A basic result of Anderson and May's (1982) early work on models of disease in natural (nonhuman) populations is that pathogen fitness is R0 = βN/(α + γ + d), where β is the horizontal transmission rate of the disease, α is the disease-induced mortality rate, d is the background mortality rate, γ is the recovery rate to the immune state, and N is host population density without the disease (see Boxes 2.1 and 2.2). In this model, pathogen strains that maximize β/(α + γ + d) competitively exclude all others. A key insight, however, is that trade-offs among fitness components prevent selection from driving horizontal transmission β to infinity, and mortality α and recovery γ to zero. For the mosquito-vectored rabbit disease myxomatosis, for example, virus strains that kill too rapidly have little chance of being transmitted, because mosquitoes do not bite dead rabbits. On the other hand, strains that kill too slowly produce such low concentrations of virus that they are also unlikely to be transmitted (Fenner 1983). Assuming that the rabbits evolve over a much longer time scale than does the virus, for such a constraint an evolutionarily stable strategy (ESS) exists at the maximum of β/(α + γ + d); compare Boxes 2.1, 2.2, and 5.1.
Spatial dynamics of populations have long been of interest to ecologists (Skellam 1951; Levin and Paine 1974; Andow et al. 1990), but recent advances in data collection and in computational power have put these concepts within the reach of many ecologists for the first time. Computational models (Wilson et al. 1993, 1995b; McCauley et al. 1993; Pacala and Deutschman 1995) suggest important and previously unexplored effects of space and discrete individuals on population dynamics. Analytic approaches that capture these effects are emerging, building on methods developed in other contexts. This chapter presents a general method for deriving approximate equations for spatial dynamics in continuous space and time that has advantages over classical and many modern approaches.
We are interested in spatial pattern formation in plant communities and in the effects of pattern on plant competition. Our goal is to find general methods for exploring this problem that are
analytically tractable, so that we can gain insight into qualitative behaviors of the system and analyze how they depend on the parameters;
sufficiently general, so that some of the same tools can be applied to answer a range of different questions about spatial dynamics in ecology; and
close enough to the characteristics of real populations that we can eventually fit the models to field data on individual behavior.
We focus on spatial point processes (Diggle 1983; Gandhi et al. 1998), continuous-time dynamical systems for discrete individuals interacting in a continuous habitat.
In the past two decades, there has been a growing recognition in ecology of the problem of scale. This generalization is especially true in the study of schooling, herding, and swarming, in which there is an inherent duality in scale between individual behavior and group and population level dynamics. Steele and others (Steele 1978 Haury et al. 1978 Levin 1992) have pointed out that our measurements and perceptions of pattern are conditioned by the perspectives we impose through our scales of description. At any scale of resolution, we average dynamics that take place on faster scales the strategy is analogous to that used by other organisms in their evolutionary responses to variability. Indeed, it is clear that not every fine-scale detail is relevant to understanding phenomena on broader scales and that inclusion of unnecessary detail only obfuscates understanding of the mechanisms underlying patterns of interest. The central problem is to determine how information is transferred across scales and exactly what detail at fine scales is necessary and sufficient for understanding pattern on averaged scales.
The problem of how information is transferred across scales cannot be addressed without modeling. In relating behaviors on one scale to those on others, one is often dealing with processes operating on radically different time scales, in which much of the detail on faster or finer scales must be irrelevant to those on slower or broader scales. Because decisions about what one can ignore require a quantitative evaluation of the manifestation of processes across scales, a quantitative approach is both unavoidable and powerful.
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