Small-scale spatial structure influences large-scale invasion rates

Abstract

Local interactions among individual members of a population can generate intricate small-scale spatial structure, which can strongly influence population dynamics. The two-way interplay between local interactions and population dynamics is well understood in the relatively simple case where the population occupies a fixed domain with a uniform average density. However, the situation where the average population density is spatially varying is less well understood. This situation includes ecologically important scenarios such as species invasions, range shifts, and moving population fronts. Here, we investigate the dynamics of the spatial stochastic logistic model in a scenario where an initially confined population subsequently invades new, previously unoccupied territory. This simple model combines density-independent proliferation with dispersal, and density-dependent mortality via competition with other members of the population. We show that, depending on the spatial scales of dispersal and competition, either a clustered or a regular spatial structure develops over time within the invading population. In the short-range dispersal case, the invasion speed is significantly lower than standard predictions of the mean-field model. We conclude that mean-field models, even when they account for non-local processes such as dispersal and competition, can give misleading predictions for the speed of a moving invasion front.

Identifying density-dependent interactions in collective cell behaviour

“…Two of the primary advantages of our IBM approach is the ability to precisely replicate the experimental initial condition; and the ease of which new mechanisms can be incorporated into, and removed from, the model. Our approach can, therefore, be applied to quantify experimental evidence for more complex mechanisms including chemotaxis [15,55], mechanotaxis [56] and generalized growth laws [53], as well as comparing more complicated choices of interaction kernel [34]. Cell aspect ratio [23] can be incorporated into the model using asymmetric choices of interaction kernels; however, this would require more detailed experimental data, such as that provided by machine vision.…”

“…The contributions of each agent to B(x) depend on the distance between x and the location of the ith agent, x i , given by r = ∥x − x i ∥. There are many possible choices of kernel [34]; however, we find that the standard choice of Gaussian leads to a good match with experimental data [4]. In this study, we choose w (b) (r) to be a Gaussian [35] of spread σ with an extremum of γ b so that…”

Modelling collective navigation via non-local communication

“…Numerous mathematical models have been proposed to describe population movements in response to external navigation cues [25][26][27][28]. Theoretical models of communication-based collective navigation are often individual-based random walk models [29,30], where the behaviour of each individual is explicitly defined; though continuum models have also been proposed [8]. A common strategy has been to abstract interactions between individuals into a generic set of attraction, repulsion and alignment interactions [10,[31][32][33][34][35].…”

Modelling collective navigation via nonlocal communication

“…Numerous mathematical models have been proposed to describe population movements in response to external navigation cues [25,26,27,28]. Theoretical models of communicationbased collective navigation are often individual-based random walk models [29,30], where the behaviour of each individual is explicitly defined; though continuum models have also been proposed [8]. A common strategy has been to abstract interactions between individuals into a generic set of attraction, repulsion and alignment interactions [31,10,32,33,34,35].…”