Abstract
Motivated by the propagation of thin bacterial films around planar obstacles, this paper considers the dynamics of travelling wave solutions to the Fisher–KPP equation u_t = u(1-u) + u_{xx} + u_{yy} in a planar strip -infty< x < infty , 0 le y le L. We examine the propagation of fronts in the presence of a mixed boundary condition (also referred to as a ‘partially absorbing’ or ‘reactive’ boundary) u_y = alpha u, with alpha >0, at y=0. The presence of boundary conditions of this kind leads to the development of front solutions that propagate in x but contain transverse structure in y. Motivated by the observation that the speed of propagation in the Fisher–KPP equation is determined (for exponentially decaying initial conditions) by the behaviour at the leading edge, we analyse the linearised Fisher–KPP equation in order to estimate the speed of the stable travelling front, a function of the width L and the imposed boundary conditions. For wide strips the speed estimate based on the linearised equation agrees well with the results of numerical simulations. For narrow channels numerical simulations indicate that the stable front propagates more slowly, and for sufficiently small L or sufficiently large alpha the front speed falls to zero and the front collapses. The reason for the collapse is the non-existence, far behind the front, of a stable positive equilibrium solution u(x, y). While existence of these equilibrium states can be demonstrated via phase plane arguments, the investigation of stability is similar to calculations of critical patch sizes carried out in similar ecological models.
Highlights
Many biological situations can be viewed as invasions of one population into another; either a new genetic trait or species out-competing an existing one for resources, or consuming a preexisting resource distributed over a spatial domain
In more general cases in which the environment for the population growth and dispersal changes unfavourably along the boundaries, rather than taking Dirichlet boundary conditions u = 0 at y = 0 and y = L, more realistic boundary conditions are the mixed-type boundary conditions u y = αu at y = 0 and u y = −βu at y = L, where α, β ≥ 0. This corresponds to situations where the Fisher–KPP equation is motivated as a continuum description arising from a microscopic model in which, when an individual crosses the boundary there is a positive probability of absorption onto the boundary rather than reflection back into the interior of the domain (Erban and Chapman 2007)
In this paper we have analysed the widely used Fisher– Kolmogorov–Petrovskii–Piskunov (FKPP) model of front propagation to look at 2D front propagation near a boundary that models a constant fractional loss of the local population density
Summary
Many biological situations can be viewed as invasions of one population into another; either a new genetic trait or species out-competing an existing one for resources, or consuming a preexisting resource distributed over a spatial domain. In more general cases in which the environment for the population growth and dispersal changes unfavourably along the boundaries, rather than taking Dirichlet boundary conditions u = 0 at y = 0 and y = L, more realistic boundary conditions are the mixed-type (or ‘reactive’, or ‘partially absorbing’) boundary conditions u y = αu at y = 0 and u y = −βu at y = L, where α, β ≥ 0 This corresponds to situations where the Fisher–KPP equation is motivated as a continuum description arising from a microscopic model in which, when an individual crosses the boundary there is a positive probability of absorption onto the boundary rather than reflection back into the interior of the domain (Erban and Chapman 2007).
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