Population Growth and Regulation

A population of 100 that grows at 0.2 per day does not reach 300 after ten days. If nothing stops it, it reaches about 739. Growth of this kind compounds: each new individual starts adding individuals of its own. No real population keeps it up for long. This lesson builds the model of unchecked growth, adds a brake, and then asks what the brake is in real populations and how you would know.

The idea: growth in proportion to size

Think of a savings account that pays interest continuously. The more money in it, the more it earns per day, so the balance grows by the same proportion each day and by a larger amount. A population with plenty of food and space behaves the same way.

Write N for the number of individuals and r for the intrinsic rate of increase, the per-capita growth rate when resources are not limiting. In a closed population (no immigration or emigration) with a per-capita birth rate b and death rate d, r = b - d. If b = 0.5 per year and d = 0.3 per year, r = 0.2 per year. The model is:

dN/dt = rN so N(t) = N0 e^(rt)

Three points trip students up. First, r has units of inverse time, per day or per year, and an r without its time unit means nothing. Second, r is a net rate, not a birth rate. Third, r is not lambda. Lambda is the factor by which the population multiplies over one time step, and for continuous growth lambda = e^r. An r of 0.2 per day gives lambda = e^0.2 = 1.22, a 22% rise per day, slightly more than 20% because of compounding. Your demography lesson shows how r and lambda come from a life table.

A constant r means a constant doubling time, ln 2 / r. With r = 0.2 per day, the population doubles every 0.693 / 0.2 = 3.47 days, whether it holds 100 individuals or 100,000.

The model assumes a closed population, no age or stage structure (every individual counts the same), and constant b and d. Those assumptions are what make it a starting point and not a description.

Adding a brake: the logistic model

As a population grows, each individual typically gets less food, space or shelter, and births fall or deaths rise. The logistic model writes this as a brake on the per-capita rate:

dN/dt = rN(1 - N/K) per-capita rate: (1/N) dN/dt = r(1 - N/K)

K, the carrying capacity, is the population size at which the model’s growth stops, in the same units as N. The per-capita rate falls in a straight line, from r when N is near zero to 0 when N = K. Total growth, dN/dt, is that per-capita rate times N: At low N there are few individuals to multiply; near K each one adds almost nothing. The product is greatest in the middle, at N = K/2, where it equals rK/4.

K is an equilibrium, not a ceiling. If N starts above K, the brake term is negative and the model predicts decline back toward K. If N starts below K, the continuous model rises smoothly toward K and never overshoots. The model also assumes the feedback is instantaneous: today’s crowding acts on today’s births and deaths.

Worked problem. Suppose an invented closed population with no age structure starts at N0 = 100 individuals, with r = 0.2 per day and K = 1,000 individuals, all held constant. These numbers are illustrative, not measured for any species.

1. Exponential projection at day 10. N(10) = 100 x e^(0.2 x 10) = 100 x e^2 = 100 x 7.389 = 738.9, about 739 individuals.
2. Logistic projection at day 10. The solution is N(t) = K / (1 + (K/N0 - 1) e^(-rt)). Here K/N0 - 1 = 9, so N(10) = 1,000 / (1 + 9 x e^-2) = 1,000 / (1 + 9 x 0.1353) = 1,000 / 2.218 = 450.9, about 451 individuals.
3. Initial logistic rate. dN/dt = 0.2 x 100 x (1 - 100/1,000) = 0.2 x 100 x 0.9 = 18 individuals per day, compared with 20 per day under the exponential model.
4. Time to K/2. Set N(t) = 500: 1,000 / (1 + 9 e^(-0.2t)) = 500, so 1 + 9 e^(-0.2t) = 2, e^(-0.2t) = 1/9, and t = ln 9 / 0.2 = 2.197 / 0.2 = 10.99 days.
5. Peak total growth. At N = 500, dN/dt = 0.2 x 500 x 0.5 = 50 individuals per day, which is rK/4 = 0.2 x 1,000 / 4.
6. Exponential doubling time. ln 2 / 0.2 = 3.47 days.
7. Check the answers against each other. At day 10 the logistic population (451) is just short of K/2, consistent with reaching 500 at day 10.99. The two models agree at first and then part: by day 20 the exponential model predicts 5,460 individuals and the logistic about 858.

r, per-capita rate 0.20 per day K, carrying capacity 1,000 individuals Day cursor 10 days

Exponential N at the cursor

739

Logistic N at the cursor

451

Rate now: exponential / logistic

148 / 50 individuals per day

Doubling time, ln 2 / r (exponential)

3.47 days

Logistic reaches K/2 at

10.99 days

Peak logistic rate, rK/4, at N = K/2

50 per day at N = 500

In this sketch: closed, unstructured population, constant r and K, no time lag. N0 = 100; the numbers are invented and illustrative, not measured for any species.

Rule to keep: both models start with the same per-capita rate r, but the logistic one lets it fall in a straight line to zero at K, so total growth peaks at K/2 and the population settles at K.

Now name what a field population breaks. Real populations have age structure: juveniles do not breed, so a population of mostly young animals grows more slowly than its N suggests. Feedback has time lags: food eaten this winter limits births next spring. And K changes with season, weather and food supply. Each of these can produce overshoot and oscillation that the continuous logistic cannot. Robert May showed in Nature in 1976 that a discrete-time version of the logistic, with growth in seasonal steps, can settle, oscillate, or fluctuate chaotically depending on r. Keep the continuous and discrete models apart.

What regulates real populations

A factor is density dependent if it changes per-capita birth or death rates as density changes. It is density independent if its per-capita effect is the same at any density. Note what that definition does not say. A predator is not automatically density dependent, and frost is not automatically density independent. Both are hypotheses to test against per-capita rates.

Two field experiments show how. Near Kluane Lake in the Yukon, Charles Krebs and colleagues added food and fenced out mammalian predators on 1-square-kilometre blocks of boreal forest, alone and together, for eight years. They reported in 1995 that snowshoe hare (Lepus americanus) density during the cycle’s peak and decline rose about twofold with predators excluded, threefold with food added, and elevenfold with both. Both factors limited hares in that forest during those years.

On patch reefs at Lee Stocking Island in the Bahamas, Mark Hixon and Mark Carr stocked newly settled blue chromis (Chromis cyanea) at different densities and crossed resident predators with transient ones. In their 1997 study, the proportion of young fish dying rose strongly with density only when both predator groups were present. With one group, they detected no density dependence, though one of those tests had low statistical power. That is density-dependent mortality within one generation and across reefs; it does not by itself show that the population is held near a level over years.

Weather and density can also act together. Soay sheep on Hirta, in the St Kilda archipelago off Scotland, crash in some winters. Tim Coulson and colleagues reported in 2001 that lambs, adult males and adult females respond differently to density and to winter weather, so two populations of equal size under identical weather can follow different paths. The population’s makeup, which the logistic leaves out, changed the outcome.

Checking a model against data

A model earns trust by predicting something it was not fitted to. Pearl and Reed’s 1920 logistic fitted United States census counts well, but a close fit to points you already have does not show that the mechanism in the equation is at work. A better test fits a model to one data set and predicts another. John Drake did this in a 2005 analysis of 281 laboratory populations of the water flea Daphnia magna, followed for 104 days under food supplies of differing variability: he fitted models to some populations and tested their extinction predictions on others. Models that let demographic chance vary with density predicted the extinctions accurately; models that left it out were severely biased and underpredicted them. That result belongs to those microcosms.

Absence of evidence is a second trap. Loïc Thibaut and Sean Connolly reanalysed population time series from the Global Population Dynamics Database and reported in 2020 that individual series, tested one at a time, gave little evidence of density dependence, while the series analysed together gave strong evidence. One short series with no detected slope does not show that a population is unregulated. These forces also set up the next two lessons, on competition and predation, whose models build directly on the logistic.

Design problem

Krebs and colleagues crossed food addition with mammalian-predator exclusion on blocks of boreal forest about 1 square kilometre each at Kluane, Yukon, over eight years. During the hare cycle’s peak and decline, densities relative to controls were about 2 times with predators excluded, 3 times with food added, and 11 times with both. Interpret the result. What does it show, what does it not show, and what would you do next?

One way to do it

Start with a null expectation. If food and predation acted independently and their effects multiplied, the combined treatment should give about 2 x 3 = 6 times control density. If they simply added on the raw scale, it would be smaller still: 1 + 1 + 2 = 4 times. The observed 11 times exceeds both, so the two factors interact. One reading is that extra food helps more when fewer hares are killed, and predator removal helps more when hares are well fed. Neither factor alone explains the decline; they limit the population together.

What it does not show: these density ratios are not r or K, and the experiment is not a test of the logistic model. The fences kept out mammals such as lynx and coyotes but not birds of prey, so “predator exclusion” means partial exclusion. The blocks were few and all in one region over one run of years, so check the replication per treatment in the paper before generalising, and do not extend the result to all hare populations or all cycles.

Next: replicate the combined and single treatments on more blocks, ideally in another region, and measure per-capita survival and reproduction against density in each treatment. That turns “density was higher” into the per-capita evidence that regulation requires.