Stable and Stationary Population Theory (Lotka System)

Before examining the general stable model where populations grow or shrink, we examine the foundational baseline of zero growth: the Stationary Population.

Definition of a Stationary Population

A Stationary Population is a special demographic state characterized by two defining conditions:

1. The total population size remains strictly constant over time: $\frac{dN}{dt} = 0$, meaning the annual rate of natural increase is identically zero ($r = 0$).
2. The age-sex distribution is completely unchanging over time, and annual births exactly equal annual deaths: $B(t) = D(t) = B_0 = \text{constant}$.
3. Net international migration is zero (closed population).
4. The Net Reproduction Rate is exactly unity: $NRR = 1.000$.

The Equivalence Between Life Table Functions and the Stationary Population

A remarkable mathematical isomorphism exists between the life table of Unit 4 and a stationary population: the life table columns describe the exact physical composition of a living stationary population.

Let a stationary population be generated by a continuous stream of $l_0$ annual births ($B = l_0$) evenly distributed across each calendar year, subjected to unchanging life table survival schedule $l(x)$:

• The number of individuals surviving to celebrate exact birthday $x$ in any calendar year is $l_x$.
• The number of deaths occurring between ages $x$ and $x+n$ during any calendar year is $_n d_x$.
• The enumerated population living in age group $[x, x+n)$ at any given moment is exactly equal to the life table person-years column:

• The total population of the stationary state is the cumulative person-years column at birth:

Fundamental Demographic Identities in a Stationary Population

From the relationship $N_{\text{stat}} = l_0 \cdot e_0$:

1. The Birth Rate of a stationary population is:

1. The Death Rate of a stationary population is:

1. The Rate of Natural Increase is:

In a stationary population, the crude birth rate and the crude death rate are identically equal to the mathematical reciprocal of life expectancy at birth:

For example, in a stationary population with a life expectancy of $e_0 = 80$ years:

If life expectancy is $e_0 = 50$ years, the birth and death rates must both equal $20.0$ per $1{,}000$.

In 1907, Alfred J. Lotka generalized the stationary model to populations undergoing continuous exponential growth or decline ($r \ne 0$).

Core Postulates of the Stable Population Model

1. The population is closed to migration ($M = 0$).
2. The force of mortality $\mu(x)$ (and hence survival function $p(x) = l(x)/l_0$) is fixed and invariant over time.
3. The force of fertility $m(x)$ (age-specific birth rate for female births) is fixed and invariant over time.

The Integral Equation of Birth Trajectories

Let $B(t)$ denote the continuous number of births occurring in the population at time $t$. Any birth occurring at time $t$ must be born to a mother aged $x \in [\alpha, \beta]$ (where $\alpha = 15$ and $\beta = 50$ are the physiological boundaries of reproduction). A mother who is aged $x$ at time $t$ was herself born at time $t - x$. The number of surviving women aged $x$ at time $t$ is:

Multiplying by the age-specific fertility rate $m(x)$ and integrating across all reproductive ages yields Lotka's Renewal Equation:

The Stable Exponential Solution

Lotka postulated that as time $t \to \infty$, the birth trajectory converges to an exponential curve:

where $r$ is the constant intrinsic rate of natural increase. Substitute this trial solution into the renewal equation:

Divide both sides by $B_0 e^{rt}$:

This is the celebrated Lotka Characteristic Equation (Euler–Lotka equation). It defines an implicit non-linear relationship between the biological vital rate schedules $\{p(x), m(x)\}$ and the intrinsic rate of growth $r$.

In a stable population, what proportion of the population is aged $x$?

Let $N(t)$ denote total population size at time $t$. Under continuous exponential growth:

The number of individuals alive at exact age $x$ at time $t$ is the survivors of those born $x$ years ago:

The proportion of the population at exact age $x$, denoted by $c(x) = \frac{N(x, t)}{N(t)}$, is:

where $b = \frac{B(t)}{N(t)}$ is the constant Intrinsic Birth Rate.

Normalization and the Intrinsic Birth Rate

Because $c(x)$ is a probability density function across all ages $[0, \infty)$, it must integrate to unity:

Substitute $c(x) = b \, e^{-rx} p(x)$:

Solving for the Intrinsic Birth Rate:

The Intrinsic Death Rate ($d$) follows from demographic accounting:

Comparison of Age Distributions:

• If $r = 0$ (Stationary): $c(x) = b \cdot p(x) = \frac{l(x)}{T_0}$. The age distribution is shaped entirely by biological mortality $p(x)$.
• If $r > 0$ (Growing Stable Population): The factor $e^{-rx}$ acts as an exponential discount factor with age. The age distribution is tilted toward younger ages, expanding the base of children and young adults (young age structure).
• If $r < 0$ (Declining Stable Population): The factor $e^{-rx} = e^{|r|x}$ expands with age, tilting the age distribution toward older cohorts and accelerating population aging.

The characteristic equation $\int_0^{\infty} e^{-rx} p(x) m(x) dx = 1$ cannot be solved in closed elementary form. To solve for $r$ numerically, Lotka expanded the exponential $e^{-rx}$ using Taylor series around $r = 0$.

Define the Net Maternity Function $\phi(x) = p(x) m(x)$. The $k$-th moment of the net maternity function about age zero is:

Notice that:

• $R_0 = \int_0^{\infty} \phi(x) \, dx = NRR$ (The Net Reproduction Rate!).
• $R_1 = \int_0^{\infty} x \, \phi(x) \, dx$ is the first moment of childbearing age.
• $\mu = \frac{R_1}{R_0} = T$ is the Mean Age of Childbearing in the Stationary Population (mean generation length).
• $\sigma^2 = \frac{R_2}{R_0} - \mu^2$ is the variance of age at childbearing.

Taylor Series Expansion of the Characteristic Equation

Expanding $e^{-rx} = 1 - rx + \frac{1}{2} r^2 x^2 - \dots$:

Divide by $R_0 = NRR$:

Taking the natural logarithm of both sides and using $\ln(1 - z) \approx -z - \frac{z^2}{2}$:

Lotka's Second-Order Approximation for $r$:

Neglecting the variance term $\frac{1}{2} r^2 \sigma^2$ yields the first-order approximation:

Substituting $r_1$ into the quadratic term yields Lotka's celebrated second-order formula:

This quadratic correction provides exceptional numerical accuracy, matching iterative numerical root-finding algorithms to within four decimal places.

In developing countries lacking reliable vital registration, Ansley J. Coale and Paul Demeny (1966, 1983) utilized stable population theory to create the Coale–Demeny Regional Model Life Tables and Stable Populations.

Coale and Demeny assembled thousands of empirical mortality schedules and identified four distinct mortality families based on historical European patterns:

1. North Family: High infant mortality and low old-age mortality.

2. South Family: Elevated adult mortality and high cardiovascular burden.

3. East Family: High tuberculosis and infectious respiratory mortality.

4. West Family: The general, balanced mortality pattern representing standard global mortality.

By crossing each mortality family across twenty-four levels of life expectancy ($e_0$ from 20 to 80 years) and thirteen intrinsic growth rates ($r$ from $-1\%$ to $+4\%$), Coale and Demeny tabulated complete sets of model stable age distributions. When an analyst conducts a census in a developing nation and observes only the proportion of children under age 15 ($P_{0-14}$) and the intercensal growth rate $r$, the analyst can look up the corresponding model stable table to deduce the unobserved life expectancy at birth $e_0$, crude birth rate $b$, and infant mortality rate $q_0$ without vital registration!

Ronald Fisher's Reproductive Value and Evolutionary Demography

A profound theoretical bridge connecting stable population theory with evolutionary biology and life history theory is the concept of Reproductive Value, formulated by the mathematical geneticist Ronald A. Fisher in 1930 in The Genetical Theory of Natural Selection.

While the life table survival function p(x) measures an individual's probability of surviving to age x, and age-specific fertility m(x) measures childbearing intensity at age x, Fisher posed a fundamental demographic question: How much does an individual of exact age x contribute to the future ancestry and genetic lineage of the population?

Mathematical Derivation of Fisher's Reproductive Value

Let a female population grow at constant intrinsic rate r with stable age distribution. Consider a cohort of females alive at exact age x. The expected future offspring that will be born to a female who has already attained age x, discounted back to age x at the continuous rate of population growth r, is defined as the Reproductive Value v(x):

where the ratio v(x) / v(0) is normalized so that the reproductive value of a newborn female infant at birth is unity: v(0) = 1.

Biological and Demographic Trajectory of Reproductive Value

When v(x) is plotted against age x, it exhibits a dramatic unimodal trajectory that differs fundamentally from both fertility and survival schedules:

1. At Birth (x = 0): By definition, v(0) = 1. A newborn girl possesses reproductive value, but her value is discounted because she faces significant risks of dying during infancy and childhood before reaching childbearing age, and her eventual reproduction lies fifteen to thirty years in the future.

2. During Childhood (Ages 1 to 14): As a girl successfully survives childhood mortality risks, her reproductive value rises continuously! By surviving the hazards of early life, she draws closer to the reproductive age span without having died, maximizing her expected future contribution.

3. At Puberty and Menarche (Ages 18 to 22): Reproductive value reaches its absolute lifetime maximum v_max, often equaling 2.5 to 3.5 times the value at birth. A young woman at age 20 has survived all childhood mortality hurdles and has her entire reproductive career immediately ahead of her.

4. Throughout Adulthood (Ages 25 to 49): As a woman gives birth to children, her remaining reproductive potential is depleted. With each passing birthday, fewer childbearing years remain, causing v(x) to decline monotonically.

5. At Menopause (Age 50+): At age 50, childbearing terminates entirely (m(x) = 0 for x >= 50), and direct reproductive value drops to exactly zero: v(x) = 0 for all x >= 50.

The Grandmother Hypothesis and Kin Selection

Why do human females experience menopause and survive for decades with zero direct reproductive value, a trait virtually unique among mammalian species? Evolutionary demographers (such as Kristen Hawkes) formulate the Grandmother Hypothesis: post-reproductive older women maximize their inclusive fitness by providing energetic subsidies, food foraging, and child-care assistance to their adult daughters. By reducing the physical burden of nursing and childcare on their daughters, grandmothers enable their daughters to shorten inter-birth intervals and bear more grandchildren, ensuring that post-reproductive longevity possesses immense indirect reproductive value under natural selection.

Coale's Sensitivity Analysis of Stable Populations to Fertility vs Mortality Changes

A celebrated question in theoretical demography, thoroughly investigated by Ansley J. Coale in 1957 and 1972, is: Which demographic vital process exerts the dominant influence on population aging—declining fertility or declining mortality?

Popular intuition frequently assumes that population aging (the expanding proportion of senior citizens) is caused by people living longer (declining mortality). However, Coale deployed stable population mathematics to prove that this intuition is largely mistaken: historically, population aging is driven almost entirely by declining fertility, not declining mortality!

Coale's Mathematical Sensitivity Derivation

Recall the stable age distribution equation:

Consider a proportional reduction in mortality that improves life expectancy at birth. In historical mortality declines (such as the control of infectious diseases in childhood), mortality reductions were concentrated among infants and young children (ages 0 to 5). When child survival improves, more newborn infants survive childhood to enter the base of the pyramid. Mathematically, improving infant survival acts as an effective increase in the birth rate! By expanding the number of surviving children, historical mortality reductions actually made populations younger, widening the base of the pyramid rather than aging it.

Now consider a reduction in fertility. When the Total Fertility Rate declines, the birth rate b drops, and the intrinsic growth rate r declines. In the stable age distribution:

The reduction in r causes the exponential discount factor e^{-rx} to become flatter across all ages x. The proportion of children at the base shrinks immediately, which arithmetically forces the proportion of adults and senior citizens at older ages to expand.

Coale proved through comparative stable population matrices that a thirty percent reduction in fertility produces a massive upward shift in median age and more than doubles the proportion of persons aged sixty-five and over. In contrast, a mortality decline that increases life expectancy by twenty years (when concentrated in early life) leaves the proportion of elderly citizens virtually unchanged. Only in modern Stage 4 societies, where mortality reductions are concentrated almost exclusively among individuals aged seventy and older, does mortality decline contribute significantly to population aging.

Convergence Rates and Numerical Spectral Radius of the Stable System

How quickly does an actual human population converge to its stable age distribution? In 1968, Nathan Keyfitz demonstrated that the speed of convergence is governed by the damping ratio:

where lambda_1 is the dominant real eigenvalue and lambda_2 is the complex eigenvalue with the next largest modulus. In typical human populations, lambda_1 is between 1.00 and 1.25 (for five-year steps), while |lambda_2| is approximately 0.60 to 0.75. The damping ratio is therefore typically between 1.4 and 1.8. The transient cyclical components decay as (1 / d)^t. Because (1 / 1.5)^4 = (2/3)^4 ~ 0.20, after four generations (approximately one century), more than eighty percent of the initial age-structural distortion has evaporated. This quantitative decay rate guarantees that stable population approximations remain highly reliable even in populations that experienced significant historical mortality or fertility shocks.