Apparent negative density-dependent dispersal in tsetse (Glossina spp) is an artefact of inappropriate analysis

Analysis of genetic material from field-collected tsetse (Glossina spp) in ten study areas has been used to predict that the distance (δ) dispersed per generation increases as effective population densities (De) decrease, displaying negative density dependent dispersal (NDDD). This result is an artefact arising primarily from errors in estimates of S, the area occupied by a subpopulation, and thereby in De, the effective subpopulation density. The fundamental, dangerously misleading, error lies in the assumption that S can be estimated as the area (Ŝ) regarded as being covered by traps. Errors in the estimates of δ are magnified because variation in estimates of S is greater than for all other variables measured, and accounts for the greatest proportion of variation in δ. The errors result in anomalously high correlations between δ and S, and the appearance of NDDD, with a slope of −0.5 for the regressions of log(δ) on log(e), even in simulations where dispersal has been set as density independent. A complementary mathematical analysis confirms these findings. Improved error estimates for the crucial parameter b, the rate of increase in genetic distance with increasing geographic separation, suggest that three of the study areas should have been excluded because b is not significantly greater than zero. Errors in census population estimates result from a fundamental misunderstanding of the relationship between trap placement and expected tsetse catch. These errors are exacerbated through failure to adjust for variations in trapping intensity, trap performance, and in capture probabilities between geographical situations and between tsetse species. Claims of support in the literature for NDDD are spurious. There is no suggested explanation for how NDDD might have evolved. We reject the NDDD hypothesis and caution that the idea should not be allowed to influence policy on tsetse and trypanosomiasis control. Author summary Genetic analysis of field-sampled tsetse (Glossina spp) has been used to suggest that, as tsetse population densities decrease, rates of dispersal increase – displaying negative density dependent dispersal (NDDD). It is further suggested that NDDD might apply to all tsetse species and that, consequently, tsetse control operations might unleash enhanced invasion of areas cleared of tsetse, prejudicing the long-term success of control campaigns. We demonstrate that NDDD in tsetse is an artefact consequent on multiple errors of analysis and interpretation. The most serious of these errors stems from a fundamental misunderstanding of the way in which traps sample tsetse, resulting in huge errors in estimates of the areas sampled by the traps, and occupied by the subpopulations being sampled. Errors in census population estimates are made worse through failure to adjust for variations in trapping intensity, trap performance, and in capture probabilities between geographical situations, and between tsetse species. The errors result in the appearance of NDDD, even in modelling situations where rates of dispersal are expressly assumed independent of population density. We reject the NDDD hypothesis and caution that the idea should not be allowed to influence policy on tsetse and trypanosomiasis control.

The central importance of S in accounting for variation in the dispersal rate () leads us 166 immediately to the main problem with the DM development, embodied in their statement (2): "The 167 average surface (S) occupied by a subpopulation can be computed as the surface area occupied by 168 the different traps used in a given survey site. When only one trap was available per site or when the 169 GPS coordinates of corresponding traps (one subsample) were not available, we computed S = 170 (D min /2) 2 where D min is the distance between the two closest sites taken as the distance between the 171 centers of two neighboring subpopulations".

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This statement is untrue and dangerously misleading. The true area (S) covered by a 173 biological subpopulation bears no relation to the area ( ) estimated to be covered by a set of traps.
174 Assuming a subpopulation is well-mixed with random mating and no intra-population genetic 175 structure (as is assumed by all methods for estimating N e ), then N e can be estimated by catching 176 flies from any area smaller than the geographic range of that subpopulation. If 3 is used (covering an area larger than the size of what can be considered a well-mixed 187 subpopulation), then the assumptions underlying the estimates of e will be violated, and the 188 estimates of e will be flawed, so introducing yet more error to e . The extent to which these 189 erroneous estimates of e will scale with -as it grows above the size of S -is not entirely clear, 190 but previous work suggests that it will not scale linearly (21) and will thus continue to produce 10 217 most distant traps in a given site, taken as the radius of the corresponding subpopulation.  (Fig 3). The other 12 sites did not provide finite estimates of e . Similarly, they 240 estimated = 0.0202 using information on genetic and geographic distances between all available 241 sites. Using the above estimates for , and e , they calculated  = 27 metres per generation, the 242 lowest among all of the 10 studies cited by DM, and the one where the estimated effective 11 243 population density, e , was the second highest. We now show that these estimates are also artefacts 244 of the way in which the traps were deployed, and the way that DM chose to analyse the data. 274 strongly correlated with log( e ), with a slope around -0.5 (Fig 1, A1, B1); log(  ) is strongly 275 correlated with log( ), with a slope around 0.5 (Fig 1, A2, B2); log( ) and log( e ) are poorly 276 correlated (Fig 1, A3, B3); log( ) and log( e ) are poorly correlated (Fig 1, A4, B4); log( e ) and 277 log( e ) are positively, but rather weakly, correlated (Fig 1, A5, B5); log( ) declines linearly with 278 increasing log( e ) (Fig 1, A6, B6). The reader is invited to make serial iterations of the stochastic 279 procedure -with each iteration using a different randomly generated error for log( ) and log( e ).

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Notice that we make no assumption about the true underlying nature of dispersal: it could be 281 NDDD, DID (density independent dispersal) or PDDD (positive density-dependent dispersal). 305 For this, e is calculated as true D e multiplied by some random factor between 0.2 and 5; these 306 errors can be made additive instead, without consequence to the conclusion. For simplicity, we 307 assume that is estimated without error ( = b). When there is error in D e , however, log(b) appears 308 uncorrelated with log(D e ) (Fig 4B). We then calculate  using Equation (1), replicating the method 309 of DM. As the error input increases from 1 (no error) to 3 (three-fold error), there emerges a 310 negative correlation between log( e ) and log(  ), with a slope tending towards -0.5 (Fig 4C). This 311 approximates the slope apparent in the DM analysis of their own real data. The reader can vary the input values of Supplementary File S3 to observe the consequences 319 for the slope of log(  ) against log( e ). Since the simulation is stochastic, the slope changes with 320 each realisation of the process but, for fold-error greater than around 1.2, the slope is invariably less 14 321 than zero. That is to say, the population appears to exhibit NDDD, despite the fact that it has been 322 set up such that dispersal is actually independent of population density. Furthermore, the 323 dominating influence of large errors in e caused by highly variable and arbitrary values of , also 324 explain the otherwise perplexing correlation we have noted between and  in DM's data. If r is increased to 3 m, such that  100 m 2 = 10 -4 km 2 , there will be less interference 392 between the traps, but we expect that there will still be some interference and we thus expect c < 393 10 flies per trap per day, and the expected catch from all six traps will thus be 10 < c < 60 flies.

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If r is increased to 100 m, as in (11)