Why No Population Can Keep Growing Forever

A single tiger cannot have a birth rate or a sex ratio, but the tigers of a forest can. Populations have properties that individuals lack, and studying them shows how numbers change and why species reproduce the way they do.

This part covers population attributes, age pyramids and density, population growth, and life history variation.

A population is a group of individuals of the same species living in a well-defined area and sharing or competing for the same resources, and it has attributes such as population density, birth rate, death rate, sex ratio and age distribution that belong to the group, not to any single individual.

Population attributes:

- Birth rate — births per individual per unit time
- Death rate — deaths per individual per unit time
- Sex ratio — the proportion of females and males
- Age distribution — the proportion of individuals of different ages
- Population density — the number of individuals in a given area or volume

Worked example. A pond has $20$ lotus plants at the start of a season, and reproduction adds $8$ more. The birth rate is

$$\frac{8}{20}=0.4\text{ offspring per lotus plant per season}$$

If $4$ of $40$ fruit flies in a laboratory die within a week, the death rate is $\frac{4}{40}=0.1$ per fly per week.

An everyday example. Census figures on India's sex ratio describe the population as a whole, since no single person has a sex ratio.

The substance. Birth and death rates are expressed per individual, which lets populations of very different sizes be compared.

An age pyramid shows the proportions of pre-reproductive, reproductive and post-reproductive individuals, and its shape — a broad base for an expanding population, a bell for a stable one and a narrow base for a declining one — reveals growth status; density can be measured as numbers, biomass, per cent cover or through indirect signs.

Age pyramids:

- Expanding — triangular, with a broad base of young individuals
- Stable — bell-shaped, with pre-reproductive and reproductive groups nearly equal
- Declining — urn-shaped, with a narrow base of young individuals

Measuring population density:

- Numbers — counting individuals, the simplest measure
- Biomass — more meaningful when individuals differ greatly in size, as with one huge banyan tree against many small Parthenium plants
- Per cent cover — the area covered by plants
- Indirect estimates — for animals that are hard to count, such as tigers, using pug marks and faecal pellets
- Relative density is often enough, such as the number of fish caught per trap

An everyday example. Surveys in Indian tiger reserves estimate tiger numbers from pug marks and camera traps rather than direct counts.

The substance. Numbers alone can mislead — a few hundred Parthenium plants and a single banyan tree carry very different amounts of living matter.

A population grows through births and immigration and shrinks through deaths and emigration; with unlimited resources it grows exponentially in a J-shaped curve, but with limited resources it grows logistically in an S-shaped curve that levels off at the carrying capacity, K.

Growth equation:

$$N_{t+1}=N_t+[(B+I)-(D+E)]$$

where B is births, I immigration, D deaths and E emigration.

Exponential growth:

- Happens when resources are unlimited
- $\frac{dN}{dt}=rN$, giving $N_t=N_0{e}^{rt}$, where r is the intrinsic rate of natural increase
- Produces a J-shaped curve

Logistic growth:

- Happens when resources are limited, as in nature
- Growth slows as the population nears the carrying capacity, K
- $\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)$, giving an S-shaped curve with lag, acceleration, deceleration and a plateau at K

Worked example. A population with $N=200$, $r=0.1$ per week and $K=1000$ grows at

$$\frac{dN}{dt}=0.1\times 200\times \frac{1000-200}{1000}=16\text{ individuals per week}$$

Under exponential growth it would grow at $0.1\times 200=20$ individuals per week.

An everyday example. Bacteria in milk left in a warm kitchen multiply rapidly at first, but growth slows as nutrients run out and waste builds up.

The substance. The logistic model is more realistic, because no environment offers unlimited resources for long.

Populations evolve life history traits that maximise their reproductive fitness in a given habitat: some organisms breed only once in their lifetime, such as Pacific salmon and bamboo, while most birds and mammals breed many times, and some species produce many small offspring while others produce a few large ones.

Breeding once versus many times:

- Breeding once — Pacific salmon and bamboo reproduce a single time and then die
- Breeding many times — most birds and mammals reproduce repeatedly over their lifetime

Many small versus few large offspring:

- Many small offspring — oysters and pelagic fishes release vast numbers of tiny young, most of which die
- Few large offspring — birds and mammals produce fewer, larger young and invest more parental care

Darwinian fitness. Each strategy is shaped by the abiotic and biotic factors of the habitat, so that the organism leaves the greatest number of surviving offspring under its own conditions.

An everyday example. A bamboo grove flowering all at once and then dying is a striking sight in the forests of north-east India.

The substance. Neither strategy is better in general — each maximises fitness only in its own kind of environment.