Does Population Growth Depend on Individual Mortality?

The finite rate of increase, usually written as $\lambda$, measures how a population changes between two census times. Mortality affects it because deaths reduce the number of individuals that remain alive to be counted. To predict the population’s rate, however, you must consider mortality alongside reproduction and, in open populations, movement into and out of the population.

What the finite rate of increase measures

For population size $N_t$ at time $t$, the finite rate over one census interval is

$$\lambda_t = \frac{N_{t+1}}{N_t}.$$

If $\lambda>1$, the population has increased over the interval; if $\lambda=1$, it has stayed the same; and if $\lambda<1$, it has declined. When the rate is constant, population size follows $N_t=N_0\lambda^t$.

This is a net measure: it records the result of all the processes that change population size. In a closed population, births and deaths are the main processes. In an open population, immigration and emigration can also change the count. Thus, an observed $\lambda$ is not simply a mortality rate.

How mortality enters a simple model

Consider a closed, one-stage population counted once per interval. Let $s$ be the fraction of individuals that survive to the next census, and let $f$ be the average number of new individuals per starting individual that reach the population count by then. The projection is

$$N_{t+1}=sN_t+fN_t, \qquad \lambda=s+f.$$

Here, $f$ must count recruits that are present at the next census, not eggs or births that fail to survive until then. If mortality rises while $f$ stays fixed, survival $s$ falls, and so does $\lambda$. But the population may still grow if reproduction is high enough.

Worked example

Suppose a population has 100 individuals at the start of a year. Eighty percent survive to the next census, and each starting individual produces an average of 0.35 recruits that are counted then. The expected next count is

$$N_{t+1}=0.80(100)+0.35(100)=80+35=115.$$

Therefore, $\lambda=115/100=1.15$: the population has grown by 15% over that interval.

Now suppose mortality rises from 20% to 35%, so survival falls from $0.80$ to $0.65$, while recruitment stays at $0.35$ per starting individual. Then

$$N_{t+1}=0.65(100)+0.35(100)=100, \qquad \lambda=1.$$

The higher mortality has reduced the finite rate from 1.15 to 1.00. If recruitment were higher—for example, $f=0.50$—the same survival of $0.65$ would give $\lambda=1.15$. Mortality can therefore reduce growth without necessarily making the population decline.

Why age structure matters

The one-stage calculation treats individuals as if they have the same survival and reproductive contribution. Many populations do not work that way: juveniles and adults can have different survival probabilities, and reproduction may occur only at certain ages or stages. In those cases, demographers use a projection matrix whose entries represent survival and reproduction between classes.

For a population with projection matrix $A$ and class-count vector $\mathbf{n}_t$, the projection is $\mathbf{n}_{t+1}=A\mathbf{n}_t$. Survival probabilities appear in the matrix alongside reproductive rates. A change in mortality can therefore affect growth differently depending on which class experiences it. Increased mortality among individuals that have not yet reproduced may have a different effect from increased mortality among highly reproductive adults.

For a constant primitive projection matrix, the dominant eigenvalue gives the long-run finite rate of increase, and the class structure converges to its stable pattern. Before that pattern is reached, the observed ratio $N_{t+1}/N_t$ can vary as the age or stage distribution changes, even if the vital rates themselves have not changed.

Mortality and continuous-time growth

In a simple continuous-time model with constant per-capita birth rate $b$ and death rate $d$, the instantaneous growth rate is $r=b-d$. Over a time interval of length one, the corresponding finite rate is $\lambda=e^r$. This relationship applies to that simple model; it does not mean that a yearly mortality percentage can always be subtracted directly from an annual birth count. Rates and probabilities use different definitions and units.

Common mistakes

• Treating mortality as the only driver. A population can grow despite substantial mortality if enough individuals reproduce and recruits survive.
• Confusing an individual death rate with $\lambda$. Mortality describes deaths; $\lambda$ describes the net change in population size over a census interval.
• Ignoring the census schedule. Whether births and deaths are counted before or after a census affects which individuals and recruits contribute to the next population count.
• Assuming every death has the same demographic effect. The effect of mortality depends partly on the age or stage of the individuals affected and their expected future reproduction.
• Ignoring migration or density dependence. Immigration can raise the observed rate, while changing competition can alter survival and reproduction as population size changes.