Closer, but I was really hoping for something more like
r0 = 1/2;
foo1 = 0;
foo2 = 1;
w = NDSolve[{D[u[r,thet,t],t] == D[u[r,thet,t],{r,2}]+(1/r) D[u[r,thet,t],r]+(1/r^2) D[u[r,thet,t],{thet,2}] +
u[r,thet,t](1-u[r,thet,t]-c*v[r,thet,t]),
D[v[r,thet,t],t] == mu*(D[v[r,thet,t],{r,2}]+(1/r)D[v[r,thet,t],r]+(1/r^2) D[v[r, thet, t], {thet, 2}]) +
a*v[r, thet, t] (1 - c*u[r, thet, t] - b*v[r, thet, t]),
u[0.01, thet, t] == thet,
u[1, thet, t] == thet,
u[r, 0, t] == r,
u[r, 2 Pi, t] == r,
v[0.01, thet, t] == thet,
v[1, thet, t] == thet,
v[r, 0, t] == r,
v[r, 2 Pi, t] == r,
u[r, thet, 0] == foo1,
v[r, thet, 0] == foo2},
{u, v}, {t, 0, 1}, {thet, 0, 2 Pi}, {r, 0.01, r0}, PrecisionGoal -> 2]
which has hopefully removed anything confusing about the boundary conditions.
That results in
NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.`. >>
NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.`. >>
which strongly hints to me that there is a zero in a denominator somewhere, either explicitly or resulting from the processing of your derivatives or you are using a variable without having assigned it a constant value..
That leads me to discovering you have used mu without assigning a value to it. I assign 1 to mu and try again. That results in
NDSolve::bcedge: Boundary condition u[1,thet,t]==thet is not specified on a single edge of the boundary of the computational domain. >>
And I notice you have used a,b,c without assigning any values to those eiter. I assign 1 to each of those. With NDsolve you can't use variables that haven't been assigned values unless they are the functions you are solving for or the parameters for those.
That still gives me the same error. But this error is perhaps more specific and easier to track down. Can you make any progress on resolving this?
If I'd been able to solve the first problem then I would have followed up with
r0 = 1/2;
foo1 = 1;
foo2 = 0;
mu = 1;
a = 1;
b = 1;
c = 1;
w = NDSolve[{D[u[r,thet,t],t] == D[u[r,thet,t],{r,2}]+(1/r)D[u[r,thet,t],r]+(1/r^2)D[u[r,thet,t],{thet,2}] +
u[r,thet,t](1-u[r,thet,t]-c*v[r,thet,t]),
D[v[r,thet,t],t] == mu*(D[v[r,thet,t],{r,2}]+(1/r)D[v[r,thet,t],r]+(1/r^2)D[v[r,thet,t],{thet,2}]) +
a*v[r,thet,t](1-c*u[r,thet,t]-b*v[r,thet,t]),
u[0.01, thet, t] == thet,
u[1, thet, t] == thet,
u[r, 0, t] == r,
u[r, 2 Pi, t] == r,
v[0.01, thet, t] == thet,
v[1, thet, t] == thet,
v[r, 0, t] == r,
v[r, 2 Pi, t] == r,
u[r, thet, 0] == foo1,
v[r, thet, 0] == foo2},
{u, v}, {t, 0, 1}, {thet, 0, 2 Pi}, {r, r0, 1}, PrecisionGoal -> 2]
and then hoped I might have been able to stitch together the two solutions. Surprisingly, this actually gives a solution, but it still complains about inconsistent conditions. But this hints I may have gotten away from your zero. The warning is probably based on the same problem as the first case. If you can find and fix that you may be close.
Can you make any sense of this and make any more progress? Or perhaps someone else can see some problem with your conditions and point out how to fix this.