Ecology: Chapter 4 — Populations

4.1 Counting #

4.1.1 What is being counted #

The population in a model is the set of individuals the count includes. A deme is a local interbreeding group. Movement of alleles among demes is gene flow, treated in Evolutionary Biology. A metapopulation is a population of local populations linked by dispersal, with local extinction and recolonisation.

Unitary organisms have a determinate body: one zygote, one countable individual. Modular organisms, including most plants and many corals, add repeated units. A genet is the genetic individual, from one zygote. A ramet is a physiological individual, a stem or a polyp that can live on its own. A count of ramets is not a count of genets, and selection sees genets while competition for light often sees ramets. Plant Biology owns the organs. The counting problem is ecological.

4.1.2 Density, dispersion, and a mark–recapture estimate #

Absolute density is numbers per unit area or volume. Relative abundance is an index: catches per trap, calls per hour. An index is enough to detect a direction if the bias stays constant, and it is not a density.

Dispersion is the pattern of locations. If individuals were placed at random, the variance among quadrat counts would about equal the mean. A variance much larger than the mean means clumping. A variance much smaller means a regular pattern. Quadrat size changes the verdict, so the index is a description at a scale. Clumping comes from behaviour, from patchy resources, or from limited dispersal. Regular spacing comes from territoriality or from plants suppressing their neighbours.

The Lincoln–Petersen estimator is the closed-population mark–recapture idea. Mark M individuals, return them, later catch n of which m are marked, and estimate N as M n / m. The assumptions are a closed population (no birth, death, or movement during the interval), marks not lost and not missed, every individual equally catchable, marking does not change catchability, and enough recaptures that m is not zero or tiny. The simple formula is slightly biased at small samples. The assumptions matter more than the bias correction.

Invented numbers, for the arithmetic only: 80 animals are marked, and a later sample of 60 contains 12 marked. The estimate is 80 × 60 / 12 = 400. If marked animals are easier to recatch, m is too high and 400 is too low. If marks are lost, m is too low and 400 is too high. Nothing in the arithmetic says any real population contains 400 animals.

Distance sampling, removal methods, and spatially explicit capture–recapture are families of estimators with their own assumptions. These notes do not operate them.

4.2 Growth #

4.2.1 Exponential growth #

When per-capita growth does not depend on density,

dN/dt = r N,

and

N(t) = N₀ e^rt.

Here r is the instantaneous per-capita rate. Positive r increases, zero holds, negative declines. In discrete time, N_{t+1} = λ N_t. When the time step matches the continuous model, λ = e^r. Doubling time is ln(2) / r, about 0.693 / r.

Invented numbers, again only for the arithmetic: if r = 0.2 per year, doubling time is ln(2) / 0.2, about 3.5 years. A population of 100 becomes about 122 after one year, because e^0.2 is about 1.22. Exponential growth describes a population that is small relative to its resources, a colonisation, or the early part of a recovery. It cannot describe a population in a finite world for long. The curve has no equilibrium except zero, and zero is unstable when r is positive.

4.2.2 Logistic growth #

The logistic, from Verhulst and later Pearl, makes per-capita growth fall linearly with N:

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dN/dt = r N (1 − N/K).

K is the carrying capacity in that environment: the N at which this population stops growing. It is not a property of the species alone. A change in food, in predators, or in climate changes K. Equilibrium at N = K is stable in this equation. N = 0 is unstable if r > 0. The absolute growth rate dN/dt is greatest at N = K/2.

With the same invented r = 0.2 per year and K = 1000. At N = 100, dN/dt = 0.2 × 100 × 0.9 = 18 per year. At N = 900, dN/dt = 0.2 × 900 × 0.1 = 18 per year. At N = 500, dN/dt = 0.2 × 500 × 0.5 = 50 per year. The population adds individuals fastest at half of K, not when it is nearest K. That fact returns in Chapter 10, because the maximum sustainable yield of this model sits at the same point, which is an awkward place to manage.

Assumptions: the population is closed; r and K are constant; density dependence is linear and immediate; there is no age structure and no chance. Real populations lag. A lag can overshoot K and oscillate. In the discrete map studied by May, written in the form x → a x (1 − x), raising the growth parameter carries the population from a stable point through cycles and, above about 3.57 in that particular form, into chaos: deterministic change that is sensitive to the starting value. The parameter a in that map is not the continuous r. Chaos in a map is not a claim that a named species fluctuates chaotically.

4.2.3 Density dependence, regulation, and the Allee effect #

A factor is density-dependent if it changes the per-capita rate as N changes. Intraspecific competition, some predation, and some disease do this. A density-independent factor changes births or deaths regardless of N: a frost, a storm, a pollutant pulse. The same weather can do both, killing a fraction outright and also reducing the food that will later be limiting.

Limitation is any reduction in the rate. Regulation is a negative feedback that tends to return the population towards an equilibrium. Andrewartha and Birch stressed weather as the determinant of numbers. Nicholson stressed density dependence. Both can be right at different scales. Weather can set the average r or the average K. Density dependence is what stops exponential growth from continuing. A population can be limited by frost every winter and still regulated by competition every summer.

Nicholson distinguished scramble, in which individuals share a resource so equally that all may fail, from contest, in which winners take enough and losers take nothing. Territorial animals are closer to contest. A scramble among insect larvae in a carcass can leave no survivors at high density and some survivors at lower density, so the relationship of recruitment to density is humped rather than merely declining.

An Allee effect is a positive relationship between per-capita growth and density at low density. Mates are hard to find, a group defence fails, a predator is not diluted, or a plant receives too little pollen. A component Allee effect in one part of fitness need not produce a demographic Allee effect in r. A strong demographic Allee effect creates an unstable threshold. Below it the population declines towards zero even though individuals are still present. That is one reason “there are still some left” is not the same as “the population will recover.”

4.3 Age, stage, and life histories #

4.3.1 Life tables and R₀ #

A cohort life table follows one generation. A static life table is a snapshot of ages and assumes a stable age distribution and rates that are not changing. Survivorship l_x is the proportion of newborns still alive at age x. Fecundity m_x is the birth rate of an individual of that age, usually counted as daughters when the table follows females. Deevey’s three shapes are a vocabulary, not a taxonomy. Type I survivorship stays high until old age, as in many large mammals. Type II falls at a roughly constant fraction, close to many birds. Type III loses most individuals young, as in many fishes, invertebrates, and plants. A tree can be type III as a seed and something else once it is a sapling. Stage then works better than age.

The net reproductive rate is

R₀ = Σ l_x m_x.

R₀ > 1 means a female more than replaces herself over a generation. R₀ is not r. The Euler–Lotka relation ties them: r is the value that satisfies Σ e^{−r x} l_x m_x = 1. Generation time is, roughly, the mean age of mothers weighted by their contribution to R₀. Two schedules with the same R₀ can have different r if one breeds earlier. Early reproduction raises r because offspring arrive sooner and themselves start breeding. That is why R₀ alone does not rank two populations’ growth.

Invented schedule, for the arithmetic: l₁ m₁ = 0.5 × 1 and l₂ m₂ = 0.25 × 4, with no other ages breeding. Then R₀ = 0.5 + 1.0 = 1.5. The cohort more than replaces itself. The number 1.5 is not a measured population, and it still does not equal λ or e^r until the timing is put into Euler–Lotka.

This demographic R₀ is not the epidemiological R₀. Microbiology uses the same symbol for the mean number of new infections from one case in a susceptible population, and it quotes no value. These notes also quote none. The symbols collide. The quantities do not.

4.3.2 Structured models, named and not filled in #

A Leslie matrix projects an age vector. Survival probabilities sit on the subdiagonal. Fecundities sit in the top row. A Lefkovitch matrix uses stages instead of ages, which suits plants, insects with a dormant stage, and any life cycle the calendar does not measure. The long-run growth rate is the dominant eigenvalue, the structured analogue of λ. Elasticity says which entry that growth rate responds to, as a proportion. A conservation action that saves eggs and one that saves breeding adults can rank in opposite orders in two species. These notes give the reason to build the matrix. They do not invent the entries for a real species.

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4.3.3 Trade-offs, r and K, and Grime’s triangle #

Resources put into reproduction are not put into survival or growth. Offspring number trades off against offspring size. Current reproduction trades off against later reproduction. Semelparity is one reproductive bout: many annual plants, many Pacific salmon, some bamboos. Iteroparity is repeated breeding. Cole asked why anything is iteroparous if one bout can use the whole body. The answer is in the schedule. If juvenile survival is low and an adult that survives can breed again, holding some resources back raises fitness. The arithmetic does not always favour spending everything.

Lack proposed that a bird’s clutch evolves towards the number that fledges the most young, which can be smaller than the number of eggs the female can form. Experiments often find that parents lay a slightly smaller clutch than the one that fledges the most in that year. The usual reading is that the parent’s own survival, and so its later broods, is part of the currency the single-year count misses.

r- and K-selection, from MacArthur and Wilson and from Pianka’s later scheme, sorts species into early, many, small offspring in disturbed places, versus late, fewer, larger offspring near capacity. It is a historical axis. Used as two bins for every species it is a caricature. A long-lived weed and a small mammal do not occupy one idea just because both breed fast.

Grime’s CSR triangle is the plant classification these notes prefer. Two axes, stress and disturbance, leave three viable corners. Competitors occupy low stress and low disturbance. Stress-tolerators occupy high stress and low disturbance: slow growth, long life, defence of captured resources. Ruderals occupy low stress and high disturbance: rapid colonisation and a large reproductive effort. High stress and high disturbance together support no strategy. A ruderal is not simply “r-selected,” and a stress-tolerator is not “K-selected.” The physiological basis of stress tolerance is Plant Physiology.

Bet-hedging appears when the environment varies. A strategy that lowers the year-to-year variance of fitness can win even if it lowers the arithmetic mean in a good year. A seed bank is the plant case: some seeds germinate now, some wait. Dormancy as a hormone balance is Plant Physiology. The population consequence, a store of individuals that buffers a bad year, is ecological, and it reappears as Chesson’s storage effect in Chapter 5.

4.4 Space and chance #

4.4.1 Metapopulations #

Levins wrote the fraction p of occupied patches as

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dp/dt = c p (1 − p) − e p.

Colonisation of empty patches is c p (1 − p). Extinction of occupied patches is e p. The equilibrium is p = 1 − e/c, so the metapopulation persists only if colonisation exceeds extinction when the species is rare in the landscape. Assumptions: many identical patches, no explicit distances, and asynchrony. If every patch crashes in the same year, recolonisation has no source.

A mainland–island structure gives the small patches a permanent source. Source–sink structure (Pulliam) is different again. A source has a local r above zero and exports individuals. A sink has a local r below zero and persists by immigration. A sink can hold a large share of the individuals. Counting animals in the sink and protecting only the sink does not protect the population. An ecological trap is a sink that animals prefer, because the cues that once marked quality now mark a dangerous site: a field that looks like habitat and is harvested, a reflective surface that looks like water. Preference is not evidence of demographic value.

The rescue effect is immigration that lowers the chance a local population goes extinct. It returns in island biogeography.

4.4.2 Three kinds of stochasticity #

Demographic stochasticity is chance in the births and deaths of individuals. It is strong at small N and averages out at large N. Environmental stochasticity is a good year or a bad year affecting the rates themselves. It does not fade as N grows. A large population can still be ended by a long run of bad conditions if there is no refuge and no other population. Genetic stochasticity is drift and the loss of variation, plus inbreeding. It is Evolutionary Biology. All three are why a population-viability analysis is a projection under stated rates and stated chance, not a discovery of a universal minimum size.

The 50/500 rule (Franklin) was a genetic rule of thumb: an effective size of about 50 to limit short-term inbreeding, and about 500 to hold quantitative variation in the longer term. It is not a law of ecology. Later genetic work has argued that both numbers are often too small for the purposes they were given. These notes will not apply 50 or 500 to a named population.

Key terms. Genet; ramet; Lincoln–Petersen estimator; exponential growth; logistic growth; carrying capacity; density dependence; regulation; Allee effect; l_x; m_x; R₀; semelparity; iteroparity; CSR; metapopulation; source–sink; demographic stochasticity.

Summary. Density needs a method and a statement of what individual is being counted. Exponential growth has a constant per-capita rate and no ceiling. Logistic growth puts the ceiling at K and the fastest absolute increase at K/2, under assumptions real populations violate by lagging, by having age structure, and by varying. R₀ is replacement over a generation and is not r, and it is not the epidemiological R₀. Life histories are trade-offs; Grime’s triangle is more honest for plants than two bins called r and K. A metapopulation persists by colonisation outrunning extinction, and a crowded sink is not a source. Small populations face chance in individuals. All populations face chance in the environment.

Self-check.

1. In the invented Lincoln–Petersen numbers (80 marked, 60 caught later, 12 marked), what is N, and how does mark loss bias it?

2. Where is dN/dt greatest in the logistic model, and which assumption fails if the population overshoots K and oscillates?

3. Why can two life tables have the same R₀ and different r? Why must this R₀ not be copied from an epidemiology chapter?

4. Place competitors, stress-tolerators, and ruderals on Grime’s two axes. Why is the fourth corner empty?

5. A sink holds most of the animals. Why can protecting only that site fail? How is an ecological trap a special case of a sink?