Fisher-KPP Equation

Overview

The Fisher-KPP equation, also known as the Fisher equation or the Kolmogorov-Petrovsky-Piskunov equation, is a fundamental nonlinear partial differential equation (PDE) in mathematical biology. It models the spread of an advantageous gene in a population or, more generally, the propagation of a biological or chemical species. Independently introduced by Ronald Fisher in 1937 and by Kolmogorov, Petrovsky, and Piskunov (KPP) in the same year, it combines diffusion and logistic growth to describe wave-like phenomena in population dynamics.

The one-dimensional Fisher-KPP equation is given by:

\[u_t = D u_{xx} + r u (1 - u)\]

Where:

• $u(x, t)$ is the density of the species or the frequency of an advantageous gene at position $x$ and time $t$.
• $D$ is the diffusion coefficient, representing spatial dispersal.
• $r$ is the intrinsic growth rate of the population.

Key Features

• Reaction-Diffusion System: Combines local reactions (logistic growth) and spatial diffusion.
• Nonlinearity: The term $r u (1 - u)$ introduces nonlinearity, leading to complex dynamics like traveling waves.
• Traveling Wave Solutions: Admits solutions of the form $u(x, t) = U(x - c t)$, representing waves propagating at constant speed $c$.

Physical Interpretation

• Population Genetics: Models the spread of a beneficial gene through a spatially distributed population.
• Ecology: Describes the invasion of a species into new territory.
• Chemical Kinetics: Represents autocatalytic chemical reactions spreading through a medium.

Properties

Traveling Wave Solutions

• Seeking solutions of the form $u(x, t) = U(z)$ with $z = x - c t$, the equation becomes an ordinary differential equation (ODE):

  \[-c \frac{dU}{dz} = D \frac{d^2 U}{dz^2} + r U (1 - U)\]

• Boundary conditions for invasion problems are:

  \[U(-\infty) = 1, \quad U(\infty) = 0\]

• Minimum Wave Speed:

  The minimum speed $c_{\text{min}}$ at which traveling waves propagate is:

  \[c_{\text{min}} = 2 \sqrt{D r}\]

Stability and Asymptotic Behavior

• Stability of Traveling Waves: Waves traveling at speeds $c \geq c_{\text{min}}$ are stable.
• Asymptotic Spread: The population front advances at speed $c_{\text{min}}$ over long times.

Linearization Near Zero

• Near $u = 0$, the equation can be linearized:

  \[\frac{\partial u}{\partial t} \approx D \frac{\partial^2 u}{\partial x^2} + r u\]

• Solutions grow exponentially if $u > 0$, leading to the invasion of the population.

Applications

• Population Dynamics: Modeling species invasion, range expansion, and the spread of epidemics.
• Genetics: Describing gene propagation in spatially structured populations.
• Ecology and Conservation Biology: Understanding the impact of habitat fragmentation and corridors on species spread.
• Chemical and Biological Waves: Studying flame propagation, neural activity, and reaction-diffusion systems.

Generalizations

• Higher Dimensions: Extension to two or three spatial dimensions:

  \[\frac{\partial u}{\partial t} = D \nabla^2 u + r u (1 - u)\]

• Heterogeneous Media: Incorporating spatially varying parameters $D(x)$ and $r(x)$.

• Time-Delayed Reactions: Introducing delays in the reaction term to model maturation time.

Model

This equation is a quadratic model

\[\dot{\mathbf{u}}(t) = \mathbf{Au}(t) + \mathbf{F}(\mathbf{u}(t) \oslash \mathbf{u}(t)) + \mathbf{Bw}(t)\]

Numerical Integration

For the numerical integration we consider two methods:

• Semi-Implicit Crank-Nicolson (SICN)
• Crank-Nicolson Adam-Bashforth (CNAB)

The time stepping for each methods are

SICN:

\[\mathbf{u}(t_{k+1}) = \left(\mathbf{I}-\frac{\Delta t}{2}\mathbf{A}\right)^{-1} \left\{ \left( \mathbf{I} + \frac{\Delta t}{2}\mathbf{A} \right)\mathbf{u}(t_k) + \Delta t\mathbf{F}\mathbf{u}^{\langle 2\rangle}(t_k) + \frac{\Delta t}{2} \mathbf{B}\left[ \mathbf{w}(t_{k+1}) + \mathbf{w}(t_k) \right]\right\}\]

CNAB:

If $k=1$

\[\mathbf{u}(t_{k+1}) = \left(\mathbf{I}-\frac{\Delta t}{2}\mathbf{A}\right)^{-1} \left\{ \left( \mathbf{I} + \frac{\Delta t}{2}\mathbf{A} \right)\mathbf{u}(t_k) + \Delta t\mathbf{F}\mathbf{u}^{\langle 2\rangle}(t_k) + \frac{\Delta t}{2} \mathbf{B}\left[ \mathbf{w}(t_{k+1}) + \mathbf{w}(t_k) \right]\right\}\]

If $k\geq 2$

\[\mathbf{u}(t_{k+1}) = \left(\mathbf{I}-\frac{\Delta t}{2}\mathbf{A}\right)^{-1} \left\{ \left( \mathbf{I} + \frac{\Delta t}{2}\mathbf{A} \right)\mathbf{u}(t_k) + \frac{3\Delta t}{2}\mathbf{F}\mathbf{u}^{\langle 2\rangle}(t_k) - \frac{\Delta t}{2}\mathbf{F}\mathbf{u}^{\langle 2\rangle}(t_{k-1}) + \frac{\Delta t}{2} \mathbf{B}\left[ \mathbf{w}(t_{k+1}) + \mathbf{w}(t_k) \right]\right\}\]

where $\mathbf{u}^{\langle 2 \rangle}=\mathbf{u} \oslash \mathbf{u}$.

Example

using CairoMakie
using LinearAlgebra
using PolynomialModelReductionDataset: FisherKPPModel

# Setup
Ω = (0, 3)  # Choose integers
T = (0.0, 5.0)
Nx = 2^8
dt = 1e-3
fisherkpp = FisherKPPModel(
    spatial_domain=Ω, time_domain=T, Δx=((Ω[2]-Ω[1]) + 1/Nx)/Nx, Δt=dt,
    params=Dict(:D => 1.0, :r => 1.0), BC=:dirichlet
)
DS = 10

# Create piecewise IC
a, b, c = (sort ∘ rand)(1:fisherkpp.spatial_dim, 3)
seg1 = ones(length(fisherkpp.xspan[1:a]))
seg2 = zeros(length(fisherkpp.xspan[a+1:b]))
seg3 = rand(0.1:0.1:0.5) * ones(length(fisherkpp.xspan[b+1:c]))
seg4 = zeros(length(fisherkpp.xspan[c+1:end]))
fisherkpp.IC = [seg1; seg2; seg3; seg4]

# Boundary conditions
Ubc1 = ones(1,fisherkpp.time_dim)
Ubc2 = zeros(1,fisherkpp.time_dim)
Ubc = [Ubc1; Ubc2]

# Operators
A, F, B = fisherkpp.finite_diff_model(fisherkpp, fisherkpp.params)

# Integrate
U = fisherkpp.integrate_model(
    fisherkpp.tspan, fisherkpp.IC, Ubc;
    linear_matrix=A, quadratic_matrix=F, control_matrix=B,
    system_input=true, integrator_type=:CNAB,
)

# Surface plot
fig3, _, sf = CairoMakie.surface(fisherkpp.xspan, fisherkpp.tspan[1:DS:end], U[:, 1:DS:end],
    axis=(type=Axis3, xlabel=L"x", ylabel=L"t", zlabel=L"u(x,t)"))
CairoMakie.Colorbar(fig3[1, 2], sf)
fig3

# Flow field
fig4, ax, hm = CairoMakie.heatmap(fisherkpp.xspan, fisherkpp.tspan[1:DS:end], U[:, 1:DS:end])
ax.xlabel = L"x"
ax.ylabel = L"t"
CairoMakie.Colorbar(fig4[1, 2], hm)
fig4

API

mutable struct FisherKPPModel <: AbstractModel

Fisher Kolmogorov-Petrovsky-Piskunov equation (Fisher-KPP) model

finite_diff_dirichlet_model(N, Δx, params)

finite_diff_mixed_model(N, Δx, params)

finite_diff_model(model, params)

finite_diff_periodic_model(N, Δx, params)

integrate_model(tdata, u0)
integrate_model(tdata, u0, input; kwargs...)

integrate_model_with_control_CNAB(
    tdata,
    u0,
    input;
    linear_matrix,
    quadratic_matrix,
    control_matrix,
    const_stepsize,
    u2_jm1
)

integrate_model_with_control_SICN(
    tdata,
    u0,
    input;
    linear_matrix,
    quadratic_matrix,
    control_matrix,
    const_stepsize
)

integrate_model_without_control_CNAB(
    tdata,
    u0;
    linear_matrix,
    quadratic_matrix,
    const_stepsize,
    u2_jm1
)

integrate_model_without_control_SICN(
    tdata,
    u0;
    linear_matrix,
    quadratic_matrix,
    const_stepsize
)