Population Growth and Regulation
© Eric R. Pianka
Intrinsic Rate of Natural Increase
A useful parameter implicit in every pair of schedules of births and deaths is the intrinsic rate of natural increase, sometimes called the Malthusian parameter. Usually designated by r, it is a measure of the instantaneous rate of change of population size (per individual); r is expressed in numbers per unit time per individual and has the units of 1/time. In a closed population the intrinsic rate of increase is defined as the instantaneous per capita birth rate, b, minus the instantaneous per capita death rate, d. [Similarly, in an open population r is equal to (births + immigration) – (deaths + emigration).] When per capita births exceed per capita deaths b > d, the population is increasing and r is positive; when deaths exceed births b < d, r is negative and the population is decreasing. In practice, r is somewhat cumbersome to calculate, as its value must be determined by iteration using Euler’s implicit equation
where e is the base of the natural logarithms and x subscripts age. Survivorship and fecundity at age x are designated with lx and mx, respectively. Provided that the net reproductive rate, R0, is near 1, r can be estimated using the approximate formula
The maximal instantaneous rate of increase per head, rmax, varies among animals by several orders of magnitude (see following Table). Small short-lived organisms such as the common human intestinal bacterium Escherichia coli have a relatively high rmaxvalue, whereas larger and longer-lived organisms such as humans have, comparatively, very low rmaxvalues. Components of rmax are the instantaneous birth rate per head, b, and instantaneous death rate per head, d, under optimal environmental conditions. Rates of reproduction and death rates evolve in concert -- when either is high the other is usually low.
(rmax, per capita per day) and Mean Generation Times (in days)
A population whose size increases linearly in time would have a constant population growth rate given by
where Nt is the number at time t, N0 is the initial number, and t0 is the initial time. But at any fixed positive value of r, the per capita rate of increase is constant, and a population grows exponentially. Its growth rate is a function of population size, with the population growing faster as N becomes larger. Suppose you wanted to estimate the rate of change of the population at an instant in time, say, at time t. As a first approximation, you might look at N immediately before and immediately after time t, say, 1 hour before and 1 hour after, and apply the preceding equation. At time t1 (say t - 1 hr) the true rate is less than, and at time t2 (say t + 1 hr) is greater than, your straight-line estimate. Differential calculus was developed to handle just such cases, and it allows us to calculate the rate of change at an instant in time. As ΔN and Δt are made smaller and smaller, ΔN/Δt gets closer and closer to the true rate of change at time t. In the limit, as ΔN and Δt approach zero, ΔN/Δt is written as dN/dt, which is calculus shorthand for the instantaneous rate of change of N at time t.
Exponential population growth is described by the simple differential equation
where, again, b is the instantaneous birth rate per individual and d the instantaneous death rate per individual (remember that r = b – d).
Using calculus to integrate the above equation shows that the number of organisms at some time t, Nt , under exponential growth is a function of the initial number at time zero, N0, r, and the time available for growth since time zero, t:
Here again, e is the base of the natural logarithms. Taking logarithms of the above equation, which is simply an integrated version of the equation before it, gives
This equation indicates that loge N changes linearly in time; that is, a semilog plot of loge N against t gives a straight line with a slope of r and a y-intercept of loge N0.
Setting N0 equal to 1 (i.e., a population initiated with a single organism), after one generation, T, the number of organisms in the population is equal to the net reproductive rate of that individual, or R0 . Substituting these values in the above equation:
Since log 1 is zero, this equation reduces to loge R0 = rT or r = loge R0/T.
Another useful population parameter closely related to the net reproductive rate and the intrinsic rate of increase is the so-called finite rate of increase, λ, defined as the rate of increase per individual per unit time. The finite rate of increase λ is measured in the same time units as the instantaneous rate of increase, and
In a population without age structure, l is thus identical with R0 [T equal to 1 in above equations].
Demographic and Environmental Stochasticity
Genetic drift can cause allele frequencies to fluctuate and can even result in a polymorphic locus becoming fixed. In small populations, processes analogous to genetic drift occur that can cause populations to fluctuate in size. Because births and deaths are not continuous but sequential discrete events, even a stable population will fluctuate up and down due to random sequences of births and deaths. If a run of three births in a row is followed by only two deaths, the population will increase by one individual, if only temporarily. Conversely, a run of two births could be followed by several deaths, causing a decrease. Very small populations can even random walk to extinction! Another type of random influence is termed environmental stochasticity -- this refers to stochastic environmental changes that affect the intrinsic rate of increase. Both demographic stochasticity and environmental stochasticity cause population sizes to fluctuate in small populations.
Verhulst–Pearl Logistic Equation
In a finite world, no population can grow exponentially for very long. Sooner or later every population must encounter either difficult environmental conditions or shortages of its requisites for reproduction. Over a long period of time, unless the average actual rate of increase is zero, a population either decreases to extinction or increases to the extinction of other populations.
So far, our populations have had fixed age-specific parameters, such as their lx and mx schedules. In this section we ignore age specificity and instead allow R0 and r to vary with population density. To do this, we define carrying capacity, K, as the density of organisms (i.e., the number per unit area) at which the net reproductive rate (R0 ) equals unity and the intrinsic rate of increase (r) is zero. At “zero density” (only one organism, or a perfect competitive vacuum), R0 is maximal and r becomes rmax. For any given density above zero density, both R0 and r decrease until, at K, the population ceases to grow. A population initiated at a density above K decreases until it reaches the steady state at K. Thus, we define ractual (or dN/dt times 1/N) as the actual instantaneous rate of increase; it is zero at K, negative above K, and positive when the population is below K.
The simplest assumption we can make is that ractual decreases linearly with N and becomes zero at an N equal to K; this assumption leads to the classical Verhulst–Pearl logistic equation:
Alternatively, by factoring out an rN, the equation can be written
Or, simplifying by setting r/K in the first equation equal to z,
The actual instantaneous rate of increase per individual, ra, decreases linearly with increasing population density under the assumptions of the Verhulst–Pearl logistic equation. In the figure below, the solid line depicts conditions in an optimal environment in which the difference between b and d is maximal. The dashed line shows how the actual rate of increase decreases with N when the death rate per head, d, is higher; equilibrium population size, N*, is then less than carrying capacity, K.
The terms rN (N / K) and zN2 represent the density-dependent reduction in the rate of population increase. Thus, at N equal to unity (an ecologic vacuum), dN/dt is nearly exponential, whereas at N equal to K, dN/dt is zero and the population is in a steady state at its carrying capacity. Logistic equations (there are many more besides the Verhulst–Pearl one) generate so-called sigmoidal (S-shaped) population growth curves. Implicit in the Verhulst–Pearl logistic equation are three assumptions: (1) that all individuals are equivalent — that is, the addition of every new individual reduces the actual rate of increase by the same fraction, 1/K, at every density; (2) that rmax and K are immutable constants; and (3) that there is no time lag in the response of the actual rate of increase per individual to changes in N.
Note that density-dependent effects on birth rate and death rate are combined by the use
of r (these effects are separated later in this section). Carrying capacity is also an
extremely complicated and confounded quantity, for it necessarily includes both renewable
and nonrenewable resources, which are variables themselves. Carrying capacity almost
certainly varies a great deal from place to place and from time to time for most organisms.
There is also some inevitable lag in feedback between population density and the actual
instantaneous rate of increase. All these assumptions can be relaxed and more realistic
equations developed, but the mathematics quickly become extremely complex and unmanageable.
Nevertheless, a number of populational phenomena can be nicely illustrated using the simple
Verhulst–Pearl logistic, and a thorough understanding of it is a necessary prelude to the
equally simplistic Lotka–Volterra competition equations. However, the numerous flaws of the
logistic must be recognized, and it should be taken only as a first approximation for small
changes in population growth, most likely to be valid near equilibrium and over short time
periods (i.e., situations in which linearity should be approximated).
Hypothetical curvilinear relationships between instantaneous rates of increase and population density. Concave upward curves have also been postulated. Notice that r in the logistic equation is actually rmax. The equation can be solved for the actual rate of increase, ractual, which is a variable and a function of r, N, and K, by simply factoring out an N:
The actual instantaneous rate of increase per individual, ractual, is always less than or equal to rmax (r in the logistic). The equation above and the Figure below show how ractual decreases linearly with increasing density under the assumptions of the Verhulst–Pearl logistic equation.
The two components of the actual instantaneous rate of increase per individual, ra, are the actual instantaneous per capita birth rate, b, and the actual instantaneous per capita death rate, d. The difference between b and d (i.e., b – d) is ractual. Under theoretical ideal conditions when b is maximal and d is minimal, ractual is maximized at rmax In the logistic equation, this is realized at a minimal density, or a perfect competitive vacuum.
To be more precise, we add a subscript to b and d, which are functions of density. Thus, bN – dN = rN (which is ractual at density N), and b0 – d0 = rmax. When bN = dN, ractual and dN/dt are zero and the population is at equilibrium. The above Figure diagrams the way in which b and d vary linearly with N under the logistic equation. At any given density, bN and dN are given by linear equations
dN = d0 + yN
where x and y represent, respectively, the slopes of the lines plotted in the Figure above. The instantaneous death rate, dN, clearly has both density-dependent and density-independent components; in the above equation and Figure, yN measures the density-dependent component of dN while d0 determines the density-independent component.
At equilibrium, bN must equal dN , or
Substituting K for N at equilibrium, r for (b0 – d0), and rearranging terms
Derivation of the Logistic Equation
The derivation of the Verhulst–Pearl logistic equation is relatively straightforward. First, write an equation for population growth using the actual rate of increase rN
Now substitute the equations for bN and dN from those given several lines above, into the equation directly above:
dN/dt = [(b0 – xN) – (d0 + yN)] N
Substituting r for (b0 – d0) and, from the above equation, r/K for
(x + y), multiplying through by N, and rearranging terms,
Density Dependence and Density Independence
Various factors can influence populations in two fundamentally different ways. If their effects on a population do not vary with population density, but the same proportion of organisms are affected at any density, factors are said to be density independent. Climatic factors often, though by no means always, affect populations in this manner (see Table below). If, on the other hand, a factor’s effects vary with population density so that the proportion of organisms influenced actually changes with density, that factor is density dependent. Density-dependent factors and events can be either positive or negative. Death rate, which presumably often increases with increasing density, is an example of positive or direct density dependence; birth rate, which normally decreases with increasing density, is an example of negative or inverse density dependence. Density-dependent influences on populations frequently result in an equilibrium density at which the population ceases to grow. Biotic factors, such as competition, predation, and pathogens, often (though not always) act in this way.
Ecologists are divided in their opinions as to the relative importance of density dependence
and density independence in natural populations (Andrewartha and Birch 1954; Lack 1954, 1966;
Nicholson 1957; Orians 1962; McLaren 1971).
Detection of density dependence can be difficult. In studies of population dynamics of Thrips
imaginis (a small herbivorous insect), Davidson and Andrewartha (1948) found that they could
predict population sizes of these insects fairly accurately using only past population sizes
and recent climatic conditions. These workers could find no evidence of any density effects;
they therefore interpreted their data to mean that the populations of Thrips were controlled
primarily by density-independent climatic factors. However, reanalysis of their data shows
pronounced density-dependent effects at high densities (Smith 1961). Population change and
population size are strongly inversely correlated, which strongly suggests density dependence.
Smith also demonstrated a rapidly decreasing variance in population size during the later
portion of the spring population increase. Furthermore, these patterns persisted even after
partial correlation analysis, which holds constant the very climatic variables that Davidson
and Andrewartha considered to be so important.
This example illustrates the great difficulty ecologists frequently encounter in distinguishing
cause from effect. There is now little real doubt that both density-dependent and
density-independent events occur; however, their relative importance may vary by many
orders of magnitude from population to population — and even within the same population
from time to time as the size of the population changes (Horn 1968; McLaren 1971).
Opportunistic versus Equilibrium Populations
Periodic disturbances, including fires, floods, hurricanes, and droughts, often result in catastrophic density-independent mortality, suddenly reducing population densities well below the maximal sustainable level for a particular habitat. Populations of annual plants and insects typically grow rapidly during spring and summer but are greatly reduced at the onset of cold weather. Because populations subjected to such forces grow in erratic or regular bursts, they have been termed opportunistic populations. In contrast, populations such as those of many vertebrates may usually be closer to an equilibrium with their resources and generally exist at much more stable densities (provided that their resources do not fluctuate); such populations are called equilibrium populations. Clearly, these two sorts of populations represent endpoints of a continuum; however, the dichotomy is useful in comparing different populations. The significance of opportunistic versus equilibrium populations is that density-independent and density-dependent factors and events differ in their effects on natural selection and on populations. In highly variable and/or unpredictable environments, catastrophic mass mortality (such as that illustrated in the following Table) presumably often has relatively little to do with the genotypes and phenotypes of the organisms concerned or with the size of their populations. (Some degree of selective death and stabilizing selection has been demonstrated in winter kills of certain bird flocks.) By way of contrast, under more stable and/or predict-able environmental regimes, population densities fluctuate less and much mortality is more directed, favoring individuals that are better able to cope with high densities and strong competition. Organisms in highly rarefied environments seldom deplete their resources to levels as low as do organisms living under less rarefied situations; as a result, the former usually do not encounter such intense competition. In a “competitive vacuum” (or an extensively rarefied environment) the best reproductive strategy is often to put maximal amounts of matter and energy into reproduction and to produce as many total progeny as possible as soon as possible. Because competition is weak, these offspring often can thrive even if they are quite small
MacArthur and Wilson (1967) designate these two opposing selective forces r- selection and K-selection, after the two terms in the logistic equation (however, one should not take these terms too literally, as the concepts are independent of the equation). Of course, things are seldom so black and white, but there are usually all shades of gray. No organism is completely r-selected or completely K-selected; rather all must reach some compromise between the two extremes. Indeed, one can think of a given organism as an “r-strategist” or a “K-strategist” only relative to some other organism; thus statements about r- and K-selection are invariably comparative. We think of an r- to K-selection continuum and an organism’s position along it in a particular environment at a given instant in time. The following Table lists a variety of correlates of these two kinds of selection.
McLain (1991) suggested that the relative strength of sexual selection depends on the life history strategy, with r-strategists being less likely to be subjected to strong sexual selection than K-strategists. Winemiller (1989, 1992) points out that reproductive tactics among fishes (and probably all organisms) can be placed on a two-dimensional triangular surface in a three-dimensional space with the coordinates: juvenile survivorship, fecundity, and age of first reproduction or generation time (see Figure below). This two-dimensional triangular surface has three vertices corresponding to equilibrium (K-strategists), opportunistic, and seasonal species. The r- to K-selection continuum runs diagonally across this surface from the equilibrium corner to the opportunistic-seasonal edge. In fish, seasonal breeders exhibit little sexual dimorphism, whereas both opportunistic and equilibrium species display marked sexual dimorphisms (Winemiller 1992).
Under situations where survivorship of adults is high but juvenile survival is low and highly
unpredictable, there is a selective disadvantage to putting all one’s eggs in the same basket,
and a consequent advantage to distributing reproduction out over a period of time (Murphy 1968).
This sort of reproductive tactic has become known as “bet hedging” (Stearns 1976) and occurs in
both r-strategists and K-strategists. Winemiller (1992) points out that a bet-hedging axis passes
across his triangular surface from the opportunistic corner endpoint to the edge connecting the
seasonal and equilibrium tactics (above Figure).
In the majority of real populations that have been examined, numbers are kept within certain bounds by density-dependent patterns of change. When population density is high, decreases are likely, whereas increases tend to occur when populations are low (Tanner 1966; Pimm 1982). If the proportional change in density is plotted against population density, inverse correlations usually result.
The following Table summarizes such data for a variety of populations, including humans (the only
species with a significant non-negative correlation). Such negative correlations are
found even in cyclical and erratic populations such as those considered in the next section.
During the past 40 years, the human population, worldwide, has more than doubled from about
3 billion people to 6.8 billion. 6,800,000,000, nearly seven thousand million, is a rather
large number, difficult to comprehend.
Each year, the human population increases by nearly 100 million, a daily increase of more
than one-quarter of a million souls. Each hour, every day, day in and day out, over 11,000
more people are born than die. These rates of increase are staggering.
Most people hold the anthropocentric opinion that earth exists primarily, or even solely, for human exploitation. Genesis prescribes: “Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth” (my italics). We have certainly lived up to everything except “replenish the earth.”
Percentage Change in Density and Population Density for a Variety
The human population explosion has been fueled by habitat destruction — we are usurping resources once exploited by other species. Tall grass prairies of North America have been replaced with fields of corn and wheat, native American bison have given way to cattle. In 1986, humans consumed (primarily via fisheries, agriculture, pastoral activities, and forestry) an estimated 40 percent of the planet’s total production (Vitousek et al. 1986). Today we consume more than half of the solar energy trapped by plants (Vitousek et al. 1997). More atmospheric nitrogen is fixed by humanity than by all other natural terrestrial sources combined. Humans have transformed nearly one-half of the earth’s land surface. More than half of all accessible surface fresh water is now used by humans. Freshwater aquatic systems everywhere are polluted and threatened. Fish and frogs are seriously threatened. All the oceans have been heavily overfished. Many species have gone extinct due to human pressures over the past century and many more are threatened and endangered. Nearly one-quarter of earth’s bird species have already been driven extinct by inane human activities such as species introductions and habitat destruction.
People everywhere today stand ready to rape and pillage their wildernesses (“wastelands”) for whatever they can be forced to yield. Raw materials, such as ore, lumber, and even sand (used to make glass), are harvested in vast quantities. Big companies enjoy privileged status, excluding the public from extensive areas, producing great ugly clear cuts, vast strip mines, deep open pit mines, instant but permanent man-made mountains, eyesores paying testimony to the avaricious pursuit of timber, precious metals, and minerals. Deforestation is nearly complete in many parts of the world. Overgrazing is rampant. Grasses and the shrub understory have been virtually eliminated over extensive areas. It is quite instructive to come upon a fenced graveyard, and to see a small patch of country as it must have been before the land rape by the pastoral industry. Native hardwoods are wasted to make charcoal and burned for firewood. Lumberjacks will soon be out of work whether or not the remaining timber is cut. Should forest habitats be saved? Is there enough left to save? This sort of pillage continues. Virtually everywhere, often with governmental subsidies and incentives, forests, deserts, and scrublands are being leveled and turned into fields for crops. Many of these fields are marginal and will soon have to be abandoned, transformed into great man-made vegetationless deserts. More dust bowls are in the making. In some regions, replacement of the drought-adapted deep rooted native vegetation with shallow-rooted crop plants has reduced evapotranspiration, thus allowing the water table to rise, bringing deep saline waters to the surface. Such salinization reduces productivity and seems to be irreversible. Some deserts have so far been able to resist the tidal wave of advancing human exploiters, but there are people who dream of the day that technological “advances,” such as water plants to move “excess” water or to distill seawater, will make it possible to develop desert regions (i.e., to replace them with vast agricultural fields, or even cities). Antonyms, such as “sustainable development,” are strung together into oxymorons by biopoliticians and developers in an attempt to make all this destruction and homogenization seem less offensive.
Most people consider basic biology, particularly ecology, to be a luxury that they can do without. Even many medical schools no longer require that premedical students obtain a biological major. But basic biology is not a luxury at all; rather it is an absolute necessity for living creatures such as ourselves. Despite our anthropocentric (human-centered) attitudes, other life forms are not irrelevant to our own existence. As proven products of natural selection that have adapted to natural environments over millennia, they have a right to exist, too. With human populations burgeoning and pressures on space and other limited resources intensifying, we need all the biological knowledge that we can possibly get. For example, in this day and age, a primer on “how to be a successful venereal microbe” has become essential reading for everyone! Ecological understanding is particularly vital. Basic ecological research is urgent because the worldwide press of humanity is rapidly driving other species extinct and destroying the very systems that ecologists want to understand. No natural community remains undisturbed by humans. Pathetically, many will disappear without even being adequately described, let alone remotely understood. As existing species go extinct and even entire ecosystems disappear, we lose forever the very opportunity to study them. Knowledge of their evolutionary history and adaptations vanishes with them: thus we are losing access to biological information itself.
Only during the last few generations have biologists been fortunate enough to be able to travel with ease to remote wilderness areas. Panglobal comparisons have broadened our horizons immensely. This is a fleeting and unique opportunity in the history of humanity, for never before could scientists get virtually anywhere. However, all too soon, there won’t be any even semipristine natural habitats left to study.
The Tragedy of the Commons
More than 30 years ago, in a set piece of rational thought that deserves much more attention than it has so far received, Garrett Hardin (1968) perceived a fly in the ointment of freedom, which he explained as follows:
The tragedy of the selfish herdsman on a common grazing land is underscored by the rush to catch the last of the great whales and the ongoing destruction of earth’s atmosphere (ozone depletion, acid rain, CO2 enhanced greenhouse effect, etc.). Global weather modification is a very real and an exceedingly serious threat to all of us, as well as to other species of plants and animals.
Over the past few hundred years, we humans have drastically engineered our own environments, creating a modern-day urbanized indoor society that simply did not exist a mere few centuries ago. Indeed, only 500 human generations ago (about 10,000 years) we were living in caves under stone-age conditions! Our only source of light other than that of the sun, moon, and stars was firelight from campfires — replaced now with television sets glimmering in the dark! (We’re using oil to generate electricity and sending fossil sunlight back out into space!) Once “primitive” hunter-gatherers who regularly walked extensive distances and worked hard to collect enough food to stay alive, we lived in intimate small clan groups planning and plotting to somehow survive winters (greed may have been an evolutionary advantage) — we struggled to escape all sorts of natural hazards long since eliminated, but replaced with new and markedly different dangers. We propelled ourselves into a complex, brand new, human-engineered urban world with artificial lights, electricity, air conditioning, computers, email, money, shopping markets (= ample cheap food for many), antibiotics, drugs, cars, airplanes, long-distance travel, overcrowding, regimentation, phones, and television.
Humans simply haven’t had time enough to adapt to all these novel environmental challenges — today, we find ourselves misfits in our own strange new world (someone dubbed us “Stone Agers in the fast lane”). Health consequences of this self-induced mismatch between organism and environment are many and varied: anxiety, depression, schizophrenia, drug abuse (including both alcohol and tobacco), obesity, myopia, diabetes, asthma and other allergies, impotency, infertility, birth defects, bullets, new microbes and resistant strains, environmental carcinogens, and other toxins. Convincing arguments suggest that we have actually produced all these maladies as well as many more. Amazingly, we have managed to create a world full of pain and suffering unknown to our stone-age ancestors just a few hundred generations ago. What a magnificent accomplishment!
People sometimes ask, "What is the carrying capacity for humans?" Six billion of us occupy roughly half of earth’s land surface, consuming over half the freshwater and using about half of earth’s primary productivity, but many of those persons are living in poverty and not getting adequate nutrition. Certainly if our population continues to double in the next 40 years as it did during the past 40, we will finally have reached our carrying capacity at 12 billion in the year 2040 — at that population density, humans will occupy all the earth’s surface (there won’t be any more wilderness or any wild animals), and we would be using every drop of freshwater as well as every photon that intercepts the surface!
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