Recently, I was reading the article "Simulating Spatial Differenetial Equations with Cellular Automata". On page 2, there is a general form that is said to hold for most biological ODE equations, namely: $$ f(u_{i,j})=\frac{\partial u}{\partial t}=m(u_{i,j})+\nabla_x^2 n(u_{i,j})+\nabla_x o(u_{i,j})~~~(4). $$
When I see it right, we have that $u=u(t,x)$ is a function of time $t$ and one dimensional space $x$.
So I really do not understand what is meant by the notation $u_{i,j}$, i.e. with time index $i$ and space index $j$.
Moreover, if $x$ is representing one-dimensional space, why don't we just write $$ \nabla_x^2 n(u_{i,j})=\frac{\partial^2}{\partial x^2}n(u_{i,j}),\qquad\nabla_x o(u_{i,j})=\frac{\partial}{\partial x}o(u_{i,j})? $$