For D = 0,
we have: a(n) = a(n-1) + a(n-0)
Which is a(n) = a(n-1) + a(n)
i.e. 0 = a(n-1) + 0
Which is just like saying a(n) = 0
So in that case you're right, a sequence of only zeros, as to what that means, I'm not sure, I guess the model breaks down a bit here.

For D < 0 interesting things happen.
For the sake of example lets say D = -1,
then we have: a(n) = a(n-1) + a(n+1)
Which is just like saying: a(n) = a(n-1) - a(n-2)
But we know that a(n-1) = a(n-2) - a(n-3)
Which means: a(n) = -a(n-3)
Therefore: a(n) = a(n-6)
So the sequence will be periodic with period 6.

Looking at the model what we've got here is that each generation has a population of the last generation's birth rate. This is problematic because positive birth rates will tend to fall in this system, leading to negative populations. This then becomes periodic because a negative birth rates of negative populations will leads more fewer negative individuals, i.e. greater populations.

I'm not sure this makes any sense to be honest. I suggest abandoning the model at this point!