original text of the thesis:
Population dynamics of the Gyrinid beetle Gyrinus marinus Gyll (Coleoptera)
With special reference to its dispersal activities (1987)

CHAPTER VIII-A. DISCUSSION AND SIMULATIONS

SUMMARY
The results of a field study of the population dynamics of the water beetle Gyrinus marinus Gyll. with the aim of tracing the effects of dispersal activities are checked and assessed by k-factor analysis as well as by stochastic simulation modelling.
The key-factor analysis establishes that survival of adult beetles from hibernation is the key-factor and highly determines population size after hibernation. Survival during larval and pupal development in spring is less, but still significantly determining the size of population fluctuations. According to the key-factor analysis survival at all other developmental and adult stages are immaterial for the fluctuations in population size.
Computer simulations show that even a small dispersal activity (three per cent of the individuals) with immigration into other populations significantly restricts the fluctuations in population size, both decreasing the chance of extinction of the population and the range of population sizes, in spite of the extra losses during dispersal activities. These effects of dispersal are most decisive when survival chances in the populations are low and when dispersal occurs at least also out of the period of reproduction. There is a rather complicated relationship between the survival chances during and out of the reproduction season, the success of reproduction, the survival from dispersal and the optimal level of dispersal activities. When the number of suitable habitats is low or when the exchange between habitats highly depends on the distances between habitats, the effects of dispersal are less than with a high number of habitats or when exchange is less distance-dependent. Dispersal as a result of high population densities restricts fluctuations in population size to a similar extent as randomly occuring dispersal, but though it longer keeps populations at higher densities as soon as the population density has become low the chance of extinction will be high. Dispersal as a direct result of adverse conditions only restricts fluctuations if dispersal occurs in most of the habitats. This situation is probably less predictable than when dispersal is due to density-dependent or random processes. A decrease of the variation in space (more synchronisation between habitats) increases the chance of extinction, whereas a decrease of the variation in time decreases it. Simulations that imitate the field situation as closely as possible show that even the very limited flight activity of Gyrinus marinus may highly decrease the chance of extinction. In simulation experiments with natural selection between Mendelian genotypes with recessive dispersal ability (M) on the one hand and without dispersal ability (B) at the other, the relative frequency of the M-genotype decreases with increasing population sizes, and increases with decreasing sizes. The M-genotype tends to low frequencies, as is known from some carabid beetles species with macropterous and brachypterous genotypes. The fluctuation in numbers of the M-genotype is relatively restricted, as compared with that of the B-genotype. The heterozygote functions as a buffer, lowering the chance of disappearance of one of the homozygote genotypes.

1. INTRODUCTION

1.1. In the previous chapters IV - VII reproduction and recruitment, as well as survival and dispersal both by flight and by swimming were analysed in the context of a comprehensive study of the population dynamics of the waterbeetle Gyrinus marinus Gyll.

In this chapter the estimates from the field study are brought together into simulation models to assess the significance of the dispersal activities found for population dynamics in general, and for Gyrinus marinus in particular. But first, using key-factor analysis, we will establish the relative importance of the consecutive stages during the development from egg to adult beetle for the changes in population size.

1.2. In Table VIII-1 the mean values and variation coefficients for reproduction, survival and dispersal from our field study are recapitulated. A substantial variation was found in the viability of eggs, the number of tenerals per oviposition, the survival of tenerals, the succes of hibernation, and the degree of dispersal both by flight and by swimming. The survival chance per week as well as egg production showed a low variation coefficient. The variation of survival during exchange by flight cannot be important, because it concerns only a small fraction of the beetles. The mortality during the different stages of development from egg until reproducing beetle can be mutually compared by k-factor analysis (Morris 1963, Varley and Gradwell 1960). The situation is complicated however, when an entire year is considered, since we have to deal then with two succeeding reproduction seasons, followed by a non-reproductive period in autumn and by hibernation.

2. KEY-FACTOR ANALYSIS

2.1. Introduction and methods
2.2.1. Key-factor analysis is a method to establish which stages during the development from adult to the next adult generation govern changes in population size (Varley and Gradwell 1960, Morris 1963, Podoler and Rogers 1975, Southwood 1978). Demographic processes among adults, such as mortality and dispersal are also considered stages in this model. The survival in each stage is defined by the chance of the individuals to reach the next one. The method requires the availability of data for a sequence of generations. For each generation the number of individuals (P_t) that is alive at the beginning of each stage (t) is calculated. The variables S_t, defined as log(P_t), and k_t, defined as S_t-1 - S_t, are calculated for each stage in each generation. The sum of the k_t-values gives the value of K (generation mortality). The mean k_t-value per stage indicates the average decrease in numbers from the beginning to the end of that stage, so that a higher mean k_t-value indicates that greater losses during that stage and thus a higher mortality is expected. As k-values are differences between two logarithms at base 10, k=0.05 corresponds with a mortality of 0.11, k=0.3 with one of 0.5 and k=1.0 with a mortality of about 0.9.
2.2.2. In our case only a small number of generations was studied, and for most generations only part of the required data could be directly estimated. Nevertheless, for most stages we could reasonably estimate a mean survival chance and a variation coefficient. These two parameters per stage suffice to simulate a number of generations, which can then be subjected to a key-factor analysis.
2.2.3. In the model the year is divided into four seasons: two reproductive seasons (spring and summer) each with a separate generation, a post-reproductive season (autumn) with the third generation which has to survive the fourth season (winter), see Fig VIII-1 and Fig VIII-2. For each season the K_S-value is the sum of the relevant k_t-values (k_t), i.e. season-mortality. K_y is the grand sum of all k_t-values of an entire year, i.e. year-mortality (K_S = k_t).
In the first two seasons two processes occur at the same time: (1) reproduction, with stages from egg production to the hardening of tenerals and (2) mortality of adult beetles, which can be divided into mortality in the population and mortality during dispersal by flight and by swimming. Only dispersal losses can be taken into account in a key-factor analysis, not dispersal itself or the resulting exchange between populations.
2.2.4. The survival per stage is simulated for each year and for each generation by drawing random values from the frequency distribution fitted to the suvival values from the field. The simulation is repeated 100 times, starting each year with 1000 females and 1000 males emerging from hibernation. The average course of numbers during the year according to this simulation is shown in Fig VIII-2.

2.3. Results of key-factor analysis
2.3.1. The 10 stages with the greatest losses are shown in Table VIII-2. On the average 43.0 per cent of the losses occur during hibernation and 26.6 per cent during larval and pupal development. The losses from dispersal by swimming or by flight are negligible except for swimming activities in spring. Note that five of the stages concern mortality in females, and only three stages concern male mortality. Neither reduction of maximal egg production (k₁) nor egg mortality (k₂) result in great losses.
2.3.2. Besides the magnitude of the mean k-value, the variation around the mean is significant because it determines the fluctuations of population size. It is clear that a high k-value with a small variation makes only a small contribution to the variation in population size. The same applies to small k-values, irrespective their variances. Therefore, the product of the mean k_t-value and its standard deviation (sd) will give a good indication of the relative contribution that each stage may give to the variation of population size. Ranking the stages again, the order is not changed in comparison to Table VIII-2 for the first five stages, see Table VIII-3.
Mortality during hibernation and, to a much smaller degree during larval and pupal development apparently govern the fluctuations in population size of adult beetles. It is important to note that the activities of adult beetles may have no influence on such losses, see below. Mortality of adults during the active seasons, and egg losses seem to be of only minor importance.
2.3.3. The contribution of the mortality in each stage to the variation in population size at the end of each season, and to that after hibernation at the end of one year-cycle is indicated by the product-moment correlation coefficients between the k-values concerned and (1) the K-factors of each period (K_P) and (2) the K-factors of the total year (K_y).
2.3.4. In our case a product-moment correlation is significant for r > 0.20 (df=98, P<0.05, Sachs 1984). The k-values of most stages are not correlated with K_P or K_y, see Table VIII-3.
Stages important to the fluctuations in population size must have a substantial K*sd-value and also have significant r-values with K_P- or K_y-values. It appears that fluctuations in the yearly population size are highly determined by hibernation and to a lesser degree by the developmental stages from egg until teneral in spring. The influence of the other stages upon the fluctuations of yearly population size is of secondary importance only.

3. EXCHANGE BETWEEN POPULATIONS

3.1. Introduction
3.1.1. Besides hibernation and development in spring, the other stages in the key-factor analysis above, including dispersal activities of adults, have only low values for k_t*sd and/or for r. Therefore, it seems immaterial for the fluctuations of population size what happens to the adults during the seasons in which they are active.
But a key-factor analysis deals only with a single population unit, whereas the significance of dispersal may be the exchange between different population units. Moreover, the key-factor analysis only deals with the variation in the costs of dispersal, not with its possible advantages: the analysis can only consider changes in numbers of individuals, not changes in the distribution of numbers of individuals over the population area.
3.1.2. The field data show a considerable variation in survival during hibernation (Table VIII-1). Thus, the future of a population unit is rather uncertain, and therefore the units can be considered unstable. Two processes may decrease the variation in population size caused by mortality during hibernation: (1) density-dependent processes and (2) exchange between (sub)populations.
According to our key-factor analysis density-dependent processes concerning the adult beetles will have little influence on the variation in population sizes. Egg-laying might be density-dependent, being influenced, for example, by changes in the quantity or quality of food. But although there is a considerable variation in the numbers of eggs laid per oviposition and per female, only minor variation was found between different populations and between different seasons (cf Chapter IV Reproduction). It is possible that we did not estimate egg production in situations in which density-dependent egg laying becomes important. However, comparing the (k_t*sd)-values of egg production with those of hibernation (cf  Table VIII-3), it is doubtful whether egg production can have a regulating influence on the population size of adult beetles.
3.1.3. Another density-dependent process may be found in larval development. In laboratory experiments with larvae of Dineutes (an American whirligig beetle species) cannibalism occured depending upon the available food (Istock 1966, 1967), but it is not clear whether larval cannibalism plays an important role in the field. Moreover, the k_t*sd-values of larval and pupal development are at least five times smaller than those of hibernation, so that larval and pupal developments cannot match themselves against hibernation in its influence upon overall fluctuations of population sizes. Like egg production, even if survival of larvae and pupae is density-dependent it is doubtful whether it can have a regulating influence on the population sizes of adults.
3.1.4. Moreover, it seems that, at least in our study area, the chance of extinction is more relevant than the chance of overcrowding. Thus, though a density-dependent egg production or larval mortality might possibly lower high numbers of individuals, it may be doubted whether small populations can be saved from extinction in that way. See also Den BOER (1986) for a discussion of density-dependent processes and their effect on population stability.
3.1.5. Southwood (1962) concludes that the amount of dispersal activity is related to the instability of the habitat. Species from instable habitats should show more dispersal than species usually living in more stable habitats. Den Boer (1968) developed the hypothesis of 'spreading of risk'. He supposes, among other things, that the dispersal of individuals between different sites or habitats would decrease the chance of extinction of the populations at the sites concerned and also lead to the (re)colonization of favourable habitats.
3.1.6. Kuno (1981) draws our attention to the fact that exchange between two populations with net reproduction asynchronously varying around R=1 will result in a relative increase of numbers in both populations (see Fig I-1). Metz et al (1983) and Klinkhamer et al (1984) show in simulations that partial exchange will also lead to a significant improvement of net reproduction ('Kuno'-effect). As the Kuno-effect can only manifest itself in the redistribution of the progeny, we can expect that it will only be traced in following generations, i.e. it occurs when the progeny is dispersing, not when the parents are dispersing (see also De Jong 1979).
3.1.7. In concordance with Southwood's postulation that dispersal is related to the instability of the habitat, we may suppose that the less predictable the conditions in the habitat are the stronger the positive effects of exchange between populations will be. Southwood implicitly assumes that the occurrence of dispersal is related both to mortality in the population and to mortality during dispersal. The worse the prospects for survival and reproduction in the population are or the better these prospects after dispersal activity, the more dispersal activity can occur. Implicitly is accepted that dispersal activity can occur more randomly when the conditions are more instable and that it will not only occur when conditions are adverse. Den Boer and Kuno suppose that dispersal not only may have the short term advantage to an individual of leaving an adverse situation, and sometimes of escaping overcrowding, but in the long term it will favour the populations (and thus the individuals or their progeny concerned) between which exchange occurs.
3.1.8. From this point of view, we can expect that exchange between Gyrinus-populations may favour persistence of the populations in the study area. The question is, of course, whether or not the estimated values for exchange, for reproduction and for survival chances in the populations as well as during dispersal will allow dispersal to improve the chance of persistence per population by contributing to a levelling of population fluctuations. In the following we will try to investigate this with the help of simulation experiments. The influence of selection upon dispersal activity will also be simulated. Apart from the particular situation of the studied populations of Gyrinus the effects of dispersal under other conditions will also be considered.

3.2. Description of the simulation models
3.2.1. Simulation models have to be considered with scepticism. The results may be greatly influenced by the organization of the model, the algorithms used, the assumptions, etc. (van der Eijk 1984). To assess the results of a simulation one must be able to understand the model, and that is why it is explained rather extensively. A flow-diagram is given in Fig. VIII-3.
3.2.2. The simulations were run on a 4 MHz personal computer and the program was writen in MicroSoft Basic. We aimed to construct a model that simulates the field situation as closely as possible, without making use of predetermined mathematical functions to describe a process. As usually in constructing complicated models a number of versions were developed, from simple to highly complicated. In general each process in the more developed versions of the model was simulated by summation and subtraction. Multiplication was used only in reproduction procedures. For example, if a number of emigrants from one habitat had to be distributed randomly over 19 other habitats, we did not use the quick method of dividing the emigrants randomly into 19 parts, but for each emigrant anew was chosen the habitat into which it will immigrate, which is how such processes occur in the field. Only for large numbers of identical emigrants the quick method was used. The consequence of such a program is that the simulation takes a great deal of time; a complicated simulation over 100 years took about three days of computer time!

3.2.3. The following features were built into the model:

• Reproduction, survival, hibernation and dispersal were considered separate processes; optionally they could influence each other.
• The initial version of the model concerned two 'species' differing in only one feature: one 'species' (M) shows dispersal, the other one (B) does not (imagine a macropterous and a brachypterous insect species respectively). In more developed versions the two 'species' became two genotypes of the same species, so that the genetic aspects of dispersal could be taken into account.
• In the initial versions no distinction was made between males and females, more developed versions differentiated males and females.
• There was a number of 'habitats' (mostly 20 in initial versions and 10 in more developed versions), between which exchange could occur.
• At the start the population size per habitat could either be determined randomly or set equal for each habitat.
• In a single year two or three generations could occur, the generation in spring always being the hibernated generation from the previous autumn. In early versions there was only one time step per generation, in later versions the model worked with steps of one month. This resulted in significant alterations in the model. For example, the time required for development from egg to teneral was no longer immaterial, because it takes more than one month, i.e. now more than one time step.
• Before the simulation started, for each period (generation or month) mean values and variation coefficients of reproduction, survival and dispersal were entered. In the more developed versions both variation in time and variation between habitats were introduced. For each habitat in each period for each process a random value X was taken from a normal distribution with the entered mean and variation coefficient as parameters: X = m*(a*b*v + 1), if m = the mean, v = variation coefficient, a =cos(2*rnd) (rnd = a random number from a homogeneous distribution between 0 and 1, b = V*(-2*ln(rnd)).
• Optionally, survival and dispersal could be made density-dependent; dispersal could also be made dependent on the occurrence of adverse conditions.
• Optionally, the variation in some process can be simulated as running synchronously between habitats (for example to simulate an overall influence of weather: if it was cold in one habitat it was more or less cold in all habitats).
• Optionally, the processes could be simulated as being interdependent.
• Optionally, the distribution of dispersing individuals over other habitats could be made dependent on the distance from the source habitat.
• Dispersing females were divided into those reproducing in the new habitat and those that have already reproduced in the old one. In versions with selection females were divided in females that were fertilized before emigration and females that were not. The program noted the genotype of the male that fertilized a dispersing female.
• The amount of eggs laid by an immigrated female was assumed to be influenced by the conditions in its new habitat but not by those in the previous one.
• To compare the different simulations the random numbers used in a given process in later versions of the model were the same for each sequence. In this way in simulations with different starting features and/or different conditions the population dynamics were considered as running through the same sequence of years with, for example, the same weather conditions at the same times.

3.3. Results
3.3.1. The effects of dispersal in relation to reproduction and survival
3.3.1.1. In many insect species no flight activities occur during reproduction (or occur only at a low intensity), see Johnson (1969: the oogenesis-flight syndrome), den Boer (1977) and van Huizen (1979)). Both flight activities and the production of eggs are considered to demand much energy and would thus generally exclude each other (e.g. Johnson 1969). On the other hand, it could be favourable for females to distribute their egg-clusters over a number of different patches (de Jong 1979). Gyrinus-females are able to combine flight activities with egg production, although they fly at a lower frequency than after the reproduction period (Chapter VI Dispersal by flight). Exchange between populations by swimming is higher during the reproduction period than afterwards (Chapter VII Dispersal by swimming).
In theoretical analyses of exchange between populations (e.g. Reddingius and den Boer 1970, de Jong 1979, Kuno 1981, Metz et al 1983, Klinkhamer et al 1984) the conclusions are restricted to annual species, or at least to species in which the individuals of consecutive generations are strictly separated in time. In par 3.1.7 we argued that dispersal will be the more important the lower the expectation of life of the individuals. Therefore, it might be expected that perennial species will show less dispersal than annual species. On the average 25 per cent of the spring generation of whirligig beetles is still alive when the summer generation emerges, but only a low percentage of the summer generation will be alive the next spring and able to reproduce a second time. It is nearly completely the autumn generation that reproduces in spring.
To trace the relationship between dispersal activities on the one hand and reproduction and survival on the other we can use the results of the simplest version of the simulation model, in which each year has only two periods, a reproductive season followed by a pre-reproductive season of the offspring. In this version two 'species' were simulated, one (M) with, the other (B) without dispersal activity. The different values of reproduction and survival per habitat and per period were, however, the same in both species, so that possible difference in the course of population size could only be due to the difference in dispersal activity. Though this simple model was inspired by the field study on Gyrinus marinus, it is not a model of the population dynamics of Gyrinus. None the less, the results obtained appear similar to those of the more complex models that better imitate the field situation of Gyrinus marinus (see below).
3.3.1.2. Some results of these simulations over 25 years are shown in Table VIII-4. The simulations distinguish between (1) a low (0.05) or (2) a high (0.50) survival chance in the reproductive season, and between situations with (a) a rather low (0.20) and (b) a rather high (0.50) survival chance in the second season. The mean value for reproduction is adjusted such that the expectation value of net reproduction for the B-population is R = 1 (no decrease or increase of population size expected). Low (about 3 %) and high (about 50 %) dispersal during and/or after the reproduction season is compared with the situation without dispersal. The survival from dispersal is 80 per cent (I/E = 0.8). In the area simulated 20 favourable habitats are available. Each habitat starts with 500 individuals of both species. The population per habitat is considered as a local population of the overall natural population from all habitats together. The situation after 25 years is given by the distribution of local population sizes over six logarithmic classes. The mean net reproduction of the overall population in the total area is given by R_g = (P_n/P₀^(1/n), Since, R = P₁/P₀*P₂/P₁*..*P_n/P_n-1 = P_n/P₀. The variation of R is the average of the variation coefficients of R-values over n (n<25) years. The range between which the total population size has fluctuated is given by the LogRange, LR = log(H) -log(L), if H and L are the highest and lowest total population size respectively (cf den Boer 1981).

3.3.1.3. The following average results were obtained from 10 runs per situation ( Table VIII-4):

Simulation (1a). Low survival chances in both seasons (no overlap of generations, annual species).

• Within 25 years species B, without dispersal, dies out in about 50 percent of the habitats. Most of the surviving populations contain less than 100 individuals.
• In all situations with dispersal the chance of extinction per local population is smaller than without dispersal, with almost no extinctions within 25 years. But the size of the dispersal effects depends distinctly on the amount and season of dispersal.
• Dispersal during reproduction has only little or no effect on population size and fluctuation range, but the chance of extinction is significantly diminished, even if only three per cent of the individuals show dispersal activities. With a high dispersal level population sizes decrease, but with less heavy fluctuations (Table VIII-4: v.c.R_g, but not LR). However, on the long term this will lead to extinction in all 20 habitats (the deterministic prediction from the average R_g<1 is that after 75 years only 3 individuals should still be present in the area).
• Dispersal that only occurs after reproduction, i.e. that is shown mainly by the offspring, has clear positive effects: Both the variation and the range of population size is reduced in comparison with the case without dispersal after reproduction, and also the mean population sizes after 25 years are higher. With a high dispersal level the populations even increase in size (Kuno-effect).
• With dispersal in the pre-reproductive season of the offspring, population sizes below 10 become rare, while population sizes above 1000 (i.e. at least twice the start value) occur more frequently. This can also be seen in Fig VIII-4, showing the fluctuations in the population size of all 20 habitats taken together from one simulation.
• A further stabilization and decrease of both variation and size-range is reached when dispersal occurs during the whole year, i.e. in both seasons. However, a high dispersal level in one season overrules a low dispersal level in the other season. We see that a low dispersal rate during reproduction together with a low or high dispersal rate out of the reproductive season gives the best stabilizing effects: R_g has most approached R = 1, both the variation and the range of population sizes are very low. This situation comes rather close to the field situation in many surface dwelling arthropods.
• In Fig VIII-4 is shown that with low dispersal rates the population sizes follow about the same course as without dispersal activity: downward trend during the first 15 years, then rapidly increasing again. Only with a high dispersal rate outside of the reproductive season is a rather stable pattern at a hardly changing level shown.

Simulation (1b). A low survival chance (0.05) during the reproductive season and a good survival chance (0.50) in the second season and during hibernation (annual species, no overlap of generations).

• Without dispersal the results are similar to those given in simulation 1a.
• Low dispersal activities lead to similar effects as in simulation 1a. However, a high dispersal level during reproduction gives good results now, whereas it has very negative effects if it occurs out of the reproductive period; in the latter case all or nearly all populations become extinct within 25 years.
• The deviating results of the high dispersal activity in both periods in this situation as compared with that of simulation 1a may be due to the already assumed relation between dispersal activity, with its own survival chance, and the survival chance in the populations. Appearently, the losses by dispersal in the second period are too high in relation to the survival chance of the emigrated individuals if they had stayed in the population.

Simulation (2a). A good survival (0.50) during the reproductive season and a lower one (0.20) in the second season of the year and during hibernation (some overlap of generations within a year, annual species).

• Without any dispersal activity a better result is obtained after 25 years than in the previous cases. The better survival during the reproductive season probably contributes to the stability of the numbers of individuals in the second season: if by chance reproduction is low in some year the surviving adults from the first period will somewhat level the fluctuations in the numbers of adults that start reproduction in the next year (on the average 10 per cent will reproduce a second time).
• Under these conditions of survival and reproduction, differences in the level of dispersal has less effect than in the cases 1a and 1b, though the occurrence of dispersal is in itself still favourable. It seems that the effect of a better survival from the first season is reinforced by dispersal in the second season. A high dispersal level can even result in overcrowding.

Simulation (2b). A good survival (0.50) in both seasons (25 % of the individuals reproduce a second time).

• Without any dispersal the situation equals that in simulation 2a.
• A high level dispersal apparently carries with it too heavy losses given the low reproductive rate, though the results are somewhat better than in case (1b). Nevertheless, the conditions of 2b in all cases allow the prediction of extinction of the species within 50 - 75 years.
• Low dispersal in the first season decreases the chance of a rapid extinction but hardly influences mean population size or its fluctuations or range.
• Low dispersal activities in both seasons or only in the second one entails the best survival possibilities for the species under the given circumstances.

3.3.1.4. Our general conclusion thus is that in most situations dispersal activities cause a decrease of the chance of extinction of the populations concerned, in spite of the losses during dispersal. Remarkable is the significant influence of dispersal by only a small fraction of the individuals, comparable to the dispersal activity by flight of whirligig beetles (cf Table VIII-1). The direct numerical effects of such a low dispersal rate are negligible, but the indirect effects of decreasing the chance of extinction and of increasing population size seem to be very important.
An important conclusion may also be that dispersal shown by the offspring out of the reproductive season has more effect than dispersal activity during reproduction. Obviously, this is due to the 'Kuno'-effect mentioned before, which works best if exchange between populations occurs before the reproductive period. The advantages of dispersal during reproduction mainly result from spreading the risks for the progeny. Therefore, it becomes comprehensible that some combinations of dispersal during both the first and second season appear to be most favourable.
3.3.1.5. To keep the simulations mutually comparable the mean values for survival and for reproduction were chosen such that the expected value of net reproduction in each case was R = 1 if there was no dispersal activity. This means, however, that because of different survival chances in the cases simulated, a fixed level of dispersal concerns different numbers of individuals. Because we wanted to adjust the mean values in the simulations above to different survival chances we had to change the numbers of tenerals per adult, to keep R = 1, by which the number of dispersing individuals in the second season is changed too.
3.3.1.6. The relation between dispersal and reproduction is rather complicated. During the reproduction season part of the reproducing females becomes redistributed over the habitats, which results in spreading of their egg batches over different habitats. This effect will be the more important the lower the number of reproducing females in comparison to the number of habitats (de Jong 1979). Another effect of exchange during the reproductive period is that differences in progeny between habitats may decrease to the extent that these are due to differences between individual females.

3.3.2. The importance of variability
In the above simulations a single variation coefficient (v.c. = 0.4) was used for all processes. This variation coefficient symbolized the spatial as well as the temporal variation, and thus all the variability in the simulated area. Therefore, in the 4 simulations of Table VIII-4 (1a - 2b) the influence of this level of variability was also tested for a lower (v.c. = 0.2) and for a higher (v.c. = 0.8) variation coefficient for both the cases of low dispersal during both seasons and that of no dispersal at all (Table VIII-5). The lower the variability the lower appears to be the chance of extinction but also the smaller the effects of dispersal. Moreover it appears that variability has less influence if the survival chances increase. The spatial and temporal variability will be treated separately later (C.3.4). These results confirm the assumed relationship between dispersal and variability, and also illustrate another form of spreading of risk, which den Boer (1968) and van Dijk (1973) mentioned, viz. spreading of the risk per individual over different reproduction seasons or generations. Note that in all cases with high variability (v.c. = 0.8) the B-populations die out rapidly (compare part 3.3.1)

3.3.3. Dispersal in relation to different spatial conditions.
3.3.3.1. Since the effects of dispersal depend both on the amount of exchange between the populations in a certain area and on the variation between populations, it is obvious that the number of habitats and populations will influence the effects of dispersal. The distances between two populations or habitats may also be important. Both aspects were studied by 10 simulations over 25 years. We again simulated a situation with low dispersal activity (1.5 % emigrated, of which the chance to immigrate elsewhere = 0.5), and with high dispersal activity (30 % emigrated, with a chance to immigrate somewhere else of 0.7). The results are summarized in Table VIII-6. See Table VIII-4 for explanation of the parameters and variables used.
3.3.3.2. As far as the chance of extinction, mean population size, and the ratio between the highest and the lowest population size are considered, exchange between a low number of habitats has less favourable effects on population sizes after 25 years than exchange between a higher number of habitats ((Table VIII-6). Dispersal at a high level gives better results in comparison with dispersal at a low level In the case of 5 habitats than in the simulations with 20 habitats.
3.3.3.3. Exchange between populations by swimming appears to be related to the distance between the habitats (Chapter VII Dispersal by swimming). If exchange between populations is made dependent on mutual distances, the simulations give a better result if dispersal is low than if it occurs at a high level; at low dispersal levels population sizes after 25 years are higher (R_r > 1) and the ratio between highest and lowest population size is smaller than at high dispersal levels, which moreover lead to R_r < 1 (see Table VIII-6).
3.3.3.4. Both Southwood and Den Boer (cf part 3.1.5-7) attribute considerable importance to variability in environmental factors in regard to dispersal activity. Variability can be considered in time (fluctuations at one site) and in space (differences between different sites at the same time). In 3.3.2. (Table VIII-5) the influence of different levels of variability in time was analysed. Whereas a lower variation in time gives both a decrease of the chance of extinction, of mean population sizes, and in the fluctuations and range of population sizes (compare Table VIII-6.e with VIII-6.a), a decrease in variation in space leads without any exception leads to extinction of the species in the entire area. The synchronisation between the habitats (Table VIII-6d) may have been too complete, however, so that in fact the habitats behaved as a single habitat. Nevertheless, the influence of somewhat less synchronisation is in the same direction, as is convincingly shown by den Boer (1981) by comparing two carabid beetles that react to a different degree to spatial heterogeneity. Decreasing heterogeneity in an area leads to unfavourable population dynamics because neither overcrowding nor extinction can be prevented by exchange between habitats.

3.3.4. The causes of dispersal activity.
3.3.4.1. Dispersal of whirligig beetles by swimming can be considered a consequence of the daily swimming activities connected with looking for food, etc. In Chapter VI arguments were given for the hypothesis that these beetles can only show flight activities when the weather is favourable for flight. In the above simulations we have shown that even the small exchange between populations as a result of flight activities may considerably decrease the chance of extinction of the populations.
This does not fit the current view (e.g. Johnson 1969) that emigration mainly occurs when conditions in the habitat become adverse, whether this be due to abiotic (e.g. drying up of wet habitats) or to biotic factors (e.g. overcrowding). If the beetles react to density with dispersal activities the exchange between populations would be density-dependent. In comparison with the above simulations in which dispersal activities occur randomly, dispersal activities would occur more contagiously distributed both over the habitats and over time. We tested by simulation whether dispersal activities that are more concentrated in space and in time (because they occur as a reaction to adverse conditions or to density) would have similar effects on the course of population size as random dispersal. These simulations are carried out according to the same procedures as above and the results are given in Table VIII-7.
3.3.4.2. Density-dependent dispersal entails the best reduction of variation in population size, and after 25 years population sizes mostly occur between 1000 and 10000. However, if dispersal were strictly density-dependent no dispersal would occur at low population sizes. This means that with population densities far below the carrying capacity a rapid extinction of all populations will occur (see also den Boer 1968, 1986). On the long term such a dispersal will result in a higher chance of extinction than if dispersal were to occur randomly.
3.3.4.3. To make dispersal a reaction to adverse conditions, a chance of 0.1, 0.2 or 0.5 respectively of the occurrence of adverse conditions is introduced into the model, together with a mean percentage of individuals that will emigrate when such conditions occur. In this way one case of low dispersal activities is obtained (on the average 1.5 percent of all individuals will emigrate from about 10 per cent of the populations per season), and two cases with high dispersal activities (almost all individuals emigrate from 20 per cent of the populations and about 50 per cent of the individuals emigrate from about 50 per cent of the populations per season).
With low dispersal activities extinctions occur, and the range of population sizes is not reduced. If nearly all individuals emigrate per season from 20 per cent of the habitats a general overcrowding is the result, i.e. the range of sizes is not at all reduced. If only 50 per cent emigrate from 50 per cent of the habitats the results are in between the cases in which a high dispersal level is either due to random or to density-dependent conditions. In general, it seems that dispersal due to adverse conditions will entail more instable situations than when dispersal is due to density-dependent reactions or when it occurs randomly.
The results of the above simulations show that there are no apriori disadvantages connected with the hypothesis of randomly occurring dispersal.

3.3.5. The role of dispersal by flight.
3.3.5.1. So far the simulations concern species of which the entire progeny emerges at the same moment and is thus present at the beginning of the next season. These simple models show some interesting consequences of dispersal activities, but what may we expect in the more complex situation described by our field studies?
3.3.5.2. From the field studies we know that flight activities of Gyrinus marinus occur at a low level of less than 5 per cent of the beetles, and that flight is restricted to days with good weather conditions. During the reproduction season females show less flight activity than males. The sexes also differ in chance of survival, which moreover decreases from April to October (Table V-1). The above simple simulations already show that a low dispersal activity decreases the chance of extinction of a population. We will now try to trace the population-dynamic effects of a low dispersal activity comparable with the flight activity of whirligig beetles. First of all males and females will be handled separately using the mean values found in the field. As the estimates of the mean numbers of tenerals per oviposition and those of survival from hibernation are the least reliable field estimates, these are adjusted such that in the model no overall increase or decrease of population sizes could be expected if no dispersal occured and all variation was neglected (deterministic net reproduction, R = 1). The length of a period of observation will be one month, so that a year will have eight periods: seven months from April to October and one winter period. A consequence of periods of one month is that the model has to follow the development from egg to teneral. Unfortunately, the RAM-memory of the personal computer used does not allow a more complicated structure in the model than simply simulating the number of tenerals per female in month j that developed from eggs in month i. Nevertheless, we think that we approached the reality of the field situation sufficiently closely with the separate monthly simulation of the processes of emigration and immigration, survival and production of tenerals, but also with the interactions between these processes. In Table VIII-8 a review is given of the mean values and variation coefficients used in this simulation model.
An important consequence of time steps of one month is, that the periods of reproduction and non-reproduction are no longer strictly separated. Egg-production occurs from April through August, but tenerals emerge from June through October. The results of the simulations with and without dispersal activity are shown in Fig VIII-5.
Without dispersal population size of all habitats together increases after 30 years to more than 8 times the starting size, which is caused by only one single, very large population. In the first 25 years the summed population size fluctuates between 10,000 and 60,000. However, after 40 years, only five of the 10 populations at the start are still present with sizes of 12, 112, 677, 10919 and 214394 (this last population had reached a peak of 557399 in the 39th year). The total number of individuals is thus mainly determined by a few populations; the others were small or rapidly went extinct (Fig VIII-5A).
With low dispersal activities the fluctuations in population size and the range of variation are relatively small. In the first 25 years the summed population size varies between 12000 and 4000. The individual populations fluctuate between 25 and 4908 individuals. After 40 years all 10 populations still exist. Instead of the trend toward increasing population sizes - as found in the case without dispersal - a slow decrease in population size appears (Fig VIII-5B). It is possible that the values for reproduction (tenerals/female) or survival during hibernation should be somewhat better than we assumed (as noted, these estimates were not as reliable as they should be (Chapter IV Reproduction and Chapter V Survival of adults).
3.3.5.3. This simulation with a more complicated model confirms the general conclusions we reached above with simple models, which indicates that these general conclusions are also valid in the specific case of Gyrinus marinus. Dispersal by flight of these beetles has a stabilizing effect upon population size, decreasing both the fluctuations of numbers and the chance of extinction. When discussing the field data about flight, we concluded that flight activities could not be of importance because we focussed only upon the absolute numbers of individuals that show flight activities (cf Chapter VI Dispersal by flight). We must now correct this earlier conclusion since it appears that even minor flight activities highly favour the permanent occurrence of Gyrinus in an area with separate populations that cannot exchange by swimming.

3.3.6. Selection of dispersal features.
3.3.6.1. Given the survival chances and reproduction values, numbers of habitats and survival chances during dispersal, as found in the field, there is an optimal level of dispersal. We consider dispersal to be optimal when the progeny (possibly after some generations) of the dispersing individuals is about equal to or greater than the progeny of the not-dispersing individuals. This implies that also random dispersal is subject to selection and will change to an optimal level. The chance of survival from dispersal activity (i.e. the net exchange between populations, or the (re)founding of populations) is crucial in this context. It is useless to consider emigration from a single population if arrival in other suitable habitats is not taken into account. Carlquist (1966), for example, shows that seeds of species of Compositae plants lose their ability to spread over long distances if no settlements occur (island situations). Similar data are given by Darlington (1943) for carabid beetles in Alpine mountains and on islands.
3.3.6.2. Flight activity of Gyrinus marinus is apparently pure dispersal activity with no other apparent function than leaving the present site. Flight ability of carabid beetles is related with wing size and development of flight muscles (den Boer et al 1980, van Huizen 1979), so let us assume for convenience that the wing-size of whirligig beetles determines the occurrence of flight activity by an individual beetle (but you may imagine another feature if you like). Assume further that wing size depends on a single recessive gene. The homozygotic genotype that can fly will be called macropterous (MM-type), the homozygotic genotype that cannot fly brachypterous (BB-type). Only the MM-type is supposed to be able to fly (the feature to fly is recessive). The choice of recessiveness is not only made for computing reasons, but also because there are reliable indications that macroptery is often recessive, at least in a number of beetle species (Jackson 1928, Lindroth 1946, Stein 1973, den Boer et al. 1980, Aukema 1986).
3.3.6.3. The genetic situation described above is built into the complicated simulation model. For emigrating females we note whether they were fertilized and if so, by what genotype of male.
The results of the simulations are shown in Fig VIII-6 and Fig VIII-7. Without dispersal activity (Fig VIII-6) there is no selection at all and the frequencies of BM, BB and MM conform the Mendelean distribution segregation (2:1:1). When dispersal activities occur the relative frequencies of the genotypes change. The BB-type makes up an increasing fraction of the population. From Fig VIII-6 it appears that the MM-type is more stable than the other two types and that the MM-type numbers are much lower than those of BB and BM. With increasing population size the relative frequency of the MM-type decreases and vice-versa.
3.3.6.4. Dispersal activity is shown in this model by only one of the three genotypes so that dispersal activity of the population as a whole is lower than in previous simulations with only one genotype. Nevertheless, the same effects of dispersal are found, see Fig VIII-7. Such limited dispersal activity cannot sufficiently eliminate the exceptional growth of one of the ten populations, although it is more restricted than in the case without dispersal.
3.3.6.5. Since we assumed that dispersal activity is a recessive feature the heterozygotic individuals preserve the feature, without being able to express it. When dispersal occurs a part of the genes responsible for dispersal activity will leave the population with the emigrating individuals. Without compensating immigration the relative frequency of these genes will decrease (cf also Carlquist 1966). However, since the feature is also hidden in the not-dispersing heterozygotic individuals, the 'dispersal'-genes can hardly be completely eliminated from the population, because no direct selection occurs on these 'hidden' genes, and there are no special reasons to suppose that indirectly there will occur some positive or negative selection of heterozygotes (e.g. via pleiotropic effects). The heterozygotic genotype will thus function as a buffer for the feature 'dispersal activity'.
3.3.6.6. In the cases described above only dispersal by flight is considered, exchange between populations by swimming being considered to be a consequence of the daily swimming activities. In fact, the situation is more complicated. In the field we generally find clusters of populations. Between the populations of the same cluster exchange by swimming frequently occurs; properly speaking, we delt with subpopulations then. Between these clusters of subpopulations only little exchange by flight is possible. In the first simulations above (Table VIII-4) it has been shown that a high level of exchange will have impressive stabilizing effects upon the subpopulation sizes. We may thus assume that each cluster of a number of subpopulations is already stabilized because of exchange by swimming, and that the complex of clusters in the whole area will be stabilized by the flight activities. When the simulation program can be run on a more powerful computer this situation of clusters of subpopulations can be built into the model and tested.

4. SOME FINAL CONCLUSIONS.

The simulations lead to the following conclusions and hypotheses:
• According to the key-factor analysis population size and its fluctuations are principally determined by survival from hibernation. None of the other processes or stages - either alone or in combination - seem to be able to compensate the high and variable losses from hibernation.
• However, by spreading the risk over a number of habitats it appears to be possible to counterbalance the losses from hibernation in a particular habitat by the differing fluctuations in numbers in other habitats, which are actualized by means of exchange of individuals between the habitats.
• Dispersal, even when it occurs at a low frequency, appears to be crucial for the survival of a species in a group of different habitats. Low frequency dispersal may suffice to prevent extinction of populations and/or lead to refounding at sites where extinction has taken place. Emigration from and immigration into habitats has a strong stabilizing influence on population size, thus decreasing the chance of extinction of the populations considered.
• If dispersal activities are connected with only one recessive genotype the numbers of the 'non-dispersing' genotypes may vary considerably, whereas the fluctuations in numbers of the 'dispersing' genotype are much smaller. The 'dispersing' genotype decreases the chance of extinction for the entire population, also for the heterozygote and homozygote fractions of the population. The relative frequency of the 'dispersing' genotype may become very low as a result of dispersal activities, but the chance that it disappears altogether is small because of the buffering effect of the heterozygote.
• The influence of dispersal activities upon the course of population size depends on the difference between the rates of emigration and immigration, i.e. on survival (s) during dispersal, and on the survival chance (Q) in the habitats. Also of importance will be the variability (v) of the processes that are responsible for the spatial and temporal fluctuations in population size, the number (n) of habitats in the area, and the distances (d) between these habitats. As we saw, it also matters whether dispersal occurs during or out of reproduction (r). Therefore, it may be expected that natural selection will stabilize the level of dispersal (e) of a species in a certain area such that the balance between the risks and the advantages of dispersal activity will be about optimal according to the above definition. These relations can be expressed by the formula:

  	Fs•Fv•Fn
  Fe =	––––––––
  	FQ•Fd•Fr

  where each F_x, if x = s,v,..r, respectively, stands for some function between dispersal activity and the parameter concerned.
• For the field populations of Gyrinus marinus studied the hypothesis cannot be rejected that the flight activities of some of the beetles depend only on favourable weather conditions, even when local conditions are favourable for survival.
• The simulations confirm both the hypothesis of 'spreading of risk' of den Boer (1968, also Reddingius and den Boer 1970) and the effects noticed by Kuno (1981). The simulations make clear that 'Kuno' effects will only occur when offspring migrate between clusters or sites, and much less or not at all when reproducing females change sites. The principal effect of spreading the risk is the significant decrease of the chance of extinction of populations.

REFERENCES
Aukema B (1986) Wing length determing in relation to dispersal by flight into wing dimorphic species of the genus Calathus (Bonelli), Coll., Carabidae. In: Carabid Beetles, their adaptations and dynamics. Ed PJ den Boer, MlL. Luff, D. Mossakowski, F. Weber. Gustaf Fisher, Stuttgart.
Boer PJ den (1968) Spreading of risk and stabilization of animal numbers. Acta Biotheor. (Leiden) 18:165-194
Boer PJ den (1977) Dispersal power and survival. Carabids in a cultivated countryside. Miscell Papers LH Wageningen 14:1-190
Boer PJ den (1981) On the survival of populations in a heterogeneous and variable environment. Oecologia (Berlin) 50:39-53
Boer PJ den (1985) Fluctuations of density and survival of carabid populations. Oecologia (Berlin) 67:322-330
Boer PJ den (1986) Density dependence and the stabilization of animal numbers. 1. The wintermoth. Oecologia (Berlin) 69:507-512
Boer PJ den, van Huizen THP, Boer-Daanje W, Aukema B, den Bieman CFM (1980) Wing Polymorphism in Ground Beetles as Stages in an Evolutionary Process (Coleoptera: Carabidae). Entom Gen. 6 (2/4): 107-134
Carlquist S (1966) The biota of long-distance dispersal. II loss of dispersibility in pacific Compositae. Evolution 20:30-48
DarlingtonPJ (1943) Carabidae of mountains and islands. Ecol. Monogr. 13:37-61
Dijk ThS van (1972) The significance of the diversity in age composition of Calathus melanocephalus L. (Col., Carabidae) in space and time at Schiermonnikoog. Oecologia (Berlin) 10:111-136
Eijk RH van der (1983) Population dynamics of gyrinid beetles I. Flight activity of Gyrinus marinus Gyll. Oecologia (Berlin) 57: 55-64 (see Chapter VI Dispersal by flight)
Eijk RH van der (1984) De invloed van vliegdispersie bij schrijvertjes op het aantalsverloop. Jaarboek 1983 Ned Entomol Ver (In dutch)
Eijk RH van der (1986a) Population dynamics of gyrinid beetles IV. Reproduction and recruitment. Oecologia (Berlin) 69:31-40 (see Chapter IV Reproduction)
Eijk RH van der (1986b) Population dynamics of gyrinid beetles III. Survival of adults. Oecologia (Berlin) 69:41-46 (see Chapter V Survival)
Eijk RH van der (1987) Population dynamics of gyrinid beetles IV. Dispersal by swimming. See Chapter VII Dispersal by swimming
Huizen T van (1979) Individual and environmental factors determining flight in carabid beetles. Miscell papers LH Wageningen 18:199-211
Istock C (1966) Distribution, coexistence and competition in whirligig beetles. Evolution 20:211-234
Istock C (1967) Transient competitive displacement in natural populations of whirligig beetles. Ecology 48 (6):929-937
Jackson DJ (1928) The inherance of long and short wings in the weevil Sitona hispdula; with a discussion of wing reduction among beetles. Trans Roy Phys Soc 55:665-735
Johnson CG (1969) Migration and dispersal of insects by flight. Methuen London, p 763
Jong G de (1979) The influence of the distribution of juveniles over patches of food on the dynamics of a population. Neth J Zool 29 (1): 33-51
Klinkhamer PGl, de Jong TJ, Metz JAJ (1984) An explanation for low dispersal rates: a simulation experiment. Neth J Zool 33(4):532-541
Kuno E (1981) Dispersal and the persistence of populations in unstable habitats: a theoretical note. Oecologia (Berlin) 49:123-126
Lindroth CH (1946) Inheritance of wing dimorphism in Pterostichus an-thracius Ill. Hereditas 32:37-40
Metz JAJ, de Jong TJ, Klinkhamer PGL (1983) What are the advantages of dispersing; a paper by Kuno explained and extended. Oecologia (Berlin) 57:166-169
Morris RF (ed) (1963) The dynamics of epidemic spruce budworm populations. Mem Entom Soc Can 31
Podoler H, Rogers D (1975) A new method for the identification of key factors from life-table data. J Anim Ecol 44:85-114
Reddingius J, den Boer PJ (1970) Simulation experiments illustrating stabilization of animal numbers by spreading of risk. Oecologia 15:245-258
Sachs L (1982) Applied Statistics. A handbook of Technics. Springer New York
Southwood TRE (1962) Migration of terrestrial arthropods in relation to habitat. Biol Rev 37:171-214
Southwood TRE (1978) Ecological methods (2nd ed). Chapman and Hall, London
Stein W (1973) Zur Vererberung des Flugeldimorphismus bei Apion virens (Herbst) (Coll., Curculionidae). Z Ang Ent 74:62-63
Varley GC, Gradwell GR (1960) Key factors in population studies. J Anim Ecol 29:399-401