Turing instabilities are not enough
Linear stability theory is often used to predict regions of pattern-forming (or ‘Turing’) instabilities. However, in the presence of multiple homogeneous equilibria, these instabilities do not guarantee that a system develops a pattern.
Here, we implement interactive versions of three local models in the paper “Turing instabilities are not enough to ensure pattern formation.”
In each case, the default is a 2D spatial domain with periodic boundary conditions. The boundary conditions can be modified by clicking → Boundary conditions, and the domain can be changed to a 1D interval by clicking → Domain and setting the dimension to be 1.
By default, an initial small random perturbation of a homogeneous equilibrium is used to generate a Turing instability which eventually leads to the solution approaching a different homogeneous equilibrium. You can alternatively click to introduce a localised perturbation, or directly input a different initial condition. Importantly all parameters and functional forms can also be changed.
Below we highlight specific parameters that give different dynamics.
Reaction–diffusion system
This reaction–diffusion simulation explores the system
\[\begin{aligned} \pd{u}{t}&=\nabla^2 u+u-v-eu^3,\\ \pd{v}{t}&=D\nabla^2 v+ a v(v + c)(v - d) + b u - e v^3. \end{aligned}\]Keller–Segel chemotaxis
This Keller–Segel simulation corresponds to the equations
\[\begin{aligned}\pd{u}{t} &= \nabla^2 u - c\vnabla \cdot(u\vnabla v) +u(b - u)(u - d),\\ \pd{v}{t} &= D \nabla^2 v + u-av. \end{aligned}\]Localised solutions can be found by setting $c=5$ and $d=0.1$. Reducing $d$ further to $0.01$ for this value of $c$ leads to spatiotemporal behaviour similar to using the purely logistic demographic term as in this alternative model.
Biharmonic equation
This biharmonic simulation corresponds to the equation
\[\pd{u}{t} = - D\nabla^2 u - \nabla^4 u + au(c - u)(u - b).\]Setting $D=1.87$ will instead lead to localised states that are near the boundary of stability. These will decay slowly for $D\leq 1.85$ but appear to remain stable for $D=1.87$.