The most significant advance in population dynamics research in recent years has been the development of spatially explicit models of population dynamics.
A number of approaches have been used to model spatial dynamics. As with temporal dynamics, spatial dynamics can be modeled using deterministic, stochastic, or chaotic functions (Hassell et al. 1991, Matis et al. 1994, Sherratt and Jepson 1993). Different spatial dynamics result from using these different types of functions.
The earliest attempts to model spatial dynamics either applied diffusion models to describe insect dispersal and population spread from population centers (Rudd and Gandour 1985, Skellam 1951,Turchin 1998) or modeled population dynamics independently among individual landscape patches, based on local conditions within each patch, and linked patches by dispersal processes (e.g., W. Clark 1979). Diffusion models assume that the environment is homogeneous and that individuals disperse independently and with equal probability in any direction. The diffusion approach is useful for modeling spatial dynamics of insects in stored grain or relatively homogenous crop systems but less useful in most natural landscapes where patchiness interrupts diffusion.
Advances in spatial modeling have been facilitated by development of powerful computers that can store and manipulate large datasets. Concurrent development of geographic positioning systems (GPS) and geographic information systems (GIS) and geostatistical software has been a key to describing insect movement (Turchin 1998) and population epidemiology (Liebhold et al. 1993) across landscapes.
A GIS is an integrated set of programs that facilitate collection, storage, manipulation, and analysis of geographically referenced data, such as topography, vegetation type and density, and insect population densities. Data for a particular set of coordinates can be represented as a value for a cell, and each cell in the matrix is given a value (Fig. 7.11). This method is called the Raster method. A second method, which requires less storage space, is the vector method in which only data representing the vertices of polygons containing data must be stored (see Fig. 7.11). Various matrices representing different map layers can be superimposed to analyze interactions. For example, a map layer representing insect population distribution can be superimposed on map layers representing the distribution of host plants, predator abundances, climatic conditions, disturbances, or topography to evaluate the effects of patchiness or gradients in these factors on the spatial dynamics of the insect population.
Geostatistics are a means of interpolating the most probable population densities between sample points to improve representation of spatial distribution over landscapes. Early attempts to characterize spatial patterns were based on modifications of s2/x, Taylor's Power Law, Lloyd's Patchiness Index, and Iwao's patchiness regression coefficients (Liebhold et al. 1993). These indices focus on frequency distributions of samples and are useful for identifying dispersion patterns (see Chapter 5), but they ignore the spatial locations of samples. Modeling spatial dynamics across landscapes requires information on the location of sampling points, as well as population-density data. The locations of population aggregations affect densities in adjacent cells (Coulson et al. 1996, Liebhold and Elkinton 1989). Development of GPS has facilitated incorporation of precise sample locations in GIS databases.
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