Population Growth

01 / UNDERSTAND & APPLY

1. An individual has an age; a population has an age distribution

A population is a group of organisms of the same species considered within a defined place and time. Population size is the number of individuals; density expresses abundance per unit area or volume. Neither should be confused with the number of species in a community. A pond containing many species is not one population merely because all organisms share the same water.

Some attributes belong to the population rather than to one organism. One individual can be a particular age, but a population has an age distribution. A population can have a birth rate, death rate and sex ratio. These summaries describe the composition and change of a group; they are not anatomical properties of one member.

Counting individuals is not always the most informative abundance measure. For organisms differing greatly in size, biomass or percentage cover may capture ecological importance better. Relative-abundance indicators such as repeated observations can be useful when a complete census is impractical, but an index is not automatically an exact population size.

Always define the sampling frame. Ten individuals in one square metre and ten in one hectare have the same counted number but very different density. A density change can reflect a change in population size, a change in the area used as denominator or both. Meaningful comparison requires consistent units and comparable boundaries.

02 / UNDERSTAND & APPLY

2. Sampling, age structure and survival

Quadrat sampling estimates density by counting organisms in defined areas and relating the sample to the whole habitat. Representative placement matters: sampling only visible clusters biases the result. A clumped distribution can produce large differences among quadrats even when the overall population is unchanged.

Mark–recapture uses the proportion of marked individuals in a later sample to estimate the population. In its simplest model, if M individuals were marked, a later sample contains C individuals and R are recaptures, N is estimated as MC/R. The interpretation assumes a sufficiently closed population, retained marks, mixing and comparable capture probabilities. A small recapture count creates a fragile estimate rather than miraculous precision.

Age structure distinguishes prereproductive, reproductive and postreproductive groups in a simplified model. A broad younger group can indicate potential for future growth, but future births also depend on survival, fertility, migration and conditions. An age pyramid is evidence about demographic structure, not an unconditional prediction.

Survivorship tracks the fraction of an initial cohort still alive at different ages. Type I patterns retain many individuals until later life, type II approximates a constant mortality probability with age, and type III shows high early mortality followed by better survival among survivors. The patterns are idealisations and must not be confused with equal absolute numbers dying in every age interval.

A life-table mortality proportion for an interval uses those alive at the beginning of that interval as the denominator. If 12 of 80 survivors die, the interval mortality is 12/80, not 12 divided by the original cohort unless that different quantity is explicitly requested. Correct denominators are the foundation of demographic reasoning.

03 / UNDERSTAND & APPLY

3. Begin with the population balance

Population size changes through births, deaths, immigration and emigration. Over a defined interval, the bookkeeping equation is Nnext = Nnow + B + I − D − E, where B, I, D and E are counts of events or individuals added and removed over that interval. This is an accounting identity when all events are counted consistently.

A closed-population model excludes immigration and emigration. It does not imply that no births or deaths occur. An open population can increase despite more deaths than births if immigration is large enough, or decline despite a natural excess of births if emigration is larger. Population change alone cannot identify the contribution of reproduction.

Distinguish total events from per-capita rates. A large population can have more births in total but a lower per-individual birth rate than a small population. A statement such as “birth rate is 0.04 per individual per year” must not be added directly to a count of 20 deaths without converting to compatible quantities.

In the simplified continuous closed model, per-capita birth and death rates b and d give net per-capita growth r = b − d. Then dN/dt = rN. The units of r are inverse time. In more detailed settings, the intrinsic rate and observed growth depend on age structure and conditions; the elementary formula is used here under stated simplifying assumptions.

04 / UNDERSTAND & APPLY

4. Constant per-capita growth produces an increasing total increment

Exponential growth assumes a constant per-capita rate in the model interval. The larger the population becomes, the more individuals contribute to the next increment, so the absolute rate rN increases when r is positive. Constant r does not mean a constant number added per year.

The continuous model gives N(t) = N₀eʳᵗ. If r > 0, size increases; if r = 0, it stays constant; if r < 0, it declines. The exponent rt must be dimensionless, so time and the rate units must match. A rate per month cannot be multiplied by time in years without conversion.

A discrete model uses Nₜ = N₀λᵗ for a fixed multiplication factor λ per interval. A 10% increase per interval gives λ = 1.10. Do not substitute the number 0.10 directly for a continuous r and expect exactly identical long-term results: the corresponding continuous rate is ln(1.10) per interval.

Exponential growth can approximate a population over a limited interval when constraints are weak. It cannot continue indefinitely in a finite environment. The limitation is not that the arithmetic is wrong; it is that resources, density effects and other assumptions eventually change.

05 / UNDERSTAND & APPLY

5. Doubling time and the meaning of a curve

For constant positive r, doubling time td = ln 2/r. The time is independent of starting population size in that model. A population of 50 and one of 500 with the same r both double in the same time, even though the latter adds ten times as many individuals.

On an ordinary population-size versus time graph, exponential increase bends upward. The slope represents absolute growth rate, not per-capita growth rate. A logarithmic transformation can make constant exponential growth appear linear because ln N = ln N₀ + rt. Changing the graph scale changes appearance, not the underlying biological process.

If equal time intervals produce equal absolute additions, the pattern is arithmetic rather than exponential. If they produce equal multiplicative factors, it is consistent with discrete exponential growth. A few noisy observations rarely establish a unique mechanism, but checking differences versus ratios is a useful first diagnostic.

The observed time needed to double in one historical interval does not prove the same doubling time will persist. Carrying capacity, environmental changes, age composition and chance can alter future growth. Calculations are conditional projections, not commitments that nature must follow the model.

06 / UNDERSTAND & APPLY

6. Density dependence creates a limiting term

For the growth model in this lesson, assume r > 0 and K > 0. The simple logistic model writes dN/dt = rN(1 − N/K), where r is the low-density per-capita growth parameter and K is the model carrying capacity. As N rises, the factor 1 − N/K reduces per-capita growth. This captures one idealised form of negative density dependence.

When N is much smaller than K, the limiting factor is close to one and the model resembles exponential growth. At N = K, net growth is zero. If N exceeds K, the model predicts negative growth toward K. A carrying capacity is an environment-dependent population scale, not an unchangeable property of the species.

Zero net growth at K does not mean births and deaths have stopped. It means additions and losses balance in the model. Similarly, K is not a physical wall that makes it mathematically impossible for a real population to exceed it. Delays and changing conditions can create overshoot and fluctuations.

The standard continuous logistic solution approaching a fixed K smoothly is S-shaped for suitable starting conditions below K. Real populations need not trace that exact curve. Seasonal resources, disturbances, age structure and delayed responses can produce departures. Calling a noisy plateau “logistic” is a model interpretation that needs evidence.

07 / UNDERSTAND & APPLY

7. Why the fastest total growth is not at the lowest density

In the logistic model, per-capita growth is highest near low density because crowding is weak. Total growth also depends on how many individuals are present. A tiny population has high per-individual potential but few contributors; a near-K population has many individuals but little net per-capita growth.

With r > 0 and K > 0, the product N(1 − N/K) reaches its maximum at N = K/2. Substitution gives the maximum total growth rK/4. This is a property of the simple model with fixed parameters, not a universal management recommendation or a statement that every population should be held at half its natural abundance.

For a concrete visualisation, imagine plotting growth rate vertically against N horizontally. The logistic rate starts at zero when N = 0, rises to a peak at K/2, then falls to zero at K. This inverted-U relationship is not the same plot as the S-shaped population-size-versus-time curve.

Graph questions often switch these axes without warning. If the vertical axis is per-capita growth, the simple model gives a declining straight line. If it is total growth, it gives a curved peak. If it is population size against time, it gives the familiar time trajectory. Name both axes before identifying the shape.

08 / UNDERSTAND & APPLY

8. Use models without making populations identical

Density-dependent effects can intensify as abundance rises, for example through limited resources or some forms of infection and competition. Density-independent disturbances are often modelled as effects whose per-capita impact does not depend strongly on density over the considered range. Real mechanisms can combine both kinds of influence.

Life histories differ in age at reproduction, offspring number, parental investment and survival schedules. Trade-offs arise because resources invested in one function cannot be spent without limit on every other function. A species producing many small offspring and one producing fewer larger offspring may have different recruitment and survival patterns.

The historical r-selected/K-selected contrast is a simplified organising framework, not a rigid rule sorting every species into one of two boxes. Traits vary along multiple dimensions and can change with environment. Use the framework only when a question specifies that conventional comparison; do not treat it as a complete modern theory of every life history.

Population predictions also face sampling error, environmental variation and random demographic events. A model may correctly express a mechanism while failing to predict every observed count. State what the model leaves out, especially migration, age structure, changing K and low-number chance effects.

09 / UNDERSTAND & APPLY

9. Five original worked demographic cases

Each result is conditional on the stated model. Keep event counts, rates, densities and estimated totals in separate units.

10 / UNDERSTAND & APPLY

10. Original practice with explained answers

These are constructed model problems. Population models are educational tools and are not recommendations for managing a real population without broader ecological evidence.