Planorbarius Corneus L.): a Modelling Approach Fred Joppã

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Limnologica 36 (2006) 17–25 www.elsevier.de/limno

Comparative studies on the dispersal of the Great Ramshorn (Planorbarius corneus L.): A modelling approach Fred JoppÃ

Laboratory for Soil Zoology and Ecology, Freie Universita¨t Berlin, Grunewaldstr. 34, 12165 Berlin, Germany

Received 16 June 2005; accepted 21 October 2005

Abstract

The Great Ramshorn (Planorbarius corneus L.) is a frequent inhabitat of central European wetlands and ﬂoodplains. The dispersal capacity of P. corneus, as a key factor to survival, was analysed using different computer model approaches: a general equation-based model and an individual-based model. Both models came to different conclusions on the maximum covered distances of P. corneus-individuals. The parametrisation of the models were based on detailed laboratory measurements and ﬁeld experiments performed with specimen of the fen wetland system of Rhin-Havelluch (Brandenburg, North-East Germany). Based on the results of the statistical analysis, the dispersal behaviour could be depicted in the equation-based model as a random walk which implied a relatively regular resource utilisation in long-term simulations. Depicting the dispersal qualities of P. corneus with a highly resolved individual-based model showed distinct local aggregations clusters of the snail-individuals. As a consequence, the resource utilisation was less regularly in the long- term simulations for the individual-based model than it was in the equation-based model. Another characteristic of the individual-based representation was that in the case of several consecutive move steps and based on the autocorrelation of the data the P. corneus-individuals could use a wider area of space. The data from the empirical measurements of short-term dispersal observations of single individuals were integrated within the different model extrapolations. This made it possible to discuss the long-term consequences, like the large dispersal range of the individuals and the spatial distribution of the population in the plane, for the persistence of P. corneus in a real landscape. r 2005 Elsevier GmbH. All rights reserved.

Keywords: Fen wetlands; Great Ramshorn; Planorbarius corneus; Dispersal; Persistence; Computer models

Introduction Germany (German Federal Ministry for the Environ- ment & Nature Conservation and Nuclear Safety, 1998). Fen wetlands are landscapes that were largely reduced Several nature conservation programmes have been by drainage in the context of agricultural development launched to conserve the biological diversity or even and land use changes. Between 1950 and 1985, about improve the quality of the vulnerable wetlands in 60% of all wetlands disappeared in the western part of Germany (Kratz & Pfadenhauer, 2002; Succow & Jeschke, 1990; Succow & Joosten, 2000). In this context, ÃTel.: +49 30 83855946; fax: +49 30 83853886. focal organisms that are increasingly used to indicate E-mail address: [email protected] (F. Jopp). wetland conditions are watersnails (Baur & Ringeis,

2002; Foeckler, Deichner, & Schmidt, 1997; Foeckler photoperiod of 13 h. The tank surface was equipped et al., 2005). Planorbid snails (Gastropoda: Planorbidae) with aquarium gravel of ca. 12 mm grain size which was are the largest family of aquatic pulmonate gastropods, geared to the roughness of the local surface pond which are nearly present on all continents and most structure in the Rhin-Havelluch (Jopp, 2003). To islands (Dillon, 2000). The largest member of this group simulate a nature-like water flow and to preserve water which is very abundant in central Europe is the Great quality, an external water filter system was applied. In Ramshorn (Planorbarius corneus L.), that lives sapro- accordance with Costil and Bailey (1998) in preliminary and phytophage and feeds on the waterground on experiments no circadian activity periodicity in snail detritus (Engelhardt, 2003). Although P. corneus still activity was found, which allowed to extrapolate day- inhabits nearly every central European flood plain and time measurements to a 24 h period. Out of the wetland, the species declines constantly (Wiese, 1991, measurement hours P. corneus-individuals were held in 2005). The reasons are not yet clear. different tanks of the same construction style and under Due to the technical problems in tracking water- the same conditions, where they were fed overnight. organisms in the field, the only examples where snail Each of 20 P. corneus-specimen was taken for observa- dispersal was observed, focussed on land-snails (Janßen tion. To assure acclimatisation, they were set 2 h before & Plachter, 1998; Popov & Kramarenko, 2004). beginning into the tank (consequently, starting positions Although it is generally agreed that (water-)snails were at random). Then, every second minute the should have small cruising radius, there are single position of each individual was recorded for at least 50 observations on water-snails that cover larger distances consecutive time steps, where runs were stopped earlier by being passively drifted away by the water current when the individual did not move anymore. In such a (Jopp, 2003). However, until now there are no detailed case, the run with the predetermined individual was coherent data on dispersal of individuals and colonisa- repeated at a later time. The measuring procedure tion capacity of water-snails yet. consisted of adjusting two orthogonal metering rules This study investigates individual dispersal of every second minute exactly over the position of the P. corneus. Laboratory measurements and field experi- observed P. corneus-individual in the fish-tank. This ments have been performed with Planorbarius-indivi- allowed to read-off directly the local x/y-positions which duals of the fen wetlands of Rhin-Havelluch were then transformed to polar coordinates to indicate (Brandenburg, North-East Germany, 60 km northwest successive step lengths and turning angles that were of Berlin). Different types of computer models were used measured against the previous movement vector (Turch- to visualise and analyse the dispersal efficiency. First, in, 1998). Therefore, when discretising movement the pathways of the individuals were analysed by means patterns that were recorded at fixed time intervals, every of a correlated random walk (CRW) (Kareiva & move step of an individual consists of one step length Shigesada, 1983) using an equation-based computer and one turning angle (Turchin, 1998). Finally, the model (see Turchin, 1998) to simulate the average discretisation process of the movement pattern leads to conditions. Then, an individual-based computer model species-specific distributions of step lengths and turning (see DeAngelis & Gross, 1992) was introduced to angles. When working with mark-recapture-, teleme- represent the high variability of the empirical data. trical or other individual-focused data gathering meth- With both model types, long-term simulations were ods on epigeic arthropods or organisms of comparable applied to analyse implications of the dispersal strategy dispersal habit, the discretisation of the dispersal data is of P. corneus-individuals to persist in complex natural a standard method (see Jopp, 2003; Jopp & Reuter, fen-wetland habitats. 2005; Turchin, 1998).

Modeling approaches Material and methods Equation-based model Data acquisition The method of Kareiva and Shigesada (1983) was used to test the hypothesis whether the animals perform Mark-recapture experiments and movement measure- a CRW. Subdividing the data set into parts of equal size, ments were performed with P. corneus-individuals in a control quantity, the net-squared displacement (NSD, ponds of the fen-wetlands of Rhin-Havelluch. Measure- see Appendix for algorithm), is calculated for the parts ments with field-grown P. corneus-individuals were of the data set. In the case that NSD grows linearly, the executed in the laboratory using flat fish-tanks of whole dispersal process behaves diffusion-like, because 100 100 10 cm. To guarantee nature-like conditions, with large N CRW processes converge against diffusion the tanks were filled with water from the fen-wetlands (Skellam, 1973; Turchin, 1998). If a linearly growing which was kept at a constant temperature and under a NSD is verified, Holmes (1993) suggests that the ARTICLE IN PRESS F. Jopp / Limnologica 36 (2006) 17–25 19 investigated dispersal process can be modeled as a homogenous distribution of the snail-individuals in the diffusion process. short-term simulation output. If this was true, then local In order to visualise the long-term consequences of aggregation clusters of individuals in the plane should space utilisation for P. corneus, an equation-based exist in the simulation output. However, these clusters computer-model, which was written in the programming should be averaged out or at least should be strongly language R (Ihaka & Gentleman, 1996), was employed reduced in the long-term simulation output (in total, to simulate a 2D-random walk (see Appendix for 1.6 108 move steps). The density of the individuals algorithm). Thirty-two P. corneus objects were released and the size of the simulation area were based on at random starting positions inside the simulation area field observations (Jopp, 2003). The time scale in all which corresponded to an area of 20 20 m, where each individual-based simulations was 2 min per move step of them made 5 104 move steps in the plane. This was and corresponded with the temporal resolution that was thousand times more per individual than the data used in the empirical measurements. To minimise recording volume in the laboratory and corresponded boundary effects, the boundary conditions were toroi- (under the same sampling rate: one move step every dal, meaning that snails that leave the simulation area in second minute) to an observation time of ca. 100 days in one direction re-emerge on the opposite side of the area, the field. To depict the long-term implications of this still moving in the same direction. The simulation area movement strategy in the plane, the simulation area was was gridded in 20 rows and 20 columns to capture the gridded in 20 rows. For statistical analyses the change in the local variance: For each of these rows the cumulative abundance of move steps per row was cumulative abundance of the total movement step recorded. positions per row was recorded. Rows were numbered ascending from bottom to top. As a geo-statistical Individual-based model measure the variance of the cumulative abundance of In a second approach, the space utilisation of P. the movement step positions per row was plotted against corneus-individuals was modeled using an individual- the spatial position of the row number (Levin & Buttel, based approach (DeAngelis & Gross, 1992). A computer 1986). model was developed using the object-orientated programming language Simula (Dahl, Myhrhaug, & Ny- gaard, 1968). First, the laboratory data of the move step lengths were classified, and a regression function was Results fitted to represent the distribution of the move step distances (y ¼ 421.52e0.756x; r2 ¼ 0.94). A movement Laboratory dispersal experiment algorithm was developed to draw random samples from this distribution and to represent the autocorrelation of In total, the 20 observed individuals moved over a the data (for algorithm, see Appendix). The general distance of 2510 cm, the maximum distance was 14.6 cm model layout follows Ho¨ lker and Breckling (2001) and in 2 min, the minimum was 0.4 cm in this time. On Breckling, Mu¨ ller, Reuter, Ho¨ lker, and Fra¨ nzle (2005): average, every individual moved 125.5 cm (median P. corneus-individuals are modeled as autonomously 111.0 cm) in the total observation time of 100 min per acting objects which are started at random positions run. The portion of the pauses in which no change of the inside a simulation representing an area of 20 20 m. To individual’s location could be observed, was overall maintain comparability with the equation-based model, 29%. If one extrapolates these dispersal performances to the individual-based model is kept very simple and a full year, based on the assumptions, that the ratio of does not take interactions among the individuals or active movement and pauses remains constant, there spatial heterogeneity into account (for a review on the would be no change in annual activity periodicity, and, potential of the individual-based approach, see that the abiotic conditions of the real habitat would Breckling & Reuter, 1996; DeAngelis & Gross, 1992; allow dispersal, then, P. corneus-individuals would cover Grimm & Railsback, 2005). In a short-time simulation a distance of 4.63 km/year. Fig. 1 shows a typical experiment each of 3200 P. corneus-individuals made distribution pattern: When arranging the data in classes 500 move steps which was corresponded to the of movement distances, it can be seen, that the shorter total amount of move steps that the individuals the distance, the more frequent it occurs. performed in the equation-based approach (in total: As to the spatial usage of the fish tank the whole 1.6 106 move steps). Then, this first approach was spatial area was used equably by the individuals, no compared with a long-term simulation experiment preferred local areas could be noticed. To verify this where 3200 individuals made 5 104 move steps, each. statistically, the whole surface of the fish tank was The experimental set-up was based on the expectation subdivided into four areas (of 25 25 cm, each) with that the built-in autocorrelation factor (see Appendix) in their orientation corresponded to the four quadrants of the individual-based model should prevent a completely the circle ((i) 1–901, (ii) 91–1801, (iii) 181–2701, and (iv) ARTICLE IN PRESS 20 F. Jopp / Limnologica 36 (2006) 17–25

Distance classes of move length Visiting frequencies per quadrant 200 160

150 120

80 100 Frequency Frequency

40 50

0 0 010155 I. II. III. IV. Distance classes (cm) Quadrants of the fish tank Fig. 1. Empirical distribution of the factor move lengths from Fig. 2. Empirical distribution of the visiting frequencies per the laboratory measurements. The data for the move length of quadrant from the laboratory measurements. After subdivid- all runs were classiﬁed and plotted against the frequency. It ing the ﬁsh tank in four different quadrants it can be seen, that can be seen, that the shorter the distance, the more frequent it these were not statistically different visited by the snails occurs in the run. (w2 ¼ 7.0734, df ¼ 3, p-value ¼ 0.0696). Therefore, the individuals did not prefer local areas during their runs.

271–3601). Then, the overall visiting frequencies of the snails during the whole measurements in-between the four quadrants was tested with the result, that there were no signiﬁcant differences (w2 ¼ 7.0734, df ¼ 3, p-value ¼ 0.0696, see Fig. 2). However, amongst single runs the individuals showed often for the endurance of 6–7 consecutive move steps a certain constancy in preferred direction and length (Fig. 3). Often, these points have a consecutive distance of ca. 1.5 cm, each. After such a homogenous sequence the individuals tend to initiate a major change of direction. With respect to these sequences of consecutive points, the movement pattern shows autocorrelation. A constancy in preferred distances and lengths that exceeds these short periods could not be observed.

NSD and diffusion After evenly dividing the whole data set into four parts, NSD was calculated (see Fig. 4a, expectation ¼ 5771, calculated ¼ 6082; y ¼ 1342x+403). Using boot- strapping, Cain (1989) has re-analysed the data of Kareiva and Shigesada (1983): he points out that the expected and the calculated NSD should not differ more than 10% lest the assumptions of a CRW would be violated. Given that in our case the difference between the expected and the calculated NSD is ca. 5% and the NSD grows linearly while calculating it from the data set, one can state that the laboratory dispersal of P. Fig. 3. Empirical walking path of one representative P. corneus-individuals is consistent with the assumption of a corneus-individual that moved 50 move steps through the ﬁsh CRW. NSD was also used to calculate the diffusion tank. The observation points, where the positions of the coefﬁcient (D ¼ 14.5 m2) for P. corneus which can be individual were recorded, are depicted as dots. ARTICLE IN PRESS F. Jopp / Limnologica 36 (2006) 17–25 21

A. Net Squared Displacement B. Random Walk, 50 move steps

6,000

Expected NSD = 5,771

Empirical NSD = 6,082 5,000

4,000

Net Squared Displacement 3,000

2,000

12 34 Parts of the Data Set

C. Random Walk, 50,000 move steps D. Visiting frequencies per row

80,000

60,000

40,000

20,000

0 rows Fig. 4. Equation-based movement analysis: (a) graphical determination of net squared displacement of the empirical data set. (b) Typical simulation output of the random walk model for one P. corneus-individual making 50 move steps. (c) Typical simulation output of the random walk model where only one P. corneus-individual makes 5 104 move steps. The whole simulation output of 32 individuals could not be depicted, anymore. The simulation area was gridded into 20 rows. (d) The analysis for spatial effects of the long-term simulation for all 32 individuals. For each row, the snails’ visiting frequency was plotted. All rows were nearly equably visited (w2 ¼ 35.165, df ¼ 19, p-value ¼ 0.01334). equated with the dispersal of the population on the individual (of 32) that started in the centre of the average (Holmes, 1993; Skellam, 1973). simulation area and made 5 104 move steps. For the sake of lucidity, only one individual is depicted in Application of the equation-based model the simulation output of Fig. 4c. Here, the move steps Fig. 4b shows an example of a typical simulated disperse randomly and diffusion-like over the whole P. corneus-individual that makes 50 move steps based on simulation area, whereas certain regions of the simula- a simple random-walk algorithm. On this short-term tion seem to be visited more frequent than others. In the time-scale there could emerge the impression of a long-term dispersal of all 32 individuals, under the preferred movement direction which is not implemented control of the diffusion-based steering mechanism, this in the source code (see Appendix). Fig. 4c shows a unevenness is leveled out, as it can be seen in the ﬁnal typical output of a long-term replication of one results of Fig. 4d. Here, the cumulative abundances of ARTICLE IN PRESS 22 F. Jopp / Limnologica 36 (2006) 17–25 the move steps of all 32 individuals are depicted inside 500 Move Steps 50,000 Move Steps the simulation area. The starting positions of the 32 individuals were at random. The distribution of the 8.0 8.0 abundances per row shows the expected shape of overall leveled frequencies in approximation. Statistically, the cumulative abundances of the move steps in-between the 7.5 7.5 20 rows are just at the border of being different (w2 ¼ 35.165, df ¼ 19, p-value ¼ 0.01334). The prior 7.0 7.0 impression of a preferred moving direction as to be seen in some short-term simulation outputs does not appear anymore in the long-term simulation output. 6.5 6.5 log10(variance) log10(variance)

Application of the individual-based model 6.0 6.0 In this approach the movement behaviour of P. corneus was depicted highly resolved. The movement 5.5 5.5 algorithm of the IBM (see Appendix) incorporated the autocorrelation of the individuals with respect to the short sequences of consecutive points as shown in Fig. 3. 51015 20 5 1015 20 As a consequence of the high resoluted simulation (in (a)rows (b) rows total, 1.6 108 move steps) the true simulation output of 3200 individuals could not be reasonably visualised, Fig. 6. Statistical comparison of the long- and short-term individual-based simulations. Simulation area corresponds to anymore. Fig. 5 shows a typical simulation output of 32 20 20 m, boundary conditions were toroidal, starting posi- P. corneus-individuals, performing 500 move steps, each. tion were chosen at random. Simulation area was gridded Fig. 6a shows the statistical analysis of the short-term into a matrix of 20 rows by 20 columns. For each row, simulation (3200 individuals, 500 move steps). Under 3200 snails’ cumulative movement steps were recorded, the same temporal resolution (one move step, every variance was calculated and log 10-transformed to standardise second minute) the movement activity of 500 move steps scales. (a) Each simulated individual makes 500 move steps. equals to an observation time over one full day. The (b) Each simulated individual makes 5 104 move steps.

variance of the cumulative move steps varied strongly amongst the rows of the simulation area: the absolute values of the variance change between 1.7 and 8.2 105 which is a variation by the ﬁve-fold extent. The average variance in the short-term simulation was 4.7 105. Fig. 6b shows the statistical analysis of the long-term simulation. In this setup the 3200 individuals perform 50,000 move steps which is corresponded to an observation time of 100 days (one move step, every second minute). Here, the variance lies between 6.8 107 and 19.8 107 which is a variation by the three-fold extent. The average variance is 10.6 107. Altogether, the variance does not differ in-between the rows in relation as much as in the former short-term output, the curve is much more smoothened and less rough as the one in Fig. 6a. But, the variance in the long-term simulation is on a much higher level (ca. 200 times as high) as in the short-term output. Note the logarithmised scale in Fig. 6a and b. An approximately equal distribution of the variances in-between the whole Fig. 5. Typical simulation output of the individual-based simulation area has neither occurred in the short-term model. Thirty-two P. corneus-individuals make 500 move nor in the long-term simulation. steps. Simulation area corresponds to 20 20 m, boundary The comparision of the application of the equation- conditions were toroidal, starting position were chosen at and the individual-based model types is given in the random. discussion part. ARTICLE IN PRESS F. Jopp / Limnologica 36 (2006) 17–25 23

Discussion r2 of 0.94 in the fitting function for the move step lengths, which is used in the individual-based dispersal Employing (correlated) random walk algorithms as a kernel). In this modeling approach, the individuals move theoretical basis for the analysis of dispersal processes is longer distances and are more unequally distributed in a standard method (Byers, 2001; Holmes, 1993). Due to the space as a diffusion process estimates and as it was the often included elaborated details and the long shown here before with the equation-based simulation development time individual-based approaches are not output. used very often in ecological studies (Jopp & Reuter, If these findings are used to assess the persistency of 2005; Turchin, 1998). On the basis of an ad hoc collected P. corneus in a fen-wetland system, first of all, one must data set of the dispersal of P. corneus-individuals the state that planorbid snails do have a large dispersal equation- and the individual-based approach were radius (here: 4.63 km/year). Thus, P. corneus has the deployed to analyse small-scale measurements and to ability to move long distances with a certain constancy extrapolate these finding to larger scales. The analysis of in moving direction and step length, as the empirical the NSD of the data set showed positively, that the part of this work has revealed. In summer times, in fen- dispersal of the planorbid-snails can be modeled on wetlands seasonal fluctuations of abiotic conditions can longer temporal scales as a simple random walk, lead to extreme situations (like local dried-up ponds). respectively as a diffusion process. A first equation- Then, the above mentioned dispersal efficiency could based simulation experiment confirmed, that long- help P. corneus to reach refugial habitats faster than lasting random walks of P. corneus in the plane do under the assumption of a pure random-walk-based have characteristics of diffusion processes (see Fig. 4). movement. As for the individual-based model, autocorrelation of Further long-term simulations that are based on short sequences of consecutive move steps was incorpo- realistical open land data on water levels, micro- rated in the model. Here, the experimental output topology, climatic sequences and open land dispersal brought different results because the long-term simula- data on P. corneus could help to achieve more precise tion of individuals led to a spatial extremely hetero- conclusions as available at the moment. geneous distribution in the plane (see Fig. 6). There was no tuning in of a qualitatively more homogeneous distribution of the cumulative move steps respectively of Acknowledgements the variance as it should be under the assumption of an operating diffusion process. I like to thank Gerd Weigmann and Broder Breckling Both modeling approaches seem to be differently for comments on earlier versions of the draft and I am suitable for the analysis of invertebrate micro-scale very grateful to the anonymous reviewers for their help. dispersal activities. Generally, equation-based methods have gained much recognition in the analysis of theoretical implications of animals’ dispersal (Hanski, 1991; Levins, 1969; Skellam, 1951; Turchin, 1998). But, Appendix A using modeling approaches with coarse averages, some authors point out that the correct estimation of local A.1. Mean squared displacement dispersion and extinction rates can be a problem (Breckling, 1992; Breckling & Mathes, 1991; Frank & The control quantity NSD (Kareiva & Shigesada, Berger, 1996). The situation depicted here of planorbid 1983), is calculated completely from the data set and snail micro-scale dispersal seems to support the above uses modified parameters of the distributions of move lengths and move angles of the data: given caveats: the calculation of NSD is done exclusively with average values (see Appendix), which accentuates c R2 nmþ 2m , obviously the diffusion-based part of the dispersal n 2 1 1 c processes. A direct reference of NSD to local spatial structures is difficult to gain. Both models come to where n is the total number of observations, m1 the different assessments of the outdoor dispersal behaviour arithmetic mean of the move lengths, m2 the squared of the animals: on the basis of a diffusion-like dispersal arithmetic mean of the move lengths, c the arithmetic kernel the exploitation of local resources will be much mean of the cosine of the move angles of the data set. more uniform. This can be seen in the different way the frequencies of the move steps per row vary in the A.2. Parameters for the equation-based model equation-based and in the individual-based long-term simulation outputs. By incorporating a stochastic The equation-based model was programmed in the and a correlation component the dispersal strategy of statistic programming language R (Ihaka & Gentleman, P. corneus could be visualised realistically (note the high 1996). The outer FOR-loop was used to create the ARTICLE IN PRESS 24 F. Jopp / Limnologica 36 (2006) 17–25 desired number of total replications of the random-walk Breckling, B. (1992). Uniqueness of ecosystems versus general- procedure. Comments are given after hashmarks. izability and predictability in ecology. Ecological Modelling, 63, 13–27. FOR (i in 1:NUMBER) { Breckling, B., & Mathes, K. (1991). Systemmodelle in der ¨ TRIAL’SAMPLE(l:4, 1) Okologie: Individuen-orientierte und kompartiment-bezo- IF (TRIAL ¼¼ 1) newx’newx+1 # step to the right gene Simulation, Anwendungen und Kritik. Verhandlungen ELSE IF (TRIAL ¼¼ 2) newx’newx1 # step to the left der Gesellschaft fu¨rO¨kologie, 19(3), 635–646. ELSE IF (TRIAL ¼¼ 3) newy’newy+1 # step upwards Breckling, B., Mu¨ ller, F., Reuter, H., Ho¨ lker, F., & Fra¨ nzle, O. ELSE (TRIAL ¼¼ 4) newy’newy1 # step downwards (2005). Emergent properties in individual-based ecological LINES ((x, newx), (y, newy)) models—Introducing case studies in ecosystem research ’ x newx context. Ecological Modelling, 186, 376–388. y’newy } Breckling, B., & Reuter, H. (1996). The use of individual-based models to study the interaction of different levels of organization in ecological systems. Senckenbergiana Mar- A.3. Parameters for the individual-based model itima, 27(3/6), 195–205. Byers, J. A. (2001). Correlated random walk equations of animal dispersal resolved by simulation. Ecology, 82, The core of the moving algorithm of the individual- 1680–1690. based model is a power law of the common form d 0.756x 2 Cain, M. L. (1989). The analysis of angular data in ecological t ¼ ch , (421.52 e ; r ¼ 0.94). The autocorrelation field studies. Ecology, 70, 1540–1543. was realised inside the movement algorithm in connec- Costil, K., & Bailey, S. E. R. (1998). Influence of water tion to the choice of move step length and direction temperature on the activity of Planorbarius corneus (L.) (angle) and is given here in pseudo-code. Uniform is a (Pulmonata, Planorbidae). Malacologia, 39, 141–150. random procedure of the form: x ¼ uniform (a, b), with Dahl, O. -J., Myhrhaug, B., & Nygaard, K., (1968). SIMULA, a and b as upper and lower limits, x is a random variable common base language, Technical Report, Norwegian for the interval (a, b). Computer Centre, Oslo.

1. Interval :¼Uniform (Intervall.OldFactor, Intervall.Old+Factor); 2. Selection :¼Uniform (Selection.OldInterval, Selection.Old+Interval);

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