Population spread in patchy landscapes under a strong Allee effect

Propagation of invasive plant species in the presence of a road

Invasive plant species pose a significant threat to biodiversity and the economy, yet their management is often resource-intensive and expensive, and further research is required to make control measures more efficient. Evidence suggests that roads can have an important effect on the spread of invasive plant species, although little is known about the underlying mechanisms at play. We have developed a novel mathematical model to analyse the impact of roads on the propagation of invasive plants. The integro-difference equation model is formulated for stage-structured population and incorporates a road sub-domain in the spatial domain. The results of our study reveal, that, depending on the definition of the growth function in the model, there are three distinct types of behaviour in front of the road. Roads can act as barriers to invasion, lead to a formation of a beachhead in front of the road, or act as corridors allowing the invasive species to invade the domain in front of the road. Analytical and computational findings on how roads can impact the spread of invasive species show that a small change in conditions of the environment favouring the invasive species can change the case for the road, allowing the invasive species to invade the domain in front of the road where it previously could not spread.

Population dynamics with spatial structure and an Allee effect

Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an individual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For example, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

Faster movement in nonhabitat matrix promotes range shifts in heterogeneous landscapes

Ecologists often assume that range expansion will be fastest in landscapes composed entirely of the highest-quality habitat. Theoretical models, however, show that range expansion depends on both habitat quality and habitat-specific movement rates. Using data from 78 species in 70 studies, we find that animals typically have faster movement through lower-quality environments (73% of published cases). Therefore, if we want to manage landscapes for range expansion, there is a trade-off between promoting movement with nonhostile matrix, and promoting population growth with high-quality habitat. We illustrate how this trade-off plays out with the use of an exemplar species, the Baltimore checkerspot butterfly. For this species, we calculate that the expected rate of range expansion is fastest in landscapes with ~15% high-quality habitat. Behavioral responses to nonhabitat matrix have often been documented in animal populations, but rarely included in empirical predictions of range expansion. Considering movement behavior could change land-planning priorities from focus on high-quality habitat only to integrating high- and low-quality land cover types, and evaluating the costs and benefits of different matrix land covers for range expansion.

Invasion pinning in a periodically fragmented habitat

Biological invasions can cause great damage to existing ecosystems around the world. Most landscapes in which such invasions occur are heterogeneous. To evaluate possible management options, we need to understand the interplay between local growth conditions and individual movement behaviour. In this paper, we present a geometric approach to studying pinning or blocking of a bistable travelling wave, using ideas from the theory of symmetric dynamical systems. These ideas are exploited to make quantitative predictions about how spatial heterogeneities in dispersal and/or reproduction rates contribute to halting biological invasion fronts in reaction-diffusion models with an Allee effect. Our theoretical predictions are confirmed using numerical simulations, and their ecological implications are discussed.