Pushing the Boundaries: Models for the Spatial Spread of Ecosystem Engineers

Lutscher, Frithjof; Fink, Justus; Zhu, Yingjie

doi:10.1007/s11538-020-00818-8

Cited by 6 publications

(3 citation statements)

References 33 publications

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“…Since that time, much attention has been devoted to various generalizations, many of which are reviewed by Du 31 . In very recent times, a variety of extensions have been documented, including a rigorous results for different boundary conditions, nonlinear or nonlocal diffusion, generalized reaction terms, with one or two phases, as well as motivation in terms of applications in ecology 32–45 . Formal results and numerical simulations have been described by us 27,46–49 and others 50 …”

Section: Introductionmentioning

confidence: 99%

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

2021

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While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher–Stefan model, a generalization of the well‐known Fisher–KPP model, characterized by a leakage coefficient κ$\kappa$ which relates the speed of the moving boundary to the flux of population there. This Fisher–Stefan model overcomes one of the well‐known limitations of the Fisher–KPP model, since time‐dependent solutions of the Fisher–Stefan model involve a well‐defined front which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting κ>0$\kappa > 0$, which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting κ<0$\kappa < 0$ leads to retreating traveling waves and an interesting transition to finite‐time blow‐up. For certain initial conditions, population extinction is also observed. Our approach involves studying time‐dependent solutions of the governing equations, phase plane, and asymptotic analysis, leading to new insight into the possibilities of traveling waves, blow‐up, and extinction for this moving boundary problem. MATLAB software used to generate the results in this work is available on Github.

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Section: Introductionmentioning

confidence: 99%

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

2021

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“…Since that time, much attention has been devoted to various generalisations, many of which are reviewed by Du [30]. In very recent times, a variety of extensions have been documented, including a rigorous results for different boundary conditions, nonlinear or nonlocal diffusion, generalised reaction terms, with one or two phases, as well as motivation in terms of applications in ecology [31][32][33][34][35][36][37][38][39][40][41][42][43][44]. Formal results and numerical simulations have been described by us [45][46][47][48][49] and others [50].…”

Section: Introductionmentioning

confidence: 99%

Travelling waves, blow-up and extinction in the Fisher-Stefan model

McCue,

El-Hachem,

Simpson

2021

Preprint

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While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher-Stefan model, a generalisation of the well-known Fisher-KPP model, characterised by a leakage coefficient κ which relates the speed of the moving boundary to the flux of population there. This Fisher-Stefan model overcomes one of the well-known limitations of the Fisher-KPP model, since time-dependent solutions of the Fisher-Stefan model involve a well-defined front with compact support which is more natural in terms of mathematical modelling. Almost all of the existing analysis of the standard Fisher-Stefan model involves setting κ > 0, which can lead to either invading travelling wave solutions or complete extinction of the population. Here, we demonstrate how setting κ < 0 leads to retreating travelling waves and an interesting transition to finite-time blow-up. For certain initial conditions, population extinction is also observed. Our approach involves studying time-dependent solutions of the governing equations, phase plane and scaling analysis, leading to new insight into the possibilities of travelling waves, blow-up and extinction for this moving boundary problem. Matlab software used to generate the results in this work are available on Github

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“…In [40,4], the authors propose a different free boundary model for species which engineers the environment to create its habitat, where the free boundary stands for the boundary of the engineered part of the environment, over which the growth of the population is governed by a Fisher-KPP function such as f (u) = u(r − au), r, a > 0, while in the rest of the available environment (un-engineered territory), the growth decays according to f (u) = −mu, m > 0; we refer to [40,4] for more details.…”

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confidence: 99%

Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition

2024

DCDS

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We study the reaction diffusion equation ut − duxx = f (u) with a monostable nonlinear function f (u) over a changing interval [g(t), h(t)], viewed as a model for the spreading of a species with population range [g(t), h(t)] and density u(t, x). The free boundaries x = g(t) and x = h(t) are not governed by the same Stefan condition as in Du and Lin [20] and other previous works; instead, they satisfy a related but different set of equations obtained from a "preferred population density" assumption at the range boundary, which allows the population range to shrink. We obtain a rather complete understanding of the longtime dynamics of the model, which exhibits persistent propagation with a finite asymptotic propagation speed determined by a certain semi-wave solution, and the density function converges to the semi-wave profile as time goes to infinity. The asymptotic propagation speed is always smaller than that of the corresponding classical Cauchy problem where the reaction-diffusion equation is satisfied for x over the entire real line with no free boundary. Moreover, when the preferred population density used in the free boundary condition converges to 0, the solution u of our free boundary problem converges to the solution of the corresponding classical Cauchy problem, and the propagation speed also converges to that of the Cauchy problem.

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Pushing the Boundaries: Models for the Spatial Spread of Ecosystem Engineers

Cited by 6 publications

References 33 publications

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

Travelling waves, blow-up and extinction in the Fisher-Stefan model

Propagation dynamics of the monostable reaction-diffusion equation with a new free boundary condition

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