Spatial Competition: Roughening of an Experimental Interface

Abstract

Limited dispersal distance generates spatial aggregation. Intraspecific interactions are then concentrated within clusters, and between-species interactions occur near cluster boundaries. Spread of a locally dispersing invader can become motion of an interface between the invading and resident species, and spatial competition will produce variation in the extent of invasive advance along the interface. Kinetic roughening theory offers a framework for quantifying the development of these fluctuations, which may structure the interface as a self-affine fractal, and so induce a series of temporal and spatial scaling relationships. For most clonal plants, advance should become spatially correlated along the interface, and width of the interface (where invader and resident compete directly) should increase as a power function of time. Once roughening equilibrates, interface width and the relative location of the most advanced invader should each scale with interface length. We tested these predictions by letting white clover (Trifolium repens) invade ryegrass (Lolium perenne). The spatial correlation of clover growth developed as anticipated by kinetic roughening theory, and both interface width and the most advanced invader’s lead scaled with front length. However, the scaling exponents differed from those predicted by recent simulation studies, likely due to clover’s growth morphology.

In many plant communities, limited dispersal aggregates conspecific individuals1. In particular, most invasive plants are clonal and propagate vegetatively2, so that invaders initially cluster among residents3. Aggregation of conspecifics has consequences for population interactions. Individual plants usually compete at the nearest-neighbor scale4,5. When different species each aggregate spatially and interact locally, intraspecific competition will predominate within clusters, while interspecific competition will localize at the interface between clusters6,7,8. This interaction geometry implies that the advance versus extinction of an invasive species may depend on development and subsequent movement of a between-species interface9,10.

An invading species’ local density declines from positive equilibrium to rarity across the width of an ecological interface11. As a competitively superior invader excludes the resident species within the interface width, the front is pushed forward. Dispersal limitation promotes spatially correlated invasive advance along the interface. These correlations, generated through lateral growth, invite application of the theory of kinetic roughening, a framework for identifying quantitative characteristics shared by different interface-growth processes12. Previous applications of the theory span materials science13, temporal pattern in parallel-computing14,15, and ecological invasion11,16.

Kinetic roughening theory predicts power-law scaling relationships governing both the development and the equilibrium statistical structure of an invader-resident interface. Our analyses emphasize scaling of both the interface width and the relative position of the “front-runner,” the most advanced invader, a metric used at both local and regional scales17,18,19. Interestingly, the exponents of scaling relationships predicted by kinetic roughening sometimes identify an interface as a member of a particular universality class. That is, quite distinct local processes may exhibit the same dependence of interface roughening on time, and the equilibrium width may exhibit the same dependence on interface length; universality implies powerful generality13. Previously, we modeled the front produced when a dispersal limited, but competitively superior, invader advances across a habitat occupied by a resident species11,20. That model’s kinetic roughening belongs to the KPZ universality class, for Kardar-Parisi-Zhang12.

We begin by analyzing spatial competition as a problem for kinetic roughening theory, and then report a field experiment testing the predictions. We let Dutch white clover (Trifolium repens) advance into plots of perennial ryegrass (Lolium perenne). We monitored the development of spatial correlations along the fronts, and estimated a series of power-law scaling relationships from roughened fronts of different lengths. The exponents implied by the observed scaling allowed us, in addition, to ask if the experimental interface belonged to the KPZ universality class12,13.