# 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 $z>0$.

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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.

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Wow, that was a really good answer. Thanks a lot. You mustve studied population dynamics ;) – DinkyDoe Jan 19 '13 at 23:02
I have another question though. How should I interpret the second, and especially the first integral. Are these expected fractions? So that $N(t)$ is actually $N_0* N(t)$ cause $N(t)$ must depend on $N_0$ right? – DinkyDoe Jan 20 '13 at 18:11
With the birth distribution $B(t)$ fixed a priori, the initial population $n_0(a)$ only affects the expected population size $N(t)$ via the $N_0(t)$ term, which simply gives the expected number of surviving members of the initial population. This is because, by fixing $B(t)$, we've effectively decoupled the births from the population state. If that seems kind of absurd, that's because it is -- in reality, new individuals are born out of old individuals, and so $B(t)$ should be a function of the population size, as in the latter part of my answer. – Ilmari Karonen Jan 20 '13 at 19:38
(continued) The problem is that, with arbitrary $\pi(a',a)$ and $B(t)$ an arbitrary function of $N_t$ (or, worse yet, the population age distribution $n_t(a)$ at time $t$), what you have is a general age-structured population model with possibly non-linear density dependence. The nice thing about such models is that they can capture a lot of realistic features of population dynamics; the not so nice thing is that, in general, it's hard to get any analytical results out of them without making major simplifications. – Ilmari Karonen Jan 20 '13 at 19:43