presently perform the modeling of real systems, on the one hand for river management (effect of dams on fish populations in a Belgium river network) and on the other hand for assessment of climate change impacts (considering the role of dispersion in a fish population of southern France). Our spatial modeling approach of population ecotoxicology can be improved by consideration of other perspectives. First, Caswell [1] proposes integration of density-dependence mech-

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anisms in order to strengthen the realism of ecotoxicological models. We have explored this approach, and the nonlinear characteristic of these systems leads to difficulties due to the high mathematical dimension of spatial models. Aggregation of variables can reduce the dimensionality even for nonlinear systems [46]. Charles et al. [47] have attempted this with brown trout in the case of nonlinear dynamics. Second, through our spatial approach, we endeavor to consider the effect of pollution on migration processes [40,43]. Indeed, the influence of avoidance or attraction behaviors on the population impacts of pollution appears to be of broad interest and warrants further study. REFERENCES 1. Caswell H. 1996. Demography meets ecotoxicology: Untangling the population level effects of toxic substances. In Newman MC, Jagoe CH, eds, Ecotoxicology: A Hierarchical Treatment. Lewis, Boca Raton, FL, USA, pp 255–292. 2. Sherratt TN, Jepson PC. 1993. A metapopulation approach to modeling the long-term impact of pesticides on invertebrates. J Appl Ecol 30:696–705. 3. Spromberg JA, John BM, Landis WG. 1998. Metapopulation dynamics: Indirect effects and multiple distinct outcomes in ecological risk assessment. Environ Toxicol Chem 17:1640–1649. 4. Chaumot A, Charles S, Flammarion P, Garric J, Auger P. 2002. Using aggregation methods to assess toxicant effects on population dynamics in spatial systems. Ecol Appl 12:1771–1784. 5. Auger P, Poggiale JC. 1996. Emergence of population growth models: Fast migration and slow growth. J Theor Biol 182:99– 108. 6. Sanchez E, Bravo de La Parra R, Auger P. 1995. Linear discrete models with different time scales. Acta Biotheor 43:465–479. 7. Hestagen T. 1988. Movements of brown trout, Salmo trutta, and juvenile Atlantic salmon, Salmo salar, in a coastal stream in northern Norway. J Fish Biol 32:639–653. 8. Clapp DF, Clark RD, Diana JS. 1990. Range, activity, and habitat of large, free-ranging brown trout in a Michigan stream. Trans Am Fish Soc 119:1022–1034. 9. Bridcut EE, Giller PS. 1993. Movement and site fidelity in young brown trout Salmo trutta populations in a southern Irish stream. J Fish Biol 43:889–899. 10. Gowan C, Young MK, Fausch KD, Riley SC. 1994. Restricted movement in resident stream salmonids: A paradigm lost? Can J Fish Aquat Sci 51:2626–2637. 11. Young MK. 1994. Mobility of brown trout in south-central Wyoming streams. Can J Zool 72:2078–2083. 12. Ovidio M. 1999. Cycle annuel d’activite´ de la truite commune (Salmo trutta L.) adulte: E´tude par radio-pistage dans un cours d’eau de l’ardenne belge. Bull Fr Peˆche Pisc 352:1–18. 13. Schuck HA. 1945. Survival, population density, growth, and movement of the wild brown trout in Crystal Creek. Trans Am Fish Soc 73:209–230. 14. Mense JB. 1975. Relation of density to brown trout movement in a Michigan stream. Trans Am Fish Soc 4:688–695. 15. Solomon DJ, Templeton RG. 1976. Movements of brown trout Salmo trutta L. in a chalk stream. J Fish Biol 9:411–423. 16. Harcup MF, Williams R, Ellis DM. 1984. Movements of brown trout, Salmo trutta L., in the river Gwyddon, South Wales. J Fish Biol 24:415–426. 17. Baglinie`re JL, Maisse G, Lebail PY, Pre´vost E. 1987. Dynamique de la population de truite commune (Salmo trutta L.) d’un ruisseau breton (France). II—Les ge´niteurs migrants. Acta Oecol— Oecol Appli 8:201–215. 18. Baglinie`re JL, Maisse G, Lebail PY, Nihouarn A. 1989. Population dynamics of brown trout, Salmo trutta L., in a tributary in Brittany (France): Spawning and juveniles. J Fish Biol 34:97– 110. 19. Baglinie`re JL, Pre´vost E, Maisse G. 1994. Comparison of population dynamics of Atlantic salmon (Salmo salar) and brown trout (Salmo trutta) in a small tributary of the River Scorff (Brittany, France). Ecol Freshw Fish 3:25–34. 20. Gouraud V, Baglinie`re JL, Ombredane D, Sabaton C, Hingrat Y, Marchand F. 1997. Caracte´ristiques biologiques de la population de truites (Salmo trutta L.) de la rivie`re Oir (Manche, Basse-

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