The minimum founding population in dispersing organisms subject to strong Allee effects

Block-pulse integrodifference equations

We present a hybrid method for calculating the equilibrium population-distributions of integrodifference equations (IDEs) with strictly increasing growth, for populations that are confined to a finite habitat-patch. This method is based on approximating the growth function of the IDE with a piecewise-constant function, and we call the resulting model a block-pulse IDE. We explicitly write out analytic expressions for the iterates and equilibria of the block-pulse IDEs as sums of cumulative distribution functions. We characterize the dynamics of one-, two-, and three-step block-pulse IDEs, including formal stability analyses, and we explore the bifurcation structure of these models. These simple models display rich dynamics, with numerous fold bifurcations. We then use three-, five-, and ten-step block-pulse IDEs, with a numerical root finder, to approximate models with compensatory Beverton-Holt growth and depensatory, or Allee-effect, growth. Our method provides a good approximation for the equilibrium distributions for compensatory and depensatory growth and offers numerical and analytical advantages over the original growth models.

An Alternative Perspective for the Theory of Biological Control

Importation biological control represents the planned introduction of a specialist natural enemy from the region of origin of an invasive pest or weed. For this study, the author considered why attempts to develop a predictive theory for biological control have been misguided and what future directions might be more promising and effective. Despite considerable interest in the theory of consumer–resource population dynamics, such theory has contributed little to improvements in the success of biological control due to a focus on persistence and equilibrium dynamics rather than establishment and impact. A broader consideration of invasion biology in addition to population ecology offers new opportunities for a more inclusive theory of biological control that incorporates the demographic and genetic processes that more specifically address the establishment and impact of introduced natural enemies. The importance of propagule size and genetic variance for successful establishment, and of contributions to host population growth, relative population growth rates, interaction strength, and coevolution for suppression of host abundance are discussed as promising future directions for a theory of biological control.

Rethinking biological control programs as planned invasions

Biological control of pests with their natural enemies essentially consists of planned invasions, with the opportunity to select both the invader and the invaded environment. Recent advances in invasion science link 'intrinsic invasion factors' (life history and behavioral traits) with invader success; connect 'extrinsic invasion factors' (abiotic and biotic aspects of the invaded environment) with environmental invasibility; and demonstrate that their interaction leads not only to ecologically driven variability but also to rapid evolutionary change in biocontrol systems. However, current theory and empirical evidence from invasion science have not yet been extensively adopted into biological control research and practice.

Sex difference and Allee effects shape the dynamics of sex-structured invasions

The rate at which a population grows and spreads can depend on individual behaviour and interactions with others. In many species with two sexes, males and females differ in key life-history traits (e.g. growth, survival and dispersal), which can scale up to affect population rates of growth and spread. In sexually reproducing species, the mechanics of locating mates and reproducing successfully introduce further complications for predicting the invasion speed (spread rate), as both can change nonlinearly with density. Most models of population spread are based on one sex, or include limited aspects of sex differences. Here we ask whether and how the dynamics of finding mates interact with sex-specific life-history traits to influence the rate of population spread. We present a hybrid approach for modelling invasions of populations with two sexes that links individual-level mating behaviour (in an individual-based model) to population-level dynamics (in an integrodifference equation model). We find that limiting the amount of time during which individuals can search for mates causes a demographic Allee effect which can slow, delay, or even prevent an invasion. Furthermore, any sex-based asymmetries in life history or behaviour (skewed sex ratio, sex-biased dispersal, and sex-specific mating behaviours) amplify these effects. In contrast, allowing individuals to mate more than once ameliorates these effects, enabling polygynandrous populations to invade under conditions where monogamously mating populations would fail to establish. We show that details of individuals' mating behaviour can impact the rate of population spread. Based on our results, we propose a stricter definition of a mate-finding Allee effect, which is not met by the commonly used minimum mating function. Our modelling approach, which links individual- and population-level dynamics in a single model, may be useful for exploring other aspects of individual behaviour that are thought to impact the rate of population spread.