The primary aim of the work reported in this paper is to elucidate the relationship between discrete and continuous deterministic representations of spatial population processes. Our experimental vehicle is a spatially explicit version of the Rosenzweig‐McArthur model with immobile prey and a diffusively dispersing predator. We find that careful formulation of the discrete representation leads to essentially complete behavioral concordance between the two representations. We examine the invasions that follow localized introduction of predators into such a system and show that the biological realism of the model predictions can be greatly enhanced by preventing in situ regrowth of predator populations from densities that should be interpreted as representing local extinction. We exploit the close concordance of behavior between continuous and discrete representations by using the discrete version to perform a wide range of numerical experiments on one‐dimensional and two‐dimensional systems, while turning to the continous version to provide approximate analytic results for the natural time and space scales of the predicted population patterns. We conclude by discussing the implications of our findings for the experimental and theoretical study of spatial population dynamics.
In this paper we report the development of a highly efficient numerical method for determining the principal characteristics (velocity, leading edge width, and peak height) of spatial invasions or epidemics described by deterministic one-dimensiohal reaction-diffusion models whose dynamics include a threshold or Allee effect. We prove that this methodology produces the correct results for single-component models which are generalizations of the Fisher model, and then demonstrate by numerical experimentation that analogous methods work for a wide class of epidemic and invasion models including the S-I and S-E-I epidemic models and the Rosenzweig-McArthur predator-prey model. As examplary application of this approach we consider the atto-fox effect in the classic reaction-diffusion model of rabies in the European fox population and show that the appropriate threshold for this model is within an order of magnitude of the peak disease incidence and thus has potentially significant effects on epidemic properties. We then make a careful re-parameterisation of the model and show that the velocities calculated with realistic thresholds differ surprisingly little from those calculated from threshold-free models. We conclude that an appropriately thresholded reaction-diffusion model provides a robust representation of the initial epidemic wave and thus provides a sound basis on which to begin a properly mechanistic modelling enterprise aimed at understanding the long-term persistence of the disease.
Summary 1.The infestation of sheep with blowfly larvae (sheep strike) is a significant animal welfare and economic problem in many regions of the world. Improved control requires greater knowledge of the population dynamics of the primary agent of strike in temperate areas, the blowfly Lucilia sericata (Diptera: Calliphoridae). 2. The abundance of L. sericata was recorded on two farms in south-west England between 1990 and 2000, and the time-series analysed to describe and explain the nature of population change within and between years. 3. To examine the catch time-series for periodicity, the data were detrended, demeaned and tapered, and plotted against a day-degree time base. Spectral analysis of periodograms showed that the clearest signal present was the low-frequency seasonal cycle. The only other significant signal in both series had a periodicity corresponding to the day-degree requirements for the entire life cycle of this insect species. This suggests that, within each season, the abundance pattern is composed of a series of semi-discrete generation peaks. 4. Significant density-dependence was detected in the seasonal change in fly abundance and this was shown to be approximately compensatory in action. 5.The results suggest that populations of L. sericata show relatively stable and predictable dynamics, with populations passing through three or four relatively discrete generations each year prior to diapause and limited by strong, apparently compensatory, density-dependence each season. 6. The results have important implications for the control of this insect parasite. In general, earlier and more intense farmer intervention, to reduce sheep susceptibility and treat struck animals during the blowfly season, would result in a lower L. sericata population and reduced strike incidence. However, at the start of the season, when fly abundance may be lower than the number of susceptible hosts, direct fly control, in addition to treatment, may be a more effective strategy.
Abstract. A comprehensive simulation model for sheep blowfly strike due to Lucilia sericata (Meigen) (Diptera: Calliphoridae), which builds on previously published versions but also incorporates important new empirical data, is used to explain patterns of lamb and ewe strike recorded on 370 farms in south-west, south-east and central England and Wales. The model is able to explain a significant percentage of the variance in lamb strike incidence in all four regions, and ewe strike in three of the four regions. The model is able to predict the start of seasonal blowfly strike within one week in three of the four regions for both ewes and lambs, and within 3 weeks in the fourth region. It is concluded that the accuracy of the model will allow it to be used to assess the likely efficacy of new control techniques and the effects of changes in existing husbandry practices on strike incidence. The model could also be used to give sheep farmers advance warning of approaching strike problems. However, the ability to forecast future strike patterns is dependent on the accuracy of the weather projections; the more long-term the forecast, the more approximate the prediction is likely to be. When applied on a regional basis, model forecasts indicate expected average patterns of strike incidence and may not therefore be appropriate for individual farmers whose husbandry practices differ substantially from the average.
The spatial distribution of blowflies of the genus Lucilia within fields in south west England was examined in 1999 and 2000. Blowflies are economically important agents of sheep myiasis in the UK and understanding local aggregation is an essential step in the development of appropriate sampling and fly control regimes. Fifty, 20 x 20 cm, non-odour-baited, sticky traps were used to catch flies, at randomized, 10 x 10 m grid co-ordinates in fields of permanent pasture. Clear aggregations were evident in all Lucilia distributions. All values of the sigma2:mean ratio were greater than 1. The catches were shown to be highly aggregated using Morisita's index of aggregation. Generalized linear modelling of binary presence/absence catch data was used to relate aggregation to microclimate and habitat. Deletion testing was used to identify significant terms in the models. In general, Lucilia blowflies were predominantly caught around the edges, in warmer and more humid areas of the field. The relationship between microhabitat and the distribution of Lucilia collected in 1999 was used as predictive model to explain the catches made in two fields in 2000. This gave a highly significant fit in one field (P = 0.001) and a relationship which approached significance in the second (P = 0.08). However, these regressions suggest that the relationships between abundance and microhabitat are complex and that 'hot spots' of blowfly catches were not necessarily found in the most extreme microclimate conditions. Nevertheless, microhabitat features do give a relatively good guide to presence or absence of Lucilia in the trap catches, thereby providing important information about the most appropriate location of traps to maximize and standardize sampling and control regimes.
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