SIGEST

Abstract

How do fish in a mountain stream, or bacteria in our intestines, continue to survive upstream, against a current? Why aren’t they washed out, eventually leading to their extinction? This "drift paradox"—the persistence of organisms which cannot "swim" against the current, yet live upstream—is part of "spatial ecology," the study of how populations of living organisms interact with the geometry and topography of their physical environment. The SIGEST selection in this issue, from the SIAM Journal on Applied Mathematics, makes groundbreaking contributions in the area of spatial ecology, including an enhanced explanation of the drift paradox. The selected paper is "The Effect of Dispersal Patterns on Stream Populations," by Frithjof Lutscher, Elizaveta Pachepsky, and Mark Lewis. As the title suggests, the paper analyzes the movement and persistence of populations in streams that can include single directional flows. This category covers a wide variety of applications, ranging from flora and fauna in rivers, to bacteria in the human digestive system, to fish in streams, just to name a few. In some cases, ecologists may be aiming to understand how to enhance survival of the population; in other cases, such as invading species, they may be aiming to understand how to prevent the population from surviving. The contribution of this paper is the modeling and analysis of the dispersal of populations in streams under a considerably more complex and realistic set of conditions than had previously been analyzed. The conditions covered by the analysis include both population growth (e.g., through reproduction) and movement of the existing population by two different methods, gradual flow and long-distance changes. The incorporation of the long-distance changes is particularly important and leads to different conclusions than had previously been reached without including this condition. In particular, the authors address and explain the drift paradox, showing that populations always can persist under high flow rates, provided the frequency of rare long-distance dispersal events is sufficient. The coverage of this range of conditions requires the authors to formulate and analyze an integrodifferential equations model, as opposed to the purely partial differential equation model that had been used in previous analysis under simpler conditions. The theory required to analyze this type of model is more difficult and is developed for the first time in this paper. The issues that are successfully investigated, under a variety of different conditions, include the minimal stream size necessary for the population to sustain itself, and the maximum stream flow under which the population can survive. The paper is very nicely written, with explanations embedded throughout it that help make it understandable to the nonexpert. It truly is an example of applied mathematics at its best: important new mathematics, leading to understanding of important applications, conducted by a team of applied mathematicians and application scientists.