Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. abstract: Unlike unconditionally advantageous "Fisherian" variants that tend to spread throughout a species range once introduced anywhere, "bistable" variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or "engineered underdominance" to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications. The University of Chicago Press and

Modeling the Manipulation of Natural Populations by the Mutagenic Chain Reaction

“…For example, an allele with perfect conversion efficiency and a selective cost of 0.41 is still expected to escape stochastic loss and sweep to fixation nearly 30% of the time in a WrightFisher population. When parameters allow for an unstable equilibrium, the MCR construct would spread in a population only if that unstable equilibrium point is sufficiently small (e.g., Barton and Turelli 2011). Thorough quantitative modeling of MCR population dynamics is strongly warranted, not only to put bounds on the frequency trajectories expected from release of an MCR, but also for possible choke points for controlling and preventing the expansion of an escaped or mutated MCR allele in a natural population.…”

Global stabilizing feedback law for a problem of biological control of mosquito-borne diseases

“…For example, [13] describes a simple model with a single differential equation, sufficient to reveal the bistable nature of the Wolbachia dynamics. Models for spatial dispersion are analyzed in [14] and [15]. In [16], [17], models are presented that assess the effect of the Wolbachia in dengue dynamics.…”

Spatial Gene Frequency Waves Under Genotype-Dependent Dispersal

“…We partition the speed of wave fronts into components due to dispersiveness, directional bias, and selection, which allows for a qualitative understanding of the impact of systematic differences in each of those factors. Furthermore, we present analytic expressions for gene frequency waves under genotype-dependent dispersal in hybrid zones, thus generalizing classical results that have proven to be of empirical relevance (Barton and Turelli 2011).…”

“…These wave fronts-at least in the simplest cases-can be calculated analytically and have proven to be of practical importance to predict rates of spatial spread, local introduction numbers necessary to initialize spatial spread, and sufficient environmental conditions that interrupt spatial spread in biocontrol applications (Barton and Turelli 2011). For the case that only the mean displacement is genotype dependent, we generalize the known wave solution to genotype-dependent dispersal (Equation 6).…”

“…To incorporate a dynamic response of population density to the heterogeneities in dispersal strategies of the population, one has to include an ecological layer into the model. Traveling wave fronts are known to speed up when moving down population density gradients and thus can be trapped in population sinks (Barton 1979;Barton and Turelli 2011). With genotype-dependent dispersal, however, the population density changes in time with the changing genetic composition of the population.…”