Fisher-KPP Equation Research Articles

The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

Chaos of the initial and boundary value problems for the reaction-diffusion equations

This paper investigates an initial and boundary value problem for the reaction-diffusion equations, which can be considered as a linearized form of the advective Fisher-KPP equations. It is demonstrated that the initial and boundary value problem is chaotic when the three parameters of the reaction-diffusion equation vary above a specific surface. However, stable solutions are obtained both on and below this surface within a particular subset of initial values. The chaos and stability of the nonhomogeneous initial boundary value problem are further studied. Finally, some numerical examples are provided to illustrate the validity of the obtained results.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

Propagation phenomena for a nonlocal reaction-diffusion model with bounded phenotypic traits

In this paper we study the stability of cylinder front waves and the propagation of solutions of a nonlocal Fisher-type model describing the propagation of a population with nonlocal competition among bounded and continuous phenotypic traits. By applying spectral analysis and separation of variables we prove the spectral and local exponential stability of the cylinder waves with the noncritical speeds in some exponentially weighted spaces. By combining the detailed analysis with the spectral expansion and the special construction of sub-supersolutions, we further prove the uniform boundedness of the solutions and the global asymptotic stability of the cylinder waves for more general nonnegative bounded initial data, and prove that the spreading speeds and the asymptotic behavior of the solutions are determined by the decay rates of the initial data. Our results also extend some classical results on the stability of planar waves for Fisher-KPP equation to the nonlocal Fisher model in multi-dimensional cylinder case.

Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen–Cahn and the Fisher–KPP equations on the manifold of fixed-rank matrices. Our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

Travelling waves in a minimal go-or-grow model of cell invasion

We consider a minimal go-or-grow model of cell invasion, whereby cells can either proliferate, following logistic growth, or move, via linear diffusion, and phenotypic switching between these two states is density-dependent. Formal analysis in the fast switching regime shows that the total cell density in the two-population go-or-grow model can be described in terms of a single reaction–diffusion equation with density-dependent diffusion and proliferation. Using the connection to single-population models, we study travelling wave solutions, showing that the wave speed in the go-or-grow model is always bounded by the wave speed corresponding to the well-known Fisher–KPP equation.

Stochastic evolution equations with Wick-analytic nonlinearities

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher–KPP equations, stochastic Allen–Cahn, stochastic Newell–Whitehead–Segel, and stochastic Fujita–Gelfand equations. By implementing the theory of C 0 − semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove the existence and uniqueness of solutions for this class of stochastic partial differential equations.

Moving Singularities of the Forced Fisher–KPP Equation: An Asymptotic Approach

Moving Singularities of the Forced Fisher–KPP Equation: An Asymptotic Approach

Spreading speeds for time heterogeneous prey-predator systems with nonlocal diffusion on a lattice

We investigate the spreading behaviour for the solutions of a non-autonomous prey-predator system on a discrete lattice. These time variations are assumed to enjoy an averaging property. This includes periodicity, almost periodicity and unique ergodicity as special cases. The spatial motion of individuals from one site to another is modelled by a discrete convolution operator. In order to take into account external fluctuations such as seasonality, daily variations and so on, the convolution kernels and reaction terms may vary with time. Our analysis of the spreading speeds of invasion of the species is based on the careful and detailed study of the hair-trigger effect and spreading speed for a non-autonomous scalar Fisher-KPP equation on a lattice. Then, we are able to compare the solutions of the prey-predator system with those of a suitable scalar Fisher-KPP equation and derive the invasion speeds of the prey and of the predator.

A Harnack inequality for a class of 1D nonlinear reaction–diffusion equations and applications to wave solutions

In this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.

Global solutions to a modified Fisher-KPP equation

Global solutions to a modified Fisher-KPP equation

Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength \beta\in\mathbb{R} . This equation exhibits a transition from pulled to pushed front behavior at \beta_{c}=2 . We prove convergence of the solutions to a traveling wave in a reference frame centered at a position m_{\beta}(t) and study the asymptotics of the front location m_{\beta}(t) . When \beta < 2 , it has the same form as for the standard Fisher-KPP equation established by Bramson: m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1) as t\to\infty . This form is typical of pulled fronts. When \beta > 2 , the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1) with c_{*}(\beta)=\beta/2+2/\beta , which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2 , the expansion changes to m_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1) , reflecting the “pushmi-pullyu” nature of the front. The arguments for \beta<2 rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case \beta>2 relies on standard pushed front techniques. The proof in the case \beta=\beta_{c} is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at \beta_{c}=2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.

Large deviations and the emergence of a logarithmic delay in a nonlocal linearised Fisher–KPP equation

We study a variant of the Fisher–KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a “Bramson-like” logarithmic delay for the linearised equation with step-like initial data. We conclude that the logarithmic delay emerges also for the solutions of the nonlinear equation. Previous papers found very precise results for the nonlinear equation with strong assumptions on the decay of the kernel. Our results are less precise, but they are valid for all thin-tailed kernels.

Rate of propagation for the Fisher-KPP equation with nonlocal diffusion and free boundaries

In this paper, we obtain sharp estimates for the rate of propagation of the Fisher-KPP equation with nonlocal diffusion and free boundaries. The nonlocal diffusion operator is given by \int_{\R}J(x-y)u(t,y)\,dy-u(t,x) , and our estimates hold for some typical classes of kernel functions J(x) . For example, if for |x|\gg 1 the kernel function satisfies J(x)\sim |x|^{-\gamma} with \gamma>1 , then it follows from [Y. Du et al., J. Math. Pures Appl. 154, 30–66 (2021)] that there is a finite spreading speed when \gamma>2 , namely the free boundary x=h(t) satisfies \lim_{t\to\infty}h(t)/t=c_{0} for some uniquely determined positive constant c_{0} depending on J , and when \gamma\in (1, 2] , \lim_{t\to\infty}h(t)/t=\infty ; the estimates in the current paper imply that, for t\gg 1 , c_{0}t-h(t)\sim \begin{cases}1&\text{ when $\gamma>3$,}\\ \ln t&\text{ when }\gamma=3,\\ t^{3-\gamma}&\text{ when }\gamma\in (2, 3),\\ \end{cases} \qquad h(t)\sim \begin{cases} t\ln t&\text{ when }\gamma =2,\\ t^{1/(\gamma-1)}&\text{ when }\gamma\in (1,2). \end{cases} Our approach is based on subtle integral estimates and constructions of upper and lower solutions, which rely crucially on guessing correctly the order of growth of the term to be estimated. The techniques developed here lay the groundwork for extensions to more general situations.

Voting models and semilinear parabolic equations

We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975 Commun. Pure Appl. Math. 28 323–31). In particular, we present ‘random outcome’ and ‘random threshold’ voting models that yield any polynomial nonlinearity f satisfying and a ‘recursive up the tree’ model that allows to go beyond this restriction on f. We compute several examples of particular interest; for example, we obtain a curious interpretation of the heat equation in terms of a nontrivial voting model and a ‘group-based’ voting rule that leads to a probabilistic view of the pushed-pulled transition for a class of nonlinearities introduced by Ebert and van Saarloos.

Stability and uniqueness of generalized traveling front for non-autonomous Fisher–KPP equations with nonlocal diffusion

This paper is concerned with the stability and uniqueness of generalized traveling front of non-autonomous Fisher–KPP equations with nonlocal diffusion. In such non-autonomous nonlocal dispersal equations, both the diffusion kernel and the reaction term generally depend on time. The existence of generalized traveling wave fronts of such equations has been studied in Ducrot and Jin (2022). By using comparison principle and part metric, we show the generalized traveling front is asymptotically stable under well-fitted perturbation and it is also unique.

Exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries

Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries, where the nonlocal diffusion operator is given by ∫RJ(x-y)u(t,y)dy-u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\displaystyle \\int _{\\mathbb {R}}J(x-y)u(t,y)dy-u(t,x)$$\\end{document}, and the kernel function J(x) behaves like a power function near infinity, namely lim|x|→∞J(x)|x|α=λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{|x|\\rightarrow \\infty } J(x)|x|^{\\alpha }=\\lambda >0$$\\end{document} for some α∈(1,2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2]$$\\end{document}. This is the precise range of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} where accelerated spreading can happen for such kernels. By constructing subtle upper and lower solutions, we prove that the location of the free boundaries x=h(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=h(t)$$\\end{document} and x=g(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=g(t)$$\\end{document} goes to infinity at exactly the following rates: limt→∞h(t)tlnt=limt→∞-g(t)tlnt=μλ,whenα=2,limt→∞h(t)t1/(α-1)=limt→∞-g(t)t1/(α-1)=22-α2-αμλ,whenα∈(1,2).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle \\lim _{t\\rightarrow \\infty }\\frac{h(t)}{t\\ln t}=\\lim _{t\\rightarrow \\infty }\\frac{-g (t)}{t\\ln t}=\\mu \\lambda ,&{} \\hbox { when } \\alpha =2,\\\\ \\displaystyle \\lim _{t\\rightarrow \\infty }\\frac{h(t)}{ t^{1/(\\alpha -1)}}= \\lim _{t\\rightarrow \\infty }\\frac{-g (t)}{ t^{1/(\\alpha -1)}}=\\frac{2^{2-\\alpha }}{2-\\alpha }\\mu \\lambda , &{} \\hbox { when } \\alpha \\in (1,2). \\end{array}\\right. } \\end{aligned}$$\\end{document}Here μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu >0$$\\end{document} is a given parameter in the free boundary condition. Accelerated propagation can also happen when lim|x|→∞J(x)|x|(ln|x|)β=λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{|x|\\rightarrow \\infty }J(x)|x|(\\ln |x|)^\\beta =\\lambda >0$$\\end{document} for some β>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >1$$\\end{document}. For this case, we prove that -g(t),h(t)=exp{[(2βμλβ-1)1/β+o(1)]t1/β}ast→∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -g (t), h(t)=\\exp \\Big \\{\\Big [\\Big (\\frac{2\\beta \\mu \\lambda }{\\beta -1}\\Big )^{1/\\beta }+o(1)\\Big ]t^{1/\\beta }\\Big \\} \\hbox { as } t\\rightarrow \\infty . \\end{aligned}$$\\end{document}These results considerably sharpen the corresponding ones in [20], and the techniques developed here open the door for obtaining similar precise results for other problems. A crucial technical point is that such precise conclusions on the propagation are achievable by finding the correct improvements on the form of the lower solutions used in [20], even though the precise long-time profile of the density function u(t, x) is still lacking.

Ahead of the Fisher–KPP front

The solution h to the Fisher–KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value of the solution ahead of the front, at time t and position ct, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the nonlinearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly explicit and irregular. The main tool of this paper is the so-called ‘magical expression’ which relates the position of the front, the initial condition, and the quantity we investigate.

Spreading Properties for Non-autonomous Fisher–KPP Equations with Non-local Diffusion

Spreading Properties for Non-autonomous Fisher–KPP Equations with Non-local Diffusion

The logarithmic Bramson correction for Fisher-KPP equations on the lattice ℤ

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice Z \mathbb {Z} . The level sets of solutions with step-like initial conditions are located at position c ∗ t − 3 2 λ ∗ ln ⁡ t + O ( 1 ) c_*t-\frac {3}{2\lambda _*}\ln t+O(1) as t → + ∞ t\rightarrow +\infty for some explicit positive constants c ∗ c_* and λ ∗ \lambda _* . This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed c ∗ c_* uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times.