Parabolic differential equations with time delay

Question

Let $d_1=1,d_2=2,a_{11}=\frac{5}{13},a_{12}=\frac{22}3,a_{21}=-2,a_{22}=\frac{6}7,\tau=\frac{5}7$, $\psi(t,x)=\cos^42x,\phi(t,x)=\frac{3}{13}x^4\sin^2 3x$, $\Omega=[0,200]$

How to solve:

$$\left\{ \begin{array}{lc} \dfrac{\partial u(t,x)}{\partial t}=d_1\triangle u(t,x)+u(t,x)\left(r_1-a_{11}u(t-\tau,x)-a_{12}v(t,x)\right),& t>0,x\in\Omega \\ \dfrac{\partial v(t,x)}{\partial t}=d_2\triangle v(t,x)+v(t,x)\left(-r_2+a_{21}u(t,x)-a_{22}v(t,x)\right),& t>0,x\in\Omega\\ \dfrac{\partial u}{\partial n}=\dfrac{\partial v}{\partial n}=0,\quad t\ge0,x\in\partial\Omega \quad(\text{Neumann conditions})\\ u(t,x)=\phi(t,x)\ge 0,\qquad v(t,x)=\psi(t,x)\ge 0, &(t,x)\in[-\tau,0]\times\Omega \end{array} \right.$$

Answer 1

Do the same discretization that you normally do for the nonlinear Heat Equation to turn $\Delta$ into $A$, the Strang matrix second order discretization (the [1 -2 1] tridiagonal matrix). Now you have a system of DDEs. Use a DDE solver on this. MATLAB's DDE23, Julia's DDE solvers, etc.