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For an infinitesimal test mass $\delta m$ on the surface of a homogeneous sphere, its Newtonian mechanical energy isThe spatially flat case corresponds to zero specific energy. Usingand dividing by $\delta m{R}^2/2$ givesSince $R$ is proportional to the scale factor $a$, this is the Newtonian derivation of the flat Friedmann equation

Now take $a(t_0)=1$ and $\rho ={\rho }_0{a}^{-4}$. PutThe contracting branch of the Friedmann equation isIntegrating $ada=-Cdt$ from $t_0$ givesThereforeon the contracting interval, reaching $a=0$ at $t=t_0+1/(2C)$.

In this discrete age-structured population model, individuals of age $a\ge 1$ present during the breeding season each produce $b_a$ newborns, so the number entering age zero isAn age-$a$ individual survives the following winter with probability $1-{\mu }_a$ and is then age $a+1$ in the next year, givingTogether with the initial age distribution, these equations determine every later cohort.

Substitute $n_{a,t}=r_a{\gamma }^t$ into the survival equation. Thenand iteration givesThe birth equation becomesA nonzero population mode therefore requires the Discrete Euler-Lotka equation

For nonnegative vital rates with at least one surviving reproductive age and enough decay to make the series finite, $\varphi$ is continuous and strictly decreasing on $\gamma >0$, tends to infinity as $\gamma \downarrow 0$, and tends to zero as $\gamma \to \infty$. The intermediate value theorem gives one positive solution, and strict monotonicity makes it unique.

The value $\varphi (1)$ is the expected lifetime offspring count under these fixed rates. Since $\varphi$ decreases, $\varphi (1)>1$ places the root at $\gamma >1$ and the population grows geometrically; $\varphi (1)<1$ places it at $\gamma <1$ and the population shrinks. Equality gives replacement, $\gamma =1$.

An exponential dispersion family has densities of the formwhere $\theta$ is the canonical parameter, $\varphi >0$ is the dispersion parameter, and the support does not depend on $\theta$. Its mean and variance are

For the given Gamma distribution, setThengivesThus the gamma laws form the Gamma exponential dispersion family.

Directly integrating, or using the cumulant-generating function of a gamma distribution,soDifferentiation at zero yields