Dispersal, settling and layer formation

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Abstract

Motivated by examples in developmental biology and ecology, we develop a model for convection-dominated invasion of a spatial region by initially motile agents which are able to settle permanently. The motion of the motile agents and their rate of settling are affected by the local concentration of settled agents. The model can be formulated as a nonlinear partial differential equation for the time-integrated local concentration of the motile agents, from which the instantaneous density of settled agents and its long-time limit can be extracted. In the limit of zero diffusivity, the partial differential equation is of first order; for application-relevant initial and boundary-value problems, shocks arise in the time-integrated motile agent density, leading to delta-function components in the motile agent density. Furthermore, there are simple solutions for a model of successive layer formation. In addition some analytic results for a one-dimensional system with non-zero diffusivity can also be obtained. A case study, both with and without diffusion, is examined numerically. Some important predictions of the model are insensitive to the specific settling law used and the model offers insight into biological processes involving layered growth or overlapping generations of colonization.

Highlights

► Derive PDE to model dispersal, settling and layer formation, through convection-dominated invasion. ► Develop a single PDE representation from the 2 PDE system. ► Develop transient and long-time limit in such systems. ► Derive successive layer formation and extent of overlap.

Introduction

In diverse contexts, populations of cells, animals or molecules disperse and invade a spatial region over time. Frequently, individuals or, as we shall call them, agents that make up the population undergo a transition from a motile to an immotile state. A steady-state spatial distribution occurs when the agents settle. Moreover, there may be multiple releases of these agents. If so, the interactions between motile and immotile agents may affect the final spatial distribution of the various releases.

Continuum modelling provides a population-level framework for investigating various alternative mechanisms for dispersal and settling, and allows for both the transient and steady-state distributions of the densities to be investigated. A useful planning outcome from such models is the ability to predict the size of the area needed for the release of a given number of animals in threatened species, but this type of modelling also has potential to increase our understanding in very different contexts, including developmental biology and malignant growth.

Reaction–diffusion systems are commonly developed to study such phenomena. Typically, the total population is divided into two sub-populations, one motile and the other an immotile or settled sub-population, giving rise to coupled partial differential equations (PDEs) describing conservation of mass. Wandering larvae settling on blades of grass [1], [2], animal and insect dispersal [3], [4], [5], and more recently the translocation of endangered species [6], [7] have been studied in this way. Dispersal, settling and proliferation with travelling wave solutions have also been investigated [8], [9], [10], [11]. Proliferation within the region being invaded is essential to sustain a travelling wave if the time interval during which invaders enter at a boundary is short. In situations where the governing equations are nonlinear but travelling wave solutions are not generated, few analytical results have appeared in the literature.

In contrast, here we are interested in modelling dispersal and settling where reproduction within the domain being invaded is not important, where diffusion may or may not be present, and where convection plays a significant role, and perhaps the dominant role. There are precedents for making a convective velocity population dependent. For example, models for collective behaviour of a swarm specify a convective velocity in terms of density, which may also account for non-local effects [12], [13]. Here we assume that the agents move in response to the settled immotile agents, so the velocity is related to settled agent density. We illustrate contexts in which our model may be relevant with two biological examples.

The Maud Island frog [6] exists in the wild only in several small, localized populations and is under significant risk of extinction. It is a K-selected species [14], which means that the species is long-lived, has delayed maturity and reproduction, produces relatively few offspring with a high probability of survival, and competes effectively for limited resources in a stable environment. These frogs occupy discrete home ranges that are small relative to the distances over which the animals disperse. Translocation experiments [15] testing strategies for enhancing the survival prospects of the species take place over times that are long enough for effective dispersal, but short enough that both reproduction and death can be neglected. Maud Island frogs are mute, but interact via chemical signals. Translocated frog populations favour moving through regions with a higher density of settled frogs to invade new territory [7]. In modelling this system it is natural to couple one or more of the dispersal mechanism and the settling rate to the population density of previously settled frogs.

A second example on a very different scale is provided by the mouse brain cortex, which contains six separate layers. Cells destined to become neurons proliferate in a region adjacent to where the layered structure forms, but not within the layers themselves. In normal development, a batch of neurons passes through each already existing layer, to form a new layer at the outer preplate—this means that the newest layers of the cortex are situated at the outermost regions whereas the oldest reside within the innermost regions. There is some evidence that in normal development, the settled neurons provide a guide to the dispersal and migration of the new batch of neurons [16], [17]. An intriguing genetic abnormality has the reversal of this layering formation, where the new group of neurons does not pass through the existing layer structure [18], [19].

In such applications, it is important to determine the size of the territory which is colonised by the settled agents. In one spatial dimension, we call this territory a layer. Each release produces a steady-state distribution for the settled agents, which increases the territory. What is the relationship between the subsequent layers, in particular how distinct are the layers, or how well mixed are the layers? We use these examples as motivation for studying a general problem.

Our most general model is introduced in Section 2.1: it allows motile agents to move by both diffusion and convection, to reproduce or die, and to differentiate into immotle agents, and the problem is formulated for arbitrary space dimension and domain geometry. Although the model originally consists of two coupled PDEs, we demonstrate how a single partial differential equation of second order with respect to both space and time derivatives can be derived for the time-integrated motile agent population (Section 2.2). Importantly, this PDE can be integrated once with respect to time under quite general circumstances.

Since reproduction of motile agents is low in our motivating examples, our further development from Section 3 onwards assumes no motile agent reproduction or death. Also, for clarity and simplicity, we adopt a particular functional form for the carrying-capacity-limited rate at which motile agents differentiate into immotile agents, although our techniques work somewhat more generally (as we illustrate in Section 6). Also for simplicity we restrict our attention to one-dimensional geometry, although some of our results extend to other geometries without difficulty. One-dimensional geometry is appropriate, for example, to the cortical development process briefly described above, since the layers formed are thin compared to their radii of curvature. For the frog translocation problem, where frogs are released from a small region, a radially symmetric model is more natural (see [20] for details).

We show in Section 3.1 that the long-time limiting behaviour of the immotile agent density can be determined from the solution of a purely spatial ordinary differential equation (ODE): there is no need to track the detailed time dependence. Such a convenient outcome is rare in spatially distributed coupled population problems. We consider first invasion of an initially empty region with both diffusion and convection active (Section 3.2) and find the zero-diffusion case to be especially simple. When diffusion is absent, elegantly simple results are obtained for a model of successive waves of invasion, leading to overlapping layers corresponding to the generations of invaders (Sections 3.3 Further observations about diffusion-free limit, 3.4 Sequential waves of diffusion-free invasion).

Our most detailed investigation of time-dependent invasion occupies Sections 4 Transient behaviour in the diffusion-free limit, 5 A single invasion wave: case studies with and without diffusion. Section 4.1 assembles some useful results for the zero-diffusion limit in one space dimension based on the method of characteristics. These results are applied in Section 4.2 to a single one-dimensional invasion event. Perhaps not unexpectedly, the quasilinear PDE on which the zero-diffusion formalism is based produces shock curves, but what is novel is the generation from the appropriate weak solutions of the PDE of solutions for the motile cell density with both continuous components and moving delta-function components.

Numerical case studies in one-dimensional invasion are discussed in Section 5, illustrating the shock and moving delta-function phenomena for the diffusion-free case (Section 5.1) and the resolution of the shocks and broadening of the delta function peaks when diffusion is active (Section 5.2). We conclude in Section 6 with a reflection on the nature and consequences of the model presented in the paper, biological insight gained, the relative insensitivity of the conclusions to some details of the model, and alternative modelling approaches.

Section snippets

General formulation of the model

Our modelling framework is deterministic and continuous, and thus assumes that the number of agents involved is large enough. One may formulate a stochastic version of the model appropriate to small populations and suitable for simulation, but apart from a few remarks in Section 6, we do not address this here.

One-dimensional invasion: the long-time limit

For applications in developmental biology and ecology, one is particularly interested in the ultimate state of a region that is invaded from the boundary by motile agents, so we consider invasion of the region x > 0 when we prescribe the value of u(0, t) for all t > 0.

Transient behaviour in the diffusion-free limit

The transient behaviour of the model in the diffusion-free limit can be studied using the method of characteristics [23]. Although such an approach is in principle straightforward, in practice we find subtleties that merit careful exploration. We discuss the auxiliary equations—a dynamical system equivalent to the PDE—and find some integrals of them in Section 4.1 and study their implications for a specific boundary condition in Section 4.2.

Our analysis above shows that the long-time limit is

A single invasion wave: case studies with and without diffusion

We consider in detail the invasion problem stated in Section 4.2 for the specific parameter values Q = T = 1 and contrast the diffusion-free time-dependent solution (Section 5.1) with the analogous problem for ε > 0 (Section 5.2).

Discussion

The model for invasion and layer formation proposed in the present paper is based on a simple mechanism for the interaction of motile and immotile agent subpopulations. However, this model captures a number of features seen in invasion and colonization processes. It also gives rise to interesting analysis and has several attractive and unusual mathematical features. The introduction of the time-integrated motile agent density w(r, t) enables the original coupled system of equations to be

Acknowledgements

The work was supported by an ARC Discovery Project grant. KAL is an ARC Professorial Fellow. Helpful comments from the referees are acknowledged.

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