Mathematical biology

Logistic travelling waves

A classic example

$\pd{u}{t}=\nabla^2 u+ru\left(1-\frac{u}{K}\right)$
with periodic boundary conditions

Survival in a harsh environment

Exploring the effects of boundary conditions

$\pd{u}{t}=\nabla^2 u+u\left(1-\frac{u}{K}\right)$
with no-flux boundary conditions

Bistability: invasion and persistence

Spatial Allee effects and changing wavespeeds

$\pd{u}{t}=D\nabla^2 u +u(u-a)(1-u)$

Schnakenberg pattern formation

Spots and stripes

$\pd{u}{t}=\nabla^2 u+a-u+u^2v$, $\pd{v}{t}=D\nabla^2v+ b-u^2v$

Brusselator pattern formation

Turing instability regions

$\pd{u}{t}=\nabla^2 u+a-(b+1)u+u^2v$, $\pd{v}{t}=D\nabla^2v+ bu-u^2v$

Gierer–Meinhardt pattern formation

Spots and stripes

$\pd{u}{t}=\nabla^2 u+a+\frac{u^2}{v}-bu$, $\pd{v}{t}=D\nabla^2v+ u^2-cv$

FitzHugh–Nagumo and excitability

Patterns, spiral waves, and chaos

$\pd{u}{t}=\nabla^2 u +u-u^3-v$, $\pd{v}{t}=D\nabla^2v+ \varepsilon_v(u-a_v v-a_z)$

Keller–Segel chemotaxis

Slime moulds and cell movement

$\pd{u}{t}=\nabla^2 u-\vnabla \cdot(\chi(u)\vnabla v)+f_u(u)$, $\pd{v}{t}=D\nabla^2v+ f_v(u,v)$

Reaction–cross-diffusion systems

Dark solitons and extended Turing spaces

$\pd{u}{t}=\vnabla\cdot(D_{uu}\vnabla u+D_{uv}\vnabla v)+a-u+u^2v,$ $\pd{v}{t}=\vnabla\cdot(D_{vu}\vnabla u+D_{vv}\vnabla v)+b-u^2v$

Cyclic competition models

Rock, paper, scissors – spiral waves!

$\pd{u}{t}=D_u\nabla^2 u+u(1-u-av-bw)$ $\pd{v}{t}=D_v\nabla^2 v+v(1-bu-v-aw)$ $\pd{w}{t}=D_w\nabla^2 w+w(1-au-bv-w)$

Dynamic bifurcations with heterogeneity

Isolated patterns and moving spikes

$\pd{u}{t}=\nabla^2 u+a+G(x)+\frac{u^2}{v}-[b+H(x)]u,$ $\pd{v}{t}=D\nabla^2v+ u^2-cv$

Banded vegetation patterns

Stripes moving towards water

$\pd{w}{t} = a-w -wn^2+v\pd{w}{x} + \nabla^2w$, $\pd{n}{t} = wn^2 - mn + \nabla^2n$

Turing instabilities are not enough

Beyond the limits of linear theory

Hills and valleys

Effects of topography on models of water and vegetation

Bacteria concentration in flow

A toy model of decay and advection in rivers