Ever since Karl Müller published his ‘colonization cycle’ hypothesis over 30 years ago (Müller 1954, 1982) there has been considerable debate amongst freshwater ecologists, and in the ecological literature in general, regarding what has now come to be known as the ‘drift paradox’ (Hershey, Pastor, Peterson & Kling 1993). Simply stated, the drift paradox arises because the upper reaches of streams remain populated by aquatic insects in the face of an apparently considerable reduction in their numbers. This reduction is due to the near-universal tendency of aquatic invertebrates to drift downstream with the current (Müller 1974; Brittain & Eikeland 1988; Allan 1995).
Recently, stable isotope (Hershey et al. 1993) and genetic (Schmidt, Hughes & Bunn 1995; Bunn & Hughes 1997) analyses have increased our knowledge of dispersal of the aquatic and adult stages of the macroinvertebrates that would seem to be most affected by drift. Some have argued that drift is simply production in excess of carrying capacity and, as such, depletion will slow as the population approaches this level (Waters 1961). However, the consensus has generally been that upstream flight (whether directed or as part of random, undirected dispersal) by some preovipositing adult females may be the key factor that resolves the paradox (Hershey et al. 1993; Allan 1995). Using an individual-based simulation approach, Anholt (1995) recently argued that upstream flight is not in itself sufficient, unless it is coupled with density dependence acting to reduce individual fitness at high local population densities. These works may resolve one paradox, but only at the cost of generating another: how do we explain the persistence of the many species that are commonly found to drift but do not have an aerial adult stage: e.g. the Turbellaria, Amphipoda and Isopoda? Here we show that reinterpretation of the results of Anholt’s model allows complementary solution of both of these paradoxes.
Anholt considered a hypothetical stream divided into 100 identical reaches, each of which begins a simulation with 10 individuals of the species under consideration. We concern ourselves only with the uppermost of these reaches, but otherwise our model follows exactly the structure presented by Anholt (1995), except for the omission of upstream movement. As we explain below, this modification is key to the correct interpretation of the model results. We use an individual-based stochastic simulation approach using discrete time-steps to describe the dynamics of the population. The model cycles through successive generations of animals, each generation being subject to drift dispersal, oviposition and density-dependent population regulation, in that order. At the start of each generation, every individual in a given reach drifts downstream and out of its original reach with independent probability 0·5. Since there is no upstream flight in our model, oviposition is only due to the stochastically calculated number of individuals remaining in the reach. The population is then changed by the integer nearest λ(10 – N), where 0 < λ ≤ 1 is the (constant) strength of density dependence, and N is the local population size in the reach after the effect of drift. This change is thus the expression of density dependence: population sizes below 10 increase, while those above 10 decrease.
Hence, Anholt describes 10 as the carrying capacity (K) of a reach in his model. He explains the mechanism by which the uppermost reach becomes empty thus: ‘… there is a finite possibility of every animal dispersing out of a reach in a single generation. When populations are relatively small, as in the simulations, this is non-negligible … Once the uppermost reach is empty, it remains empty because there is no drift in from upper reaches and no upstream movement of any kind’ (Anholt 1995). After this happens, the next highest reach becomes more vulnerable, as it no longer receives an influx of drifting organisms. In time it, too, will reach extinction. This pattern will continue until the population has been swept out of the whole system. Anholt is correct: in such a system, eventual extinction is inevitable because there is always the finite chance that all the animals in the highest reach drift simultaneously. However, as hinted by the quotation above, this probability is strongly dependent on population size, and for this reason we do not follow Anholt in using population half-life as a measure of extinction likelihood, but instead run our simulations for the full duration of a population’s existence (within computational limits detailed in the legend to Table 1). We performed a number of simulations, assuming that there was no movement of adults or larvae other than drift in the downstream direction. In Table 1, we show the expected number of generations until extinction of the highest reach in relation to the carrying capacity of that reach. We illustrate our results with an example generated setting λ to 0·25, the minimum considered by Anholt, but our results are qualitatively similar for other values of λ. The number of generations to extinction in the first reach increases very quickly with increasing carrying capacity. Indeed, for carrying capacities above 40, the mean number of generations to extinction of the uppermost reach is greater than 15 000: for annually reproducing species, this extends beyond the last glaciation. Even a bivoltine species, which would be expected to have experienced approximately 30 000 generations since the last ice-age, could manage to remain extant in the top reach for this period with a carrying capacity of less than 50 individuals. Hence, although Anholt’s model is technically correct, if the reach carrying capacities are above 40, our results suggest that extinction of a species from upstream reaches due to an absence of upstream movements may only happen over geological timescales (> 1000 years, Richards 1982).
|Reach carrying capacity (K)||Mean generations to extinction (mean ± SD)|
|10||15 ± 13·31|
|20||135 ± 126·90|
|30||1 739 ± 1 744·55|
|40||25 736 ± 20 790·69|
|50*||88 957 ± 24 315·56|
If we consider that a local population is defined by interactions such that members of this population interact strongly with each other, but not with members of other populations, we can see that the spatial scale of movement will have a strong influence on the size and density of a local population. Anholt thus suggests that a suitable scale for such a local population is that of the reach, but because he examined population persistence in terms of population half-life he implicitly assumed that it is the relative rate of depletion, not the absolute numbers in the population that are important. Using Table 1, we can see that a local population of 10 individuals is likely to go extinct within around 15 generations, we therefore suggest that, rather than pertaining to a whole reach, a carrying capacity of 10 drift organisms should apply to a much smaller stream section. Although carrying capacity itself is very difficult to measure, we feel that it is unlikely that any given reach will have a carrying capacity as low as 40 individuals, except perhaps for invertebrate predators such as stonefly or dragonfly larvae, which do not tend to be prevalent in the drift. Instead, we suggest that Anholt’s model should be re-interpreted as applying on a different spatial scale. Rather than pertaining to a whole reach, a carrying capacity of 10 drift organisms should apply to a much smaller stream section. Using a conservative population density of 250 individuals per m2 for a commonly drifting species such as the mayfly Baetis rhodani or the amphipod Gammarus pulex, and a stream width of 1 m, we suggest a more suitable spatial scale for the original model would be around 20 cm of streambed. That is, a 20-cm length of streambed would hold a local population of around 50, a value that is over twice the reach carrying capacity considered by Anholt. From Table 1, if there is no upstream movement between neighbouring 20 cm sections we would expect that the local population of 50 animals in the top 20 cm of the system will persist for thousands of generations.
However, we can now see that defining local populations with carrying capacities of 50 results in the strong assumption of no upstream movement on a scale of only 20 cm. This seems hard to justify in the face of ample evidence of upstream movement by aquatic stages of many drifting species (for reviews see Söderström 1987; Williams & Williams 1993). For example, estimates for distances of upstream movements in Gammarus pulex (which has no aerial dispersal stage) can be up to 14 m per day (Elliott 1971b). Similarly, larvae of Baetis rhodani, a common drifter with an aerial adult phase, have been estimated to travel up to 5 m per day upstream (Elliott 1971b). This is the key finding of our study: such within-stream movement could recolonize upstream sites easily, halting the downstream cascade of extinction described previously. Hence, the second paradox also disappears: long-distance flight by aerial adults is not required to prevent extinction of upstream reaches; it can be achieved with very small movement of individuals along the substrate.
For commonality with Anholt (1995) we used the following combination of parameter values, setting the probability of drift to P = 0·5 and carrying capacity to K = 50. However, since we argue that a carrying capacity of 50 individuals translates into approximately 20 cm of our hypothetical stream, this parameter combination could be argued to be a poor representation of the drift behaviour of a taxon such as Gammarus; setting P equal to 0·5 suggests that 50% of individuals do not drift out of their natal ‘reach’ (in this case 20 cm) within their lifetime. Consider an increase in the carrying capacity by a factor of 50, thus representing 10 m of streambed (K = 2500), a length of stream characteristic of both the spatial scale of individual drift events (Elliott 1971a) and daily benthic movement (Elliott 1971b). The appropriate value of P is difficult to estimate and, indeed, the use of a geometric distribution may not be the most appropriate in all situations. However, even if P is as high as 0·98 (98% of individuals leave their natal reach during the time between birth and their first reproductive episode), then the number of generations needed before extinction of the upper reach is still very high. In a series of 100 simulations (λ = 0·25) of the top 10 m of our hypothetical stream (i.e. with K = 2500), the minimum time to extinction for any population was above our computational ceiling of 100 000 generations. Mean time to extinction (as illustrated in Table 1) will be considerably higher than this minimum.
Anholt’s model has recently been used by Kopp, Jeschke & Gabriel (2001) in their theoretical treatment of the drift paradox. More importantly, however, Anholt’s (1995) scaling assumptions and emphasis on aerial dispersal have been perpetuated by Kopp et al.’s formalization of drift compensation ideas using invasion analysis (Kopp et al. 2001). Kopp et al. use stochastic simulations, as we have used here, but despite agreeing that time to extinction is a better measure of persistence than population half-life, they only consider reaches each with a carrying capacity of between 10 and 20 individuals. Using our scaling arguments above, these equate to streambed distances of less than 10 centimetres.
As a result of our findings, we argue that interpretations of the importance of different types of dispersal in lotic systems should be reconsidered, and that concerns regarding the amount of drift and the distances travelled may be largely irrelevant when considering the drift paradox. Instead, relatively small upstream movements, in combination with density dependence, greatly influence how quickly a streambed will become denuded of animals. Thus, although drift may well be an important determinant of benthic population dynamics, it can also now be viewed less as a problem for population stability, and more of a concern to individual fitness – bringing our understanding of drift more into line with current evolutionary theories.
A number of further points must be made. Like Anholt, we do not require that movement is biased in an upstream direction, only that some movement occurs in this direction. Again, like Anholt, our arguments above are dependent on there being density dependence such that there is reduced individual fitness at high population densities. Without density dependence, drift without (biased) upstream movement, would inevitably lead to much greater population densities at downstream sites than at upstream ones. However, with density dependence acting on our population, our arguments above still hold even if drift tends to move individuals hundreds of metres downstream, whereas upstream movements are only of the order of centimetres. The tendency of drift to move the population downstream will be counteracted by a higher individual fitness experienced in the lower population densities upstream.
Critically, our simulations indicate that random instream movements (coupled with a realistic consideration of spatial scale) are capable of retarding depletion due to drift such that populations may persist over geological timescales, and thus perhaps longer than the ‘life’ of their habitat. In a recent study, Speirs & Gurney (2001), using a quite different modelling approach, conclude that a balance between diffusive (random) movements and advection (drift) means that populations are able to persist indefinitely. In practice, it matters little whether the populations could exist indefinitely in an unchanging habitat (as Speirs & Gurney predict) or whether depletion is so slow that it only has a quantifiable effect over thousands of years (as we predict here). Hence, there is strong agreement between Speirs & Gurney and ourselves: upstream flight is not required to explain persistence of drifting organisms – small-scale, random undirected movements on the streambed are enough.
It should be noted that Speirs & Gurney (2001) take issue with the form of density dependence used in Anolt’s model. It was necessary for our study that we kept our model structure identical to that of Anolt, in order to allow a fair comparison. However, we agree with Speirs & Gurney that there has been an over-emphasis on dispersal mechanisms and a corresponding lack of consideration of methods of population regulation. To advance this theory further, the spatial scale of density dependence must be examined more closely; we defined this in terms of the ‘local population density’. The extent of this local population needs to be quantified, and will depend on the movement patterns of individuals. Hence, we hope that this paper will provide further impetus towards greater study of within-stream movements other than drift.
In one sense we agree with Anholt (1995) that the drift paradox can be resolved by a combination of upstream movement and density dependence. However, we have demonstrated above that the spatial scale of upstream movement must be considered in greater detail. Indeed, we show here that small movements along the substrate of the order of a few centimetres are all that is required, rather than flight over a scale of kilometres, as has previously been thought. In short, random movements, most probably related only indirectly to downstream drift, are able to compensate for large-scale advection of animals via water flow. This in turn resolves the ‘second drift paradox’, as it provides a means of drift compensation for animals with no aerial adult phase.