Ecological Invasion, Roughened Fronts, and a Competitor's Extreme Advance: Integrating Stochastic Spatial-Growth Models

Both community ecology and conservation biology seek further understanding of factors governing the advance of an invasive species. We model biological invasion as an individual-based, stochastic process on a two-dimensional landscape. An ecologically superior invader and a resident species compete for space preemptively. Our general model includes the basic contact process and a variant of the Eden model as special cases. We employ the concept of a "roughened" front to quantify effects of discreteness and stochasticity on invasion; we emphasize the probability distribution of the front-runner's relative position. That is, we analyze the location of the most advanced invader as the extreme deviation about the front's mean position. We find that a class of models with different assumptions about neighborhood interactions exhibit universal characteristics. That is, key features of the invasion dynamics span a class of models, independently of locally detailed demographic rules. Our results integrate theories of invasive spatial growth and generate novel hypotheses linking habitat or landscape size (length of the invading front) to invasion velocity, and to the relative position of the most advanced invader.Comment: The original publication is available at www.springerlink.com/content/8528v8563r7u2742

Approximability of sparse integer programs

The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2-approximation algorithm. This is the first polynomial-time approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)-approximation for the same problem by Bansal et al