Modeling the Effects of Constant and Variable ...

Environmental Entomology Advance Access published May 5, 2015 POPULATION ECOLOGY

Modeling the Effects of Constant and Variable Temperatures on the Vital Rates of an Age-, Stage-, and Sex-Structured Population by Means of the SANDY Approach G. NACHMAN1,2 AND T. GOTOH3

Environ. Entomol. 1–14 (2015); DOI: 10.1093/ee/nvv056

ABSTRACT We present a general and flexible mathematical model (called SANDY) that can be used to describe many biological phenomena, including the phenology of arthropods. In this paper, we demonstrate how the model can be fitted to vital rates (i.e., rates associated with development, survival, hatching, and oviposition) of the two-spotted spider mite (Tetranychus urticae (Koch)) exposed to different constant temperatures ranging from 15 C to 37.5 C. SANDY was incorporated into an age-, stage- and sex-structured dynamic model, which was fitted to cohort life-tables of T. urticae conducted at five constant temperatures (15, 20, 25, 30, and 35 C). Age- and temperature-dependent vital rates for the three main stages (eggs, immatures, and adults) constituting the life-cycle of mites were adequately described by the SANDY model. The modeling approach allows for simulating the growth of a population in a variable environment. We compared the predicted net reproductive rate (R0) and intrinsic rate of natural increase (rm) at fluctuating temperatures with empirical values obtained from life-table experiments conducted at temperatures that changed with a daily amplitude (60, 63, 66, 69, and 612 C) around an average of 22 C. Results show that R0 decreases with increasing amplitude, while rm is more robust to variable temperatures. An advantage of SANDY is that the same simple mathematical expression can be applied to describe all the vital rates. Besides, the approach is not confined to modeling the influence of a single factor on population growth but allows for incorporating the combined effect of several limiting factors, provided that the combined effect of the factors is multiplicative. KEY WORDS life-table, population growth, developmental rate, intrinsic rate of natural increase, Tetranychus urticae The purpose of life-table studies carried out under controlled laboratory conditions is to obtain important information about a species’ vital parameters such as age-specific survival and oviposition rates, developmental times, sex ratio, generation time, population growth rate etc. (Pielou 1969). Such in vitro experiments are especially useful when dealing with small arthropod species, such as insects and mites, which often have overlapping generations and, at the same time, are difficult to mark and recapture in the field. Except for relatively few studies of thermoperiodism where individuals were exposed to alternating temperatures (see, e.g., Foley 1981, Beck 1983, Fantinou et al. 2003, Renault et al. 2004, Mironidis and Savopoulou-Soultani 2008, Ragland and Kingsolver 2008, Vangansbeke et al. 2013), the majority of life-table studies are conducted at constant temperatures. To avoid any adverse effects of biotic factors, such as food limitation, natural enemies, and diseases, the biotic environment is usually optimized. Thus, results obtained from life-table 1 Section of Ecology and Evolution, Department of Biology, University of Copenhagen, Universitetsparken 15, DK 2100 Copenhagen Ø, Denmark. 2 Corresponding author, e-mail: [email protected]. 3 Laboratory of Applied Entomology and Zoology, Faculty of Agriculture, Ibaraki University, Ami, Ibaraki 300-0393, Japan.

studies allow us to predict how a population is expected to develop if it is exposed to environmental conditions similar to those applied in the laboratory. In nature, however, environmental conditions differ from those at which life-table data have been obtained. Natural ecosystems are usually characterized by considerable variations in environmental factors, which are likely to have profound effects on a population’s vital rates and therefore also on its realized growth rate. Consequently, environmental variability has to be taken into consideration when modeling the dynamics of natural populations (see, e.g., Hardman et al. 2013). The challenge is therefore to develop mathematical methods that permit extrapolation of life-table data obtained under controlled and constant laboratory conditions to field conditions. A substantial number of papers have addressed this issue by presenting models that link vital rates to the ambient temperature (Developmental rate: Logan et al. 1976; Schoolfield et al. 1981; Taylor 1981; Wagner et al. 1984a,b; Hagstrum and Milliken 1991; Worner 1992; Briere et al. 1999; Ikemoto 2005; Son and Lewis 2005; Re´gnie`re et al. 2012; Fand et al. 2014; Survival rate: Son and Lewis 2005, Re´gnie`re et al. 2012, Amarasekare and Sifuentes 2012, Fand et al. 2014; Oviposition rate: Re´gnie`re et al. 2012, Fand et al. 2014). Although the various models used to fit data for development, survival, and

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ENVIRONMENTAL ENTOMOLOGY

oviposition may provide good agreement with empirical data, they are usually developed with the purpose of fitting specific data sets and may therefore lack generality. Besides, models used to describe developmental rates often do not fit data for the full range of temperatures (e.g., Campbell et al. 1974) or, alternatively, become rather complex (e.g., Sharpe and DeMichele 1977, Schoolfield et al. 1981, Yan and Hunt 1999). Finally, no model takes into account that several factors may simultaneously affect a vital rate. In this paper, we propose a general mathematical approach that can be used to extrapolate laboratory-generated life-table data to predict the dynamics of a population living in a variable environment. We apply a simple mathematical expression (called SANDY4), which is so flexible that it can be used to fit many biological relationships including vital rates. The SANDY approach allows for combining the simultaneous influence of two or more factors on these rates, for instance if the survival rate of an individual depends on the ambient temperature as well as on its physiological or biological age (Taylor 1981, van Straalen 1983). First, we will present a generic dynamic model of an arthropod population in which survival, developmental, and oviposition rates depend on the age, stage, and sex of individuals. Secondly, we will apply the model to generate the dynamics of a single cohort where all individuals initially are of the same age. Thirdly, we will introduce SANDY to describe the vital rates in the dynamic models, allowing us to predict how a population will develop over time in a constant or variable environment. We demonstrate how specific life-table data of the two-spotted spider mite Tetranychus urticae (Koch) obtained at constant temperatures in the range between 15 and 35 C can be fitted by means of SANDY. Finally, we will use the dynamic model to predict the growth rate of a population exposed to variable temperatures and compare the predictions with experimental data obtained by changing the temperature on a daily basis between 22 6 0 C, 22 6 3 C, 22 6 6 C, 22 6 9 C, and 22 6 12 C. Materials and Methods A Dynamic Model of a Stage-, Age-, and SexStructured Population. Consider a stage-structured population consisting of N individuals atXtime t of S Nðs; tÞ, which N(s,t) belong to stage s, i.e., NðtÞ ¼ s¼1 where S is the number of stages. Each stage is divided into a number of age classes. An individual has age 0 when it enters a new stage and for each time step (e.g., Dt ¼ 1 day) its age is updated with Dx ¼ Dt until the individual either moves to another stage or dies. In the following we will for simplicity use the term “hatching” to describe the transition from one stage to the next. Thus, for each time step an individual may either survive (with probability ps) or die (with probability1ps). Provided it survives, it may either hatch to the next

4 The model was acknowledgment).

dedicated

to Sandra

(Sandy)

Walde

(see

stage (with probability ph) or remain in the same stage (with probability 1ph). Mathematically, the dynamics of individuals in stage s at time t can be written as Nðg; s; xÞ ¼ ps ðg; s; x  1Þð1  ph ðg; s; x  1ÞÞNðg; s; x  1Þ

(1) where N(g,s,x) denotes the expected number of individuals of gender g belonging to stage s and have age x (x  1). ps(g,s,x1)(1ph(g,s,x1)) is the proportion of individuals in the previous age class that neither die nor hatch during the time step. The expected number of individuals entering stage s during a time step is found as Nðg; s; 0Þ ¼

1 X ps ðg; s  1; x  1Þph ðg; s  1; x  1Þ x¼1

Nðg; s  1; x  1Þ (2) The expected number of males (g ¼ 1) and females (g ¼ 2) entering the first stage (i.e., eggs) are obtained as Nð1; 1; 0Þ ¼

1  X

 1  pf ðx  1Þ Fðx  1ÞNð2; S; x  1Þ

x¼1

(3a) Nð2; 1; 0Þ ¼

1 X

pf ðx  1ÞFðx  1ÞNð2; S; x  1Þ (3b)

x¼1

where S denotes the last stage (i.e., the adult stage), F(x1) is the mean fecundity of females of age x1 and pf(x1) is the proportion of female offspring produced by females in the age class. ps, ph, pf, and F in the above model are likely to depend on the ambient temperature and/or the individuals’ age. However, in arthropods development and aging are typically associated with the ambient temperature, so instead of using the chronological age (x) of individuals, we apply their biological age (Q), which depends on the cumulated thermal energy. The biological age of individuals with chronological age x is calculated as QðxÞ ¼

x1 X

DQi

(4)

i¼0

where DQi is the increment in biological age during time interval Dti where the individuals experienced temperature Ti. DQi will be  0 when the temperature is above a lower threshold called Tb and will increase with T, but only up to an optimum temperature and then decline steeply to become 0 when the temperature exceeds an upper threshold called Tu. Q is measured in day-degrees (Gilbert and Gutierrez 1973, Mironidis and Savopoulou-Soultani 2008). Developmental variability, i.e., individuals entering a stage at the same time may not leave it again synchronously (Wagner et al. 1984b,

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NACHMAN AND GOTOH: MODELING TEMPERATURE EFFECTS ON VITAL RATES

Schaalje and van der Vaart 1989, Son and Lewis 2005), is modeled by assuming that the likelihood that an individual completes development during Dt increases with its biological age. We assume that Q(x) cannot exceed an upper limit Qmax above which no further development will take place. This means that Q(x)/Qmax corresponds to the physiological age of individuals (van Straalen 1983, Re´gnie`re and Logan 2003). Modeling the Dynamics of a Single Life-Table Cohort. Equations 1–3 predict the dynamics of a population where new individuals are recruited to the population at each time step, which means that the model can be applied to populations with overlapping generations. In contrast, a cohort life-table is constructed by following the same individuals from birth to death. It means that all individuals will be of the same age measured from birth whereas individuals of the same age may belong to different stages. In the following, we replace age within a stage (x) with t, which is the time elapsed since the cohort was started with newly laid eggs. Their age is by definition x ¼ t ¼ 0 days. At time t the cohort consists of N(t) individuals of which N(s,t) belong to stage s. The number of individuals in stage s to time tþ1 is found as Nðs; t þ 1Þ ¼ ps ð1  ph ÞNðs; tÞ þ Hðs  1; tÞ  Hðs; tÞ: (5) where ps(1ph)N(s,t) is the number of individuals in stage s surviving from day t to day tþ1 without hatching to the next stage. H(s1,t) and H(s,t) denote the number of individuals hatching from stage s1 to stage s and from stage s to stage sþ1, respectively. The number of eggs produced by the cohort at time t is modeled as Nð1; 0Þ ¼ FðtÞNf ðtÞ:

(6)

where Nf(t) is the number of surviving females at time t and F(t) is the age-dependent fecundity rate per female. The vital rates ps, ph, and F are likely to depend on the ambient temperature and/or the biological age of individuals at time t. However, as individuals in stage s may not have entered the stage at the same time (except for the egg stage), the mean biological age of individuals in stage s at time tþ1 is calculated as Qðs; t þ 1Þ ¼

ðQðs; tÞ þ DQðsÞÞps ð1  ph ÞNðtÞ : Nðs; t þ 1Þ

variable. To illustrate this, let y denote the value of a given response function and let ymax be the maximum value y takes if all n factors affecting y were optimal. In accordance with the multiplicative Mitscherlich model (Johns and Vimpany 1999, Harmsen 2000, Nijland et al. 2008), we write y as a product of limiting factors y ¼ ymax f1 ðÞf2 ðÞ    fn ðÞ ¼ ymax

n Y

fj ðÞ

(8)

j¼1

where fi() is the effect of the jth factor on y. fi() can take values between 1 and 0. Thus, if fj() ¼ 1 for all n factors, the maximum value of y is attained, whereas if one or more values of fj() are < 1, y will be smaller than ymax. Finally, y will be 0 if just one of the limiting factors is 0. The use of a multiplicative approach is justified by its successful application in proportional hazards models (see e.g., Cox 1972, Breslow 1975). The factors affecting y could be external factors, e.g., the ambient temperature, humidity, amount and quality of food, the influence of conspecifics, competitors or natural enemies. f() can also be used to represent internal factors such as the effect of biological age, body size, or nutritional state on y. SANDY: An Effect-Response Model. In the following, we introduce SANDY as a general model linking the response variable f() to a factor whose current value is V. The generic version of the model reads  f ðVÞ ¼ C

V  Vmin Vmax  Vmin

a 

Vmax  V Vmax  Vmin

b (9)

f(V) should take values between 0 and 1 when Vmin  V  Vmax and to be either 0 or 1 when V is outside this interval. a and b are two parameters that determine the shape of the function, and C is a constant that scales f(V) to ensure that 0  f(V)  1. To fulfill this requirement, C is seen to be  C¼

a aþb

a 

b aþb

b (10)

when both a and b = 0, otherwise C ¼ 1. If both a and b > 0, the function has a maximum (equal to 1) when

(7)

based on the assumption that the surviving individuals increase their biological age with DQ(s) during Dt, while the biological age of newcomers to a stage is 0. The Concept of Limiting Factors. The four vital rates used in the dynamic model (ps, ph, pf, and F) are specific for a given gender, stage, and age-class of the species in focus and may depend on the ambient environment as well as the conditions to which individuals have been exposed earlier in their life. We assume that each of these rates has a maximum for an optimum combination of the factors that simultaneously affect the response

3

Vopt ¼

aVmax þ bVmin : aþb

(11)

Otherwise, the maximum is located at Vopt ¼ Vmin when a ¼ 0 and b > 0, and at Vopt ¼ Vmax when a > 0 and b ¼ 0. If both a and b are 0, f(V) will be equal to 1 for all Vmin < V < Vmax, and either 0 or 1 outside this range, depending on what process f(V) represents. If the function has a maximum given by equation 11, the curve can be either symmetric (if a ¼ b) or asymmetric (if a=b), and the shape of the function can be either convex (if a  1; b  1), bell-shaped (if a > 1; b > 1), or combinations of these two shapes (i.e., if a 

ENVIRONMENTAL ENTOMOLOGY

4

1; b > 1 or if a > 1; b  1). Monotonically increasing or decreasing curves can be generated by setting y to 1 when V  Vopt (for an increasing relationship) or when V  Vopt (for a decreasing relationship) and can be linear, convex, concave, or sigmoid. Thus, monotonically increasing curves are concave (if 0 < a < 1; b ¼ 0), linear (if a ¼1; b ¼ 0), convex (if a > 1; b ¼ 0), or sigmoid (if a > 1; b > 0). In cases where f(V) has a minimum in the interval between Vmin and Vmax, the shape parameters a and b are both negative. The minimum of f(V) is then located at Vopt. Figure 1 shows some examples of the flexibility of the SANDY model to describe various shapes of f(V). Using SANDY to Model Vital Rates. In most life-table studies the main environmental factor is temperature (T), which can have both an acute and a cumulated effect. Their combined effect is modeled as y ¼ ymax f1 ðTÞf2 ðQÞf3 ðÞ

(12)

where f1(T) denotes the acute effect and f2(Q) the cumulated effect, manifested through the biological age of an individual. f3() denotes the combined effect of other factors than temperature (e.g., humidity, food etc). In the following, we assume that these additional factors are all optimal so f3() is set to 1. We therefore have 

  T  Tmin a Tmax  T b y ¼ ymax C1 Tmax  Tmin Tmax  Tmin  c   Q  Qmin Qmax  Q d C2 Qmax  Qmin Qmax  Qmin

(13)

where a, b, c, and d are constants. y will be 0 when T  Tmin and T  Tmax. The same applies if Q  Qmin or Q  Qmax when both c and d are positive. Q at age x is found by means of equation 4, where the increase in day-degrees during age interval Dti (DQi) is positive in the interval between the lower and upper temperature thresholds (Tb and Tu), otherwise DQi will be 0. Thus, DQ is modeled as     T  Tb q Tu  T r DQ ¼ ymax C Dt Tu  Tb Tu  Tb

(14)

where T denotes the average temperature during Dt and q and r are two shape parameters analogous to a and b. We set ymax to unity at the optimum temperaqTu þrTb ture, i.e., whenT ¼ Topt ¼ qþr , which means that DQ ¼ Dt. Therefore, biological age (Q(x)) is equal to chronological age (x) when the ambient temperature is optimal (i.e., T ¼ Topt), otherwise Q(x) will be less than the chronological age. Constant Temperature Experiments. Experimental data comprise survival rate and developmental time from egg to adulthood, female oviposition rate, and longevity of T. urticae (Gotoh et al. in press). The experiments were carried out in 2010 and used mites from a laboratory culture established in 2001 from specimens

collected from Citrullus lanatus (Thunb.) at Takikawa, Hokkaido, Northern Japan (43 560 N;141 900 E). Life-table parameters were determined under laboratory conditions using the common bean (Phaseolus vulgaris L.) as the host plant. Duration of development from egg to adult was measured at 11 constant temperatures ranging from 15 C to 37.5 C at intervals of 2.5 C. Cohort life-tables were established at five constant temperatures (15, 20, 25, 30, and 35 C) using eggs laid within the previous 24 h on leaf discs. The initial numbers of eggs were 67 (15 C), 72 (20 and 25 C), and 73 (30 and 35 C). Eggs that hatched to larvae were counted either every day (at 15 C, 20 C, and 25 C) or every half day (at 30 C and 35 C), and surviving immatures (larvae and five nymph stages) were counted once or twice every day until they had moulted to become adults. The adults were sexed and the males discarded. The females were transferred individually to new leaf discs, where they were allowed to mate with two males from the stock culture for 48 h, and their survival and oviposition were recorded every day until all had died. Data Analysis. At each temperature, the net reproductive per generation (R0) was calculated as Xrate 1 R0 ¼ l m and the generation time (G) as X1x¼0 x x lx mx xx G ¼ Xx¼0 , where lx is the proportion of female 1 lm x¼0 x x eggs in the cohort that survive until age x and mx is the average number of female eggs produced by a female of age x (Pielou 1969). The proportion of female eggs in the cohort was estimated from sex ratio determined when immatures moult to adults. The intrinsic rate of naturalX increase (rm) was found by iteration as the solu1 tion to x¼0 lx mx erm x ¼ 1. The finite rate of increase (k) was found ask ¼ erm . We used SANDY to fit 1) the sex ratio, 2) developmental rate for egg to adult, 3) survival rate from egg hatch to adulthood, 4) the oviposition period of adult females, 5) the total number of eggs produced per female, 6) the generation time (G), 7) the net reproductive rate (R0), and 8) the intrinsic rate of natural increase (rm) as temperature-dependent processes. Modeling Daily Survival, Hatching, and Oviposition Rates. SANDY was also used to model the daily survival rate (ps), hatching rate (ph), and oviposition rate (F) of T. urticae in order to predict these rates from the ambient temperature (T) and the mean biological age of individuals (Q) at age t for individuals in stage s. To reduce the number of parameters necessary to describe the full life-table of T. urticae at any temperature we assumed: (a) Daily survival probabilities of eggs and immatures (ps) depend on the ambient temperature but are independent of age. ps is therefore modeled as    T  Tmin a Tmax  T b : ps ¼ smax C Tmax  Tmin Tmax  Tmin

(15)

(b) Daily hatching probabilities of eggs and immatures (ph) increase monotonically with biological age. Hatch rate is 0 when Q ¼ 0 and reaches a

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NACHMAN AND GOTOH: MODELING TEMPERATURE EFFECTS ON VITAL RATES

1

A

B

C

D

E

F

5

y

0.8 .6 0.4 0.2 0 1

y

0.8 0.6 0.4 0.2 0 1

y

0.8 0.6 0.4 0.2 0 5

10

15

20

25

30

V

5

10

15

20

25

30

35

V

Fig. 1. Six examples demonstrating curves generated by means of the SANDY model (equation 9). In all examples, ymax ¼ 1; Vmin ¼ 10, and Vmax ¼ 30. (A) a ¼ b ¼ 0; (B) a ¼ b ¼ 0.5; (C) a ¼ b ¼ 1; (D) a ¼ b ¼ 5; (E) a ¼ 2, b ¼ 1; (F) a ¼ 4, b ¼ 0.5 (y is set to ymax when V  Vopt).

maximum (ph ¼ hmax) when Q ¼ Qmax, so setting Qmin ¼ 0 and d ¼ 0 in equation 13, the hatch rate at a given temperature is modeled as  ph ¼ hmax

Q Qmax

c

become 1 when Q ¼ Qmax. d(Q) is therefore modeled as  dðQÞ ¼ hmax

:

(16)

(c) Adults do not hatch to the next stage, but instead they may die due to aging. Thus, the survival probability of adults depends on both the ambient temperature and their biological age. Survival probability of an adult of age Q at temperature T is modeled as

Q Qmax

c

where hmax is interpreted as the maximum death rate of adults (i.e., hmax ¼ 1). In combination, the daily survival probability of adults is modeled as       T  Tmin a Tmax  T b Q c ps ¼ smax C 1 Tmax  Tmin Qmax Tmax  Tmin

(17) ps ¼ sðTÞð1  dðQÞÞ where s(T) is the survival probability at temperature T when the age-dependent death rate (d(Q)) is 0. s(T) is modeled as equation 15, while d(Q) is assumed to increase monotonically with age and

(d) The daily fecundity rate of adult females depends on both the ambient temperature and the females’ biological age. Fecundity rate is 0 when Q  Qmin and Q  Qmax and peaks at intermediate

ENVIRONMENTAL ENTOMOLOGY

6

values of Q. For simplicity, we set Qmin ¼ 0 so that fecundity is modeled as 

  T  Tmin a Tmax  T b Tmax  Tmin Tmax  Tmin  c   Q Q d C2 1 Qmax Qmax

F ¼ Fmax C1

(18)

Tmin and Tmax for fecundity are likely to be different from those used for survival. (e) Development is described by equation 14. For simplicity we assume that the lower and upper thresholds for development (Tb and Tu) coincide with the lower and thresholds for survival (Tmin and Tmax), so that Tb ¼ Tmin and Tu ¼ Tmax. We used Excel (Microsoft Office 2003) to generate life-tables of T. urticae at five different constant temperatures (15, 20, 25, 30, and 35 C) by using the dynamic cohort model (equations 5–7) to simulate the fate of a single cohort of eggs followed from birth until the last adult female dies. ps, ph, and F in the dynamic life-table model were replaced by equations 15–18. For each time step (corresponding to one day), the chronological age of the individuals was updated by one day and their biological age increased withDQðsÞallowing us to calculate the values of Q(s,tþ1), N(s,tþ1), H(s,tþ1), and F(tþ1) from their values at day t. The parameters describing the functions ps and ph for each stage, as well as F for the adult females, were estimated by means of the Solver tool in Excel as the values that yielded the smallest sum of squared deviations between the observed and predicted number of surviving individuals at day t and the number of eggs produced per female at day t. Ideally, experimental data should include measurements obtained at temperatures spanning the entire range from the lower (Tmin) to the upper (Tmax) threshold. If this is not the case, it may be difficult to obtain accurate and sometimes even realistic threshold values (Campbell et al. 1974, Manly 1990, Re´gnie`re et al. 2012). To avoid unrealistically low or high threshold values, Tmin was constrained to be  0 C and Tmax to be  45 C. Alternatively, Tmin and Tmax could be found from existing literature. The maximum rates smax and hmax were constrained to be  1. We checked the output of Solver by visually inspecting the fitted curves. It is recommended to initialize Solver with different values to reduce the risk that the estimated parameter values represent local minima. Variable Temperature Experiments. To study the effect of variable temperatures on the vital rates, a series of experiments was conducted where the average temperature was 22 C but the daily amplitude was 60, 63, 66, 69, or 612 C. The amplitude experiments were conducted in 2013. As the T. urticae stock used for the first series of experiments with constant temperatures went extinct in 2012, the mites used in the amplitude experiments came from a culture established

in 2006 on specimens collected from strawberry (Fragaria x ananassa Duch. var Nyoho) at Hitachi, Ibaraki, Central Japan (36 490 N; 140 600 E). There were six replicates for each amplitude treatment, but for the experiments using 63, 66, 69, and 612 C, three of the replicates were started with the highest temperature on the first day, while the other three were started at the lowest temperature. These two groups, nested within each amplitude, were denoted H and L, respectively. Each replicate was started by placing 3–5-d-old female mites on 20 leaf discs with 5 mites per disc where they were allowed to lay eggs for 24 h. The offspring on each disc were followed until they reached adulthood, when they were sexed and the sex ratio calculated. For each disc, the percentage of eggs that hatched to larvae, the percentage of eggs that reached adulthood, and the time to develop from egg to adult were recorded for each gender separately. Female longevity and daily oviposition were recorded by following individual mated females from when they moulted until they died. The number of females per replicate was 20 (see Gotoh et al. (2014) for more details). Data Analysis. For each amplitude treatment, the empirical net growth rate (R0), the generation time (G), and the intrinsic rate of increase (rm) were calculated. The expected values of R0 and rm were calculated from the daily survival rates (lx) and oviposition rates (mx) obtained by using the dynamic life-table model parameterized with values obtained from the constant temperature experiments. The daily rates were obtained by assuming that the temperature changed instantaneously from day to day starting at day 0 with either the minimum or the maximum temperature of the chosen amplitude. Results Constant Temperatures. At 15 C, the proportion of females in the cohort was 0.60 (36 females:24 males), 0.58 (40 females:29 males) at 20 C, 0.65 (46 females:25 males) at 25 C, 0.52 (25 females:32 males) at 30 C, and 0.57 (36 females:27 males) at 35 C. As the sex ratios did not differ significantly among temperatures (v2 ¼ 2.35; df ¼ 4; P ¼ 0.67), the overall proportion of females was found by pooling data for the five temperatures as 0.585 (193 females:137 males). This sex ratio was used to estimate the number of female eggs in the cohorts. The sex ratio of the offspring produced by the adult females varied with temperature and was highest (about 75% females) at intermediate temperatures (Fig. 2A). Figure 2B shows that the developmental rate (the reciprocal of the time from egg to adult) peaked at temperatures around 35 C, while the proportion of larvae that survived until adulthood dropped steeply as the temperature approached the upper threshold (Fig. 2C). The duration of the oviposition period declined with increasing temperature (Fig. 2D). Lifetime oviposition per female peaked at intermediate temperatures and was low both at 15 and 35 C (Fig. 2E). Generation time declined with temperature

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NACHMAN AND GOTOH: MODELING TEMPERATURE EFFECTS ON VITAL RATES 0.25

A 0.6 0.4 0.2 0 1

C

0.8 0.6 0.4 02 0.2 200 0

ggs/female Eg

E 150 100 50

Net reprod uctive rate

0 100

G

80 60 40 20 0 10

15

20

25

30

35

40

Temperature ( o C)

Development/day

B

0.8

0.20

growth (day-1) Generattion time (days) Ovipo Intr. rate of g osition period (Days)

oportion surviving Pro

Proportion females

1

7

0.00 40

0.15 0.10 0.05

D

30 20 10 0 50

F

40 30 20 10 0 0.4

H

0.3 0.2 0.1 0 10

15

20

25

30

35

40

Temperature ( o C)

Fig. 2. Results of life-table experiments with T. urticae conducted at five different constant temperatures and P. vulgaris L. as host plant. Dots represent the observed average values (6 SD) and lines the predicted relationships based on equation 9. (A) Proportion of female offspring produced by the females used in the cohort life-tables: ymax ¼ 0.7432, Tmin ¼ 0 C, Tmax ¼ 45 C, a ¼ 0.6880, b ¼ 0.6661. (B) Developmental rate from egg to adult: ymax ¼ 0.1834 d1, Tmin ¼ 0 C, Tmax ¼ 43.61 C, a ¼ 2.2225, b ¼ 0.3938. (C) Proportion of eggs surviving to adulthood: ymax ¼ 0.9269, Tmin ¼ 1.32 C, Tmax ¼ 35.02 C, a ¼ 0.0077, b ¼ 0.0251. (D) Duration of female oviposition period: ymax ¼ 26.05 d, Tmin ¼ 0 C, Tmax ¼ 45 C, a ¼ 1.4491, b ¼ 3.2106. (E) Lifetime oviposition per female: ymax ¼ 129.98 eggs, Tmin ¼ 9.91 C, Tmax ¼ 45 C, a ¼ 2.4921, b ¼ 4.7862. (F) Generation time (G): ymin ¼ 8.5146 d, Tmin ¼ 9.98 C, Tmax ¼ 41.58 C, a ¼ 1.1618, b ¼ 0.1187. (G) Net reproductive rate (R0): ymax ¼ 95.79, Tmin ¼ 9.51 C, Tmax ¼ 45 C, a ¼ 3.0407, b ¼ 5.6595. (H) Intrinsic rate of natural increase (rm). Parameters for the finite rate of increase model (k ¼ erm ): ymax ¼ 1.4187 d1, Tmin ¼ 0 C, Tmax ¼ 35.78 C, a ¼ 0.5189, b ¼ 0.0678.

to reach a minimum at 38.6 C (Fig. 2F). Figure 2G shows the empirical and predicted values of R0. Finally, as rm can take negative values (corresponding to 0 < R0 < 1), and because SANDY does not allow for negative values of the response variable, SANDY was instead fitted to the empirical values of k ¼ erm . The expected values of rm at temperature T were subsequently derived as ^r m ¼ ln^k (Fig. 2H).

As seen from Fig. 3, the dynamic life-table model (equations 5–7) provides a good fit to the experimental life-table data of T. urticae. The parameter estimates are given in Table 1. The constraints imposed on Tmin (0 C) and Tmax (45 C) were encountered in three cases. Figure 4 shows the observed and predicted net reproductive rate (R0) and the intrinsic rate of natural

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ENVIRONMENTAL ENTOMOLOGY

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Days Fig. 3. Observed (markers) and predicted (lines) life-tables of T. urticae conducted at five constant temperatures using P. vulgaris as host plant. Circles: Eggs; Triangles: Immatures (larvae and nymph stages); Squares: Adult females; Crosses: Oviposition rates per day and female. Heavy lines: Predicted number of surviving individuals; Broken line: Predicted ovipostion rate. Parameters describing the predicted values are given in Table 1.

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NACHMAN AND GOTOH: MODELING TEMPERATURE EFFECTS ON VITAL RATES

Table 1. Estimated parameter values for eggs, immatures, and adult females of T. urticae Parameter smax hmax Qmax Fmax Tmin Tmax a b c d q r

Eggs

Immatures

Adults

Fecundity

1 0.8787 1.4882 – 0.2183 45 0.0103 0.0055 52.6617 0b 4.4803 1.4202

1 0.7396 2.3124 – 0.2407 35.3071 0.1146 0.1666 32.0328 0b 4.9653 2.2824

1 1 23.0845 – 0 35.3232 0.1443 0.1216 3.0112 0b 1.6605 0.0005

– – 20.2540a 11.4786 10.2052 45 2.5724 3.2979 1.0438 6.7590 – –

smax is the maximum survival rate per day, hmax the maximum hatch rate per day (eggs and immature) or the maximum daily mortality rate (adults), Qmax the biological age at which a life stage is completed, Fmax the maximum oviposition rate per day, Tmin and Tmax the lower and upper thresholds for survival and oviposition, a and b the shape parameters associated with the current ambient temperature (T), c and d the shape parameters associated with the biological age (Q), q and r the shape parameters associated with rate of development (increase in biological age). The parameters determining the sex ratio of the offspring are given in the legend to Fig. 2. a Qmax for fecundity is assumed to be the same as for adult survival. b Hatching (eggs and immatures) and age-dependent mortality (adults) are assumed to increase monotonically with age, so d is set to 0.

increase (rm). The maximum observed and predicted values of R0 are both around 95 and occur around 21 C. The maximum predicted value of rm is about 0.338 d1 (as compared with the observed value of 0.346 d1) and occurs at a temperature around 28 C. The model predicts a steeper decline in rm when the temperature exceeds 28 C than was actually observed. Variable Temperatures. R0 decreased with the temperature amplitude (Fig. 5A). The observed values of R0 tended to be higher when the maximum temperatures occurred at the first day (Group H) compared with when the minimum temperatures occurred first (Group L). The opposite pattern was observed for the generation time (Fig. 5C). On the other hand, the observed rm-values tended to increase with amplitude to reach a maximum when the amplitude was 69 C (Fig. 5B). rm for Group H was higher than for Group L but only at the two highest amplitudes. The generation time (G) declined from 23.3 d at 22 60 C to 16–17 d at 22 6 9 and 22 6 12 C. The predicted values in Fig. 5 were obtained from the dynamic life-table model and the parameter values in Table 1. The predicted values of R0 were almost identical for Group H and L, so only one line is shown, whereas the predicted values of rm and G were slightly different for the two groups, except at the smallest amplitudes. When the temperature was set to 22 60 C, the observed values of R0 and rm were 102.83 and 0.199 d1, respectively. The corresponding values obtained from the first series of experiments can be found from the graphs fitted to the data in Fig. 2. Thus, at 22 C the predicted value of R0 is 95.78 (Fig. 2G) and of rm 0.243 d1 (Fig. 2H), indicating that the

9

differences between the observed and predicted values in Fig. 5, at least partly, can be attributed to genetic differences between the two strains of T. urticae. To elucidate the difference between the two strains further, we compared the percentage of female offspring, the developmental rate from egg to adult of females, percentage surviving from egg to adult, the oviposition period of adult females, and the total number of eggs produced per female of the Ibaraki strain with the corresponding values of the Hokkaido strain. We used the fitted graphs in Fig. 2A–E to obtain the predicted values at 22 C. As seen from Table 2, the Ibaraki strain was characterized by having a lower sex ratio and developmental rate, a longer oviposition period, and a higher total fecundity per female than the Hokkaido strain. The survival rate from egg to adult was more or less the same for both strains. A lower sex ratio will, all other things being equal, reduce both R0 and rm, while a longer oviposition period and a higher oviposition rate will increase R0 and rm. However, a lower developmental rate will decrease rm without affecting R0. Therefore, the relatively low rm of the Ibaraki strain is attributed primarily to a longer developmental time from egg to adult compared with the Hokkaido strain (12.20 d vs. 10.86 d). Discussion The purpose of this paper is to demonstrate that a fairly simple mathematical expression, called the SANDY model, is sufficiently flexible to serve as a general model in many ecological contexts where a factor, or a combination of factors, affects a response variable (Fig. 1). The rationale of the model is that the response variable y has a maximum (termed ymax) when all factors affecting y are optimal. The more the factors deviate from their optimal values, the lower the value of y. When a single factor takes values outside a lower threshold (denoted Vmin) or an upper threshold (denoted Vmax) y will become 0, irrespective of the values of all the other factors. Another assumption of the SANDY approach is that the combined effect of two or more limiting factors is multiplicative. This assumption corresponds to the Baule’s extension of Mitscherling’s law of diminishing return when a combination of different factors affects plant growth (Harmsen 2000). Multiplicative models have been used by Duthie (1997) and Guyader et al. (2013) to account for the combined effect of temperature and wetness on foliar diseases. The multiplicate approach assumes that the factors act independently of each other. Therefore, it is the simplest assumption to make and serves as a starting point if there are no experimental data available to disclose possible interactions between the factors. Unfortunately, life-table studies where more than one factor has been varied at a time, as in a factorial design, are rare. An example is the study by Weisse et al. (2013), where they exposed the rotifer Cephalodella acidophila to all combinations of pH (with four levels), temperature (with three levels), and food concentration (with three levels), in total 36 different combinations. Their

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ENVIRONMENTAL ENTOMOLOGY 100

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Fig. 4. (A) The net reproductive rate (R0) and (B) The intrinsic rate of increase (rm) of T. urticae at five constant temperatures using P. vulgaris as host plant. The predicted values (lines) are obtained by means of the dynamic life-table model using the parameter values given in Table 1.

analysis, based on a three-way ANOVA, showed a strong interaction between the factors, but the analysis assumed that the factors were additive and not multiplicative. In the present paper, we have focused on a single environmental factor, namely, temperature, though this factor may affect vital rates in two ways: either directly, e.g., by causing instant mortality if individuals are exposed to extreme temperatures, or indirectly via their biological age, which represents the cumulated effect of temperatures experienced earlier in life. In arthropods, the aging process progresses with the temperature when the temperature is above a lower threshold (Tb) (Maiorano et al. 2012). Early models of development (e.g., Campbell et al. 1974) assumed that development rate increases proportionally with temperature, whereas more sophisticated models also incorporated a negative effect of high temperatures (e.g., Curry et al. 1978, Yan and Hunt 1999, Maiorano et al. 2012, Fand et al. 2014). These models assume that the upper threshold for development (Tu) coincides with the upper limit for survival (Tmax), which is in agreement with empirical data (Logan et al 1976, Re´gnie`re and Logan 2003, Son and Lewis 2005). There are also some data that point at Tb and Tmin being more or less the same (see, e.g., Fand et al. 2014), but not always. Thus, Ullah et al. (2011, 2012) found that the lower thresholds for development and survival were quite close to each other in the spider mites Tetranychus merganser and Tetranychus macfarlanei, but not in the closely related species Tetranychus kanzawai. We should expect that species adapted to sustain cold periods can set development at stand-by

without concurrently increasing mortality significantly, implying that Tmin will be markedly lower than Tb. In such cases, our assumption of Tb ¼ Tmin will be wrong and two different values should be used instead of one. In the case of T. urticae, the lower thresholds for development at survival were estimated to be close to 0 C for all three stages (Table 1). This agrees quite well with empirical data for developmental rate (Fig. 2B), whereas data for survival (Fig. 2C) provide no clue to Tmin as the drop in survival rate occurs outside the range of empirical data. However, as Tmin cannot exceed Tb and individuals are expected to die when exposed to freezing temperatures, it seems likely that both Tmin and Tb are close to 0 C. We used biological age as a measure of the cumulated day-degrees an individual has achieved while in a given stage. An individual’s biological age (Q) will be the same as its chronological age (x) if development takes place at the optimal temperature, but lower if development occurs at suboptimal temperatures. Biological age increases until it reaches a maximum value (denoted Qmax), which corresponds to the chronological time it takes to complete development in a stage if temperature is optimal. We modeled hatching and mortality rates as increasing functions of an individual’s biological age. It means that individuals born at the same time will not necessarily develop through the life stages at the same time. This corresponds to the concept of developmental variability (Schaalje and van der Vaart 1989, Re´gnie`re and Logan 2003). An advantage of the SANDY model is that three of its five parameters (ymax, Vmin, and Vmax) are easy to

2015

NACHMAN AND GOTOH: MODELING TEMPERATURE EFFECTS ON VITAL RATES

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Fig. 5. (A) The net reproductive rate (R0), (B) the intrinsic rate of natural increase (rm), and (C) the generation time (G) of T. urticae when the temperature varied around 22 C with a daily amplitude of 60, 63, 66, 69, and 612 C. Symbols show the observed values of R0, rm, and G when mites were exposed to the maximum temperature of the amplitude on the first day (Group H, squares), and when the minimum temperature was used first (Group L, circles). The lines show the predicted values (Group H: full line: Group L: broken line) based on the dynamic life-table model using the parameter values in Table 1. The predictions for R0 were almost identical for Group H and L, so only one line is shown.

Table 2. Differences between the Ibaraki and Hokkaido strains of T. urticae with respect to life-table parameters Parameter

Average 6 SE (Ibaraki strain)

n

Estimated values (Hokkaido strain)

% Females Developmental rate (d1) % Survival from egg to adult Oviposition period (d) Total eggs per female

61.80 6 1.37 0.0820 6 0.0001 86.42 6 0.97 23.78 6 0.47 162.43 6 3.20

905 1448 1649 120 120

74.24 (Fig. 2A) 0.0921 (Fig. 2B) 91.78 (Fig. 2C) 19.23 (Fig. 2D) 129.97 (Fig. 2E)

Values for the Ibaraki strains were obtained from the amplitude experiment conducted at 22 60 C. n is the number of individuals used to calculate the average and standard error (SE) of each parameter. The estimated values were obtained from the SANDY model fitted to data from the first series of experiments using mites from the Hokkaido strain.

interpret in a biological context while the two other parameters (a and b) determine the shape of the relationship between the effect variable V and the response variable y. In combination, Vmin, Vmax, a, and b define a species’ niche with respect to the factor V. Thus,

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stenotopic species are are characterized by a small difference between Vmax and Vmin and large value of a and b, while the opposite applies to eurytopic species. The same parameters defining the niche width also determine the value of V where the species has an optimum with respect to factor V (Vopt). In contrast, Yan and Hunt (1999) used Vopt directly as a parameter in a model describing developmental rate. Their model has fewer parameters than SANDY, but at the expense of generality. Likewise, the model of Amarasekare and Sifuentes (2012) is limited to cases where the temperature-dependent survivorship curve is symmetric around the optimal temperature. Sigmoid curves produced by SANDY are qualitatively similar to curves produced by the Weibull distribution (Weibull 1951) except that the former yields y ¼ 1 for V  Vopt, whereas the latter asymptotically approaches 1 for V ! 1. Thus, in cases where a biologically meaningful Vopt cannot be defined as in, e.g., functional response models, the Weibull distribution may be preferable. The Weibull distribution has been widely used to describe the cumulative distribution of developmental time (Wagner et al. 1984b, Re´gnie`re 1990, Son and Lewis 2005). There are, however, some limitations to the type of relationships SANDY can fit. Thus, it can only be applied to unimodal functions and to functions where y  0 for all values of V. The former limitation may not be a problem, as the great majority of biological relationships are unimodal, whereas the latter limitation can be circumvented by modeling ey instead, where y can take both positive and negative values. We used this approach to model rm via the finite rate of increase, i.e., k ¼ erm (Fig. 2H) In this paper, we have applied SANDY as a simple curve-fitting tool, for instance to predict how the net reproductive rate (R0) and the intrinsic rate of increase (rm) vary with temperature (Fig. 2). Curve-fitting allows for estimating values of the dependent variable (y) for arbitrary values of the independent variable (V) in the interval between Vmin, and Vmax, though values obtained by extrapolation outside the range for which observations are available should be done with caution. However, our main purpose of developing the SANDY model was that we wanted a simple mechanistic model to describe how a population, consisting of individuals at different stages and ages, is expected to develop over time in a fluctuating environment. Here, we have demonstrated that the SANDY model can be used in combination with a dynamic model of an age-, stage- and sex-structured population to build a model describing age- and temperature-dependent survival, hatching, and oviposition rates of T. urticae (Fig. 3). When the dynamic model is used to simulate populations in a variable environment, new cohorts are started at each time step (e.g., every day), which means that the generations will overlap and individuals will experience different environmental conditions throughout their life-time. Skovga˚rd and Nachman (in press) used the SANDY approach to model the dynamics of stable flies (Stomoxys calcitrans) and their natural enemy, the parasitic wasp Spalangia camaroni in a Danish stable where the temperature varies at a daily basis.

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In this paper, we used a time scale of one day when modeling life-table data, both because this is appropriate in a constant environment and because experimental data are recorded on a daily basis. However, in an environment with fluctuating temperatures other time scales may be more relevant, e.g., hours or minutes (Worner 1992, Maiorano et al. 2012, Ikemoto and Egami 2013). In the amplitude experiments characterized by regular temperature changes, a time scale of one hour would allow for incorporating a gradual transition from one temperature level to another instead of assuming that the transitions occur instantaneously when a time scale of one day is used. From a modeling point of view, the individuals will be exposed to extreme conditions for a longer time if the temperature change occurs stepwise compared with a gradual transition. As a consequence of modeling the temperature transitions as instantaneous, the predicted influence of fluctuations on R0 and rm will be exaggerated compared with observed values, in particular when the amplitude is large and the transition periods are long. As our purpose was not to obtain quantitative agreement between observed and predicted values but rather to demonstrate the approach, we did not adjust for this bias by using a shorter time unit. In the amplitude experiment, where the temperature was set to 22 6 0 C, we expected values of R0 and rm similar to those obtained by fitting SANDY to data from the constant temperature experiments. However, this was not the case, as the observed and predicted values of R0 were 102.83 and 95.78, respectively (Fig. 2G), while they were and 0.199 d1 and 0.243 d1 for rm (Fig. 2H). In comparison, when the dynamic lifetable model, parameterized from values in Table 1, was used to predict R0 and rm, the values were R0 ¼ 89.29 and rm ¼ 0.257 d1 (Fig. 4). As the two series of experiments were conducted under similar conditions, we attribute the differences in R0 and rm to genetic differences between the two stock populations of T. urticae. The Hokkaido culture was founded in 2001 and used in 2010 for the constant temperature experiments, while the Ibaraki culture was founded in 2006 and used in 2013 for the amplitude experiments. The Ibaraki strain had a lower sex ratio and a longer developmental time than the Hokkaido strain, whereas the latter has a shorter oviposition period and a lower lifetime egg production per female (Table 2). In combination, these lifehistory traits indicate that the former strain has been subject to K-selection and the latter to r-selection, using r- and K-selection as relative terms (see, e.g., Pianka 1974, Mueller and Ayala 1981, Stearns 1992). Whether the differences between the two strains existed already before they were collected in nature or have evolved later as a result of keeping the mites in culture for so many years, or a combination of both causes, cannot be decided. However, it seems likely that the laboratory populations have undergone some genetic changes as adaptations to the stocking conditions (Mueller and Joshi 2000, Zygouridis et al. 2014). Figure 5 shows that R0 declined with increasing temperature amplitude, whereas rm was relatively constant except for the most extreme amplitude. We also found

that generation time (G) declined with amplitude (Fig. 5C), which implies that the decrease in R0 is balanced by a decline in generation time as the relationship between R0, rm, and G is rm ¼ lnR0 =G (Birch 1948). This indicates a trade-off between developmental rate and reproductive output (Stearns 1992, Koene and Matt 2004). It is important to stress that the analysis of the effects of temperature fluctuations on the life-history parameters are based on fluctuations around 22 C and with equal exposure to the upper and lower temperature, so the results are not likely to apply in general. Thus, the variable temperatures may either accelerate or decelerate development depending on the average and amplitude of the temperature (Tanigoshi et al. 1976, Hagstrum and Milliken 1991, Worner 1992, Fantinou et al. 2003). Figure 2B shows that the curve relating developmental rate to temperature is concave up to about 27 C whereupon it becomes convex. This means that moderate temperature amplitudes around 22 C will yield a higher average developmental rate compared with when the temperature is constant. On the other hand, if the temperature fluctuates around, e.g., 28 C or other temperatures located at the convex part of the developmental curve, the average developmental rate is expected to decrease with increasing amplitude (Ikemoto and Egami 2013). As developmental rate is not the only factor contributing to rm, the dynamic lifetable model allows us to identify the threshold temperature at which a variable temperature does not increase rm compared with the value attained at a constant temperature. We found that variable temperatures will increase the growth rate when the temperature is below 24.5 C and otherwise decrease the growth rate. Fantinou et al. (2003) found that alternating temperatures reduced the lower temperature threshold for immature stages of Sesamia nonagrioides (Lepidoptera: Noctuidae) which, all other things being equal, will improve performance at low temperatures. Vangansbeke et al. (2013) compared the net reproductive rate and the intrinsic rate of increase of T. urticae at two constant temperature regimes (15 and 20 C) and one variable temperature regime (8 h at 5 C and 16 h at 20 C) corresponding to an average temperature of 15 C. The highest R0 and rm were found when the temperature was constantly at 20 C (107.7 and 0.192 d1) and the lowest values when the temperature was constantly at 15 C (67.0 and 0.102 d1) with the variable temperature regime in between (99.0 and 0.136 d1). When we applied the dynamic life-table model to predict R0 and rm we found R0 ¼ 88.37 and rm ¼ 0.242 d1 for 20 C and R0 ¼ 30.90 and rm ¼ 0.0812 d1 for 15 C, while for fluctuating temperatures the model predicted R0 ¼ 37.35 and rm ¼ 0.143 d1. Though the predicted values differ quantitatively from the values found by Vangansbeke et al. (2013), which is expected as the experimental procedure and mite strains were not identical, the patterns are consistent with respect to how variable and constant temperatures affect R0 and rm.

2015

NACHMAN AND GOTOH: MODELING TEMPERATURE EFFECTS ON VITAL RATES Acknowledgments

The ideas presented in this paper date back to 1997, where Dolf Harmsen, Sandra (Sandy) Walde, John Hardman, Phil Lester, and the first author spent three inspiring days together in a cottage belonging to Queens University’s field station (Ontario, Canada). During the stay, Sandy Walde celebrated her birthday and the model was therefore named after her. We want to thank Maj-Britt Pontoppidan, University of Copenhagen, for her constructive comments to the manuscript.

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