Population size at time $t$. I want to make a formula for the population size $N(t)$, 
with the following ingredients: Let $N_t$ be the random variable denoting number of individuals at time $t$. I then call $$ N(t) =\mathbb{E}[N_t]=\int_{0}^{\infty}\mathbb{P}(N_t>z)dz$$
Let $N_0$ the the population-size at time $0$. Furthermore let $B(t)$ be the birthrate at time $t$, and $\pi(a',a)$ the probability that an individual of age $a$ survives upto $a'\geq a$. Assume that $$\lim_{a'\to\infty}\pi(a',a)=0 $$ Also let $n_0(a)$ be the initial age distribution. 
So far all my attempts to write down a formula failed...I think im missing some simple logical insight. 
So actually I want to have a formula for $\mathbb{P}(N_t>z)$ for all $z>0$.
 A: If you just want $N(t)$ given $B(t)$, you don't need to calculate the actual distribution of $N_t$.  Instead, just integrate the birth rate $B(t)$ from $0$ to $t$, multiplied by the probability of each born individual surviving up to time $t$:
$$N(t) = \int_0^t B(\tau)\,\pi(t-\tau,0)\,d\tau + N_0(t),$$
where $N_0(t)$ is the expected number of individuals from the initial population still surviving at time $t$:
$$N_0(t) = \int_0^\infty n_0(a)\,\pi(a+t,a)\,da.$$
Note that this is all assuming that the birth distribution $B(t)$ is given a priori.  If it is instead a function of the population size, things get a lot more complicated.  However, in the special (and somewhat unrealistic) case where the birth rate is a linear function of the actual population size $N_t$, we can use the linearity of the expected value operator to express it as a function of the expected population size $N(t)$ instead (e.g. as $B(t) = m(t) + b(t)N(t)$ where $m(t)$ and $b(t)$ are respectively the immigration rate and the per capita birth rate at time $t$) and obtain a delay equation of the form:
$$N(t) = \int_0^t (m(\tau)+b(\tau)N(\tau))\,\pi(t-\tau,0)\,d\tau + N_0(t),$$
In general, the solution to such equations depends strongly on the survival kernel $\pi$.  For some simple choices, like $\pi(a',a)=e^{\mu(a-a')}$, the delay equation above is actually equivalent to a simple ordinary differential equation, in this case:
$$\frac{d}{dt}N(t) = m(t) + (b(t)-\mu)N(t).$$
However, as far as I know, there's no easy and straightforward way to solve this problem for arbitrary $\pi$; in general, you may have to resort to approximate numerical integration.
