(i) Birth rate and death rate. An individual may have births and deaths, but a population has birth rates and death rates. These are per capita figures — the change in numbers with respect to the members of the population — so they require a number of individuals to divide by. A single lotus plant either produces offspring or does not; the figure 0.4 offspring per lotus per year describes a group.

(ii) Sex ratio. An individual is either a male or a female. A ratio such as 60 per cent females and 40 per cent males can exist only across a collection of individuals.

(iii) Age distribution. An individual has one age at a time. A distribution is the per cent of individuals in each age or age group, plotted as an age pyramid, whose shape shows whether the population is growing, stable or declining.

(iv) Population density (N). Density is a property of a group in an area — measured as total number, per cent cover, biomass or a relative index. A single organism has no density.

The general principle. Each of these is a statistical property: it emerges only from many individuals considered together. This is why natural selection, although it acts on individuals, is studied at the population level — and why population ecology links ecology to population genetics and evolution.

The distinction. The how-type questions seek the mechanism behind a process; the why-type questions seek the significance of the process.

Applied to the cactus. How does a cactus have thorns? The answer is developmental and anatomical: the thorns are modified leaves, formed from leaf primordia whose growth is arrested and whose cells become sclerified rather than expanding into a blade. Why does a cactus have so many thorns? The answer is ecological and evolutionary: thorns are the most common morphological means of defence against herbivores, and a plant in a dry habitat that cannot afford to lose tissue or water benefits doubly, since reduced leaf area also reduces transpiration.

Why both are needed. The mechanism without the significance leaves the fact arbitrary — you would know how a thorn is built but not why any plant should build one. The significance without the mechanism leaves the fact unexplained — you would know what thorns are for but not how a leaf becomes one, nor how the trait could change. Biology is unusual among sciences in having both kinds of answer available for the same observation, because living things have a history as well as a machinery.

The chapter's other examples work the same way: why are night-blooming flowers generally white (visibility to nocturnal pollinators) versus how does a bee know which flower has nectar (scent and visual cues and learning).

Why density is the right measure. Whatever ecological process we wish to investigate in a population — the outcome of competition with another species, the impact of a predator, or the effect of a pesticide application — we always evaluate it in terms of any change in population size. The number killed is not the outcome that matters: a pesticide that kills a million insects has failed if the survivors reproduce back to the same density within a fortnight, and a small kill may matter greatly if it holds the population below the level at which it damages the crop. Density also allows before-and-after comparison, and comparison between fields of different size.

How they would measure it. Rarely by counting every individual. Population size need not necessarily be measured in numbers only, and we are mostly obliged to estimate population sizes indirectly, without actually counting or seeing them. Reasonable choices here would be a relative density — the number of insects caught per trap per night, on the same principle as the number of fish caught per trap in a lake — or counts on a fixed number of sampled plants, or the proportion of plants showing damage.

What must be controlled. Density changes anyway, because of natality, mortality, immigration and emigration, so an untreated control area is essential. Otherwise a decline caused by weather or by a natural predator would be credited to the pesticide.

What the shapes mean. The shape of an age pyramid reflects the growth status of the population. A very broad base with a narrow top means a large proportion of young individuals and relatively few old ones — a growing population. Nearly vertical sides mean each age group is of similar size — a stable population.

What has changed. Two things must have happened together. The birth rate has fallen, which narrows the base; and survival has improved, which fills out the middle and upper age groups so that individuals now live to occupy them. A stable pyramid is not a population that has stopped changing — it is one in which births and deaths have come into balance.

The prediction for the next fifty years. The population will roughly hold its size, but its composition will keep shifting: the large cohorts that were young in the first pyramid move upwards, so the proportion of older individuals rises. If the birth rate continues to fall, the base narrows further and the pyramid begins to take the declining form, with a narrow base and a wide middle.

The caution to add. An age pyramid describes only natality, mortality and age structure. Immigration and emigration also change density, and in a human population they can change it faster than births do — so a pyramid forecasts, it does not determine.

(a) Banyan and carrot grass: per cent cover or biomass. If 200 carrot grass plants stand against a single huge banyan with a large canopy, saying the banyan's density is low relative to the carrot grass amounts to underestimating the enormous role of the banyan in that community. Number treats a seedling and a century-old tree as equal; cover and biomass do not.

(b) Bacteria in a petri dish: biomass or an indirect index. Total number is not an easily adoptable measure if the population is huge and counting is impossible or very time-consuming. Biomass, turbidity or colony-forming units per unit volume all work, and all scale with the quantity that matters.

(c) Fish in a lake: a relative density — the number caught per trap. For certain ecological investigations there is no need to know the absolute densities; relative densities serve the purpose equally well. If the trapping effort is kept constant, catch per trap tracks the real density faithfully, and that is enough to detect change.

(d) Tigers in a reserve: an indirect estimate from pug marks and fecal pellets. The tiger census in our national parks and tiger reserves is often based on exactly this. The animals are few, secretive and dangerous to approach, so direct counting is not an option; the signs they leave are.

The principle behind all four. Population density is a question about the ecological weight of a species in its habitat, and the appropriate unit is whatever best reflects that weight — number, biomass, cover, or an index — not whatever is easiest to state.