| Because of limited mobility and localized interactions, most organisms do not interact equally with all parts of their environment but instead with a limited neighborhood. The resulting spatial correlations affect population dynamics. The discreteness of organisms can also affect population dynamics. Because population size cannot change by less than one, and size-changing events such as births and deaths occur at distinct times, population dynamics are noisy. For large populations, this so-called “demographic stochasticity” is often ignorable, but when population size is small, either throughout the system or in a region, noise can have important consequences.; This dissertation explores the combined effects of spatial correlations and population discreteness. Chapter II discusses the limitations of many traditional physics techniques in analyzing ecological models. Chapters III and IV consider grid-based models. Every grid point can be vacant or occupied by an individual, and individuals interact according to simple, probabilistic rules. In chapter III, I develop approximate equations for the population mean and variance, including the effects of demographic stochasticity, by ignoring all but very short-range spatial correlations (a moment closure scheme). I apply this to a grid model and obtain expressions for population mean and variance. In chapter IV, I develop an empirical moment closure scheme based on observed spatial correlations. This leads to expressions for population mean and variance that are both simpler and more accurate, as well as to probability distributions for how long the population will take to reach a given, low level.; Subsequently, I turn to the effects of population discreteness on the spread of newly introduced species. In chapter V, I analyze a common class of one-dimensional, single-species invasion models and find three effects of population discreteness and demographic stochasticity on invasion speed. The result is that for very general conditions, the equation for mean population density predicts an invasion speed which is too fast. Studies of chemical waves suggest that the net result of particle discreteness and fluctuations is to introduce an effective concentration cutoff. Cutoffs work for ecological models only when per capita offspring numbers are low. I discuss the reasons for this in ch. VI. |
