The Importance of Being Indivisible: Ecological Invasions as Random Walks Instead of Waves

Invasions are one of the most easily identified spatial phenomena in ecology, and have inspired a rich variety of theories for ecologists and naturalists consideration. However, a number of arguments over the sensitivities of invasion rates to stochasticity, density-dependence, localization, and discreteness persist in the literature. The standard mathematical approach to invasions is based on Fisher’s analysis of travelling waves solutions for spread of an advantageous allele. In this paper, I present an alternative theory based on Ellner’s premise that species invasions are best interpreted not as waves, but as random walks, and that the discreteness of living organisms is fundamentally important. Using a density-dependent invasion model in a stationary environment with indivisible (atomic) individuals where reproduction and dispersal are stochastic and independent, I will prove 3 key properties of Ellner’s invasions previously suggested by simulation analysis: (1) greater dispersal stochasticity quickens invasions; (2) greater demographic stochasticity slows invasions, and (3) negative densitydependence slows invasions. The proofs are done using stochastic-dominance methods for ranking probability distributions, although one important result is left as a conjecture. We also explain why, in contradiction with continuous theory, an infinitely wide atomic invasion in two-dimensions can travel faster than a finite-width invasion and a one-dimensional invasion. The paper concludes with a classification of invasion dynamics based on dispersal kernel tails, providing proofs for atomic invasions with demographic stochasticity and negative density dependence that exponentially bounded kernels have speeds that can be bounded independent of carrying capacity, cubic-tailed and heavier kernels always accelerate, and the speeds for intermediate kernels depend on the carrying capacity.