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Population Dynamics

Encyclopedia of Global Change

Daniel Goodman

Population Dynamics 

The science of population dynamics treats the change in numbers of individuals in a population. The changes in numbers are often important to human welfare and environmental management. We may wish to achieve population growth for harvested populations and endangered populations; we may wish to achieve population declines for pest populations or agents of infectious disease; we may wish to limit population fluctuations that might affect an ecological balance; and we may be concerned to predict human population numbers as a guide to social and economic planning.

The analysis of change in population numbers may operate at the level of the simple count of numbers, but it is often informative to decompose the change in numbers into effects owing respectively to birth and death rates for different age classes of individuals. The decomposition may reveal which age classes are the most promising targets for intervention, and this in turn may suggest the types of environmental management most likely to achieve the desired result.

The extrapolation of population sizes into the future, and the relation between the overall population growth rate and the component rates, are inherently mathematical issues. The estimation of population numbers and birth and death rates from limited data involves substantial statistical challenges. Understanding how environmental conditions affect the component birth and death rates encompasses much of the science of ecology.

Mathematics of Change in Population Numbers

If the instantaneous average per capita birth rate among individuals in a population is b offspring per individual per unit time, and the instantaneous average per capita death rate is μ deaths per individual per unit time, then the instantaneous per capita rate of change in numbers of individuals in the population is

r = bμ


individuals per individual per unit time. The instantaneous rate of change in numbers of individuals in the total population then is rN, where N is the current number of individuals.

If these per capita birth and death rates, b and μ, remain constant, then the population will grow exactly exponentially, so that

Nt = N0ert


where Nt is the population size after an elapsed time t, N0 is the initial population size, and e is the base of the natural logarithms (roughly 2.7).

Exponential growth, like compound interest, is a self-accelerating process: the rate of growth of the total population itself grows exponentially with time. Thus, when conditions allow approximate constancy of the per capita birth and death rates, and the birth rate is larger than the death rate (so that r is positive), the population in time will grow explosively. And, for similar constancy, with the death rate larger than the birth rate, the population decline will be exponential, and the population will decline steadily to extinction.

This potential for exponential growth or decline gives the dynamics of populations a capability for extreme volatility. At times, this potential is expressed in actuality, when we see epidemics, outbreaks of pest species, and population crashes. Much of the time, however, many natural populations are relatively stable, with their numbers varying within a limited range. So the constancy of conditions that fosters the dynamics of exponential growth is more the exception than the rule.

Generally, the highly volatile dynamics are perceived as undesirable from the standpoint of human welfare, so we are motivated to learn to recognize, to predict, and perhaps to control the determining circumstances. Also, there is the intellectual challenge of wanting to understand how some populations in nature achieve comparative stability, contrary to the dynamics predicted by our most elementary model of population growth.

Population Composition

One factor that can cause departures from constancy of per capita birth and death rates in a population is population composition—age structure and sex ratio. If, for example, a population is composed mostly of individuals of prime reproductive age, then for a time the birth rates in the population will be abnormally high; but, as this parental generation ages, and as its offspring (which are initially too young to reproduce themselves) come to make up a larger fraction of the population, the per capita birth rates will decline for a while.

The effects of population composition can routinely make enough of a difference in predicting the details of population growth rates that these effects need to be taken into account in assessing human population growth or in assessing the dynamics of harvested renewable resources, where modest differences in rates can have substantial economic or public health impacts. But the effects of population composition on population dynamics do not make the difference between stability and instability of the sort that concerns us in the bigger ecological picture.

In fact, if the birth and death rates of each age class are constant over time (but not age), then, regardless of the initial composition, the effects of age composition in temporarily increasing or decreasing the rate of population growth will be cyclic, with a cycle time approximately of the same duration as the population's generation time, but with the magnitude of the departure from constant growth rate diminishing in each successive generation. The age composition of such a population will eventually become constant, at which time the per capita birth and death rates, averaged over the entire population, will become constant, and then the population grows exponentially in accordance with equation (2). The equation linking the age-specific birth and death rates with the eventual exponential population growth rate that this population will display, if those rates continue to hold, is

Population Dynamics


where x designates age, bx is the birth rate to individuals age x, and μx is the death rate of individuals age x. The mathematics of dealing with this forbidding-looking equation, and the practicalities of obtaining estimates of the age-specific birth and death rates from real data, are treated in the discipline of mathematical demography.

Biology of Change in Population Numbers

The crucial terms appearing in the mathematical formulas for population dynamics are biological characteristics of individuals in the population. At best, these properties conform approximately to the assumptions of the mathematical model, in which case the dynamics of the real population will approximate the dynamics we deduce from the mathematics. With respect to the key assumption of constancy of the per capita age-specific birth and death rates, the biological reality may depart very greatly from the simple assumptions, and then the dynamics of the population may be vastly different from the dynamics deduced from the simple model. Indeed, the differences can even be qualitative, such as stable versus unstable. It is therefore up to us to examine the plausibility of the assumptions of our mathematical models, judging them against what we know about biological mechanisms. [See Modeling of Natural Systems.]

Even when we know that a model is seriously unrealistic, we may still be able to draw worthwhile conclusions from investigating the mathematical behavior of the model, provided we keep the conclusions in proper perspective. For example, we might construct a series of simplified models, where each model captures the essence of one particular mechanism while ignoring other mechanisms. Then we could use this series of models to learn the dynamic consequences of each mechanism, which might not be nearly as understandable in a more complicated model in which all the mechanisms were operating simultaneously.

Equation (2) has revealed the dramatic dynamic consequences of constant per capita birth and death rates. Next we need to consider the reasonableness of the assumption of such constancy, the circumstances under which it might apply, the types of departure from constancy that we might expect to result from different biological mechanisms, and the kinds of dynamics that result from each major pattern of departure from constancy.

Constant Conditions

The per capita birth and death rates reflect how favorable are the conditions that individuals in the population experience. This will be a function of factors such as nutritional status as influenced by food supply per capita, availability of shelter, incidence of disease, rates of predation, and the physiological tolerance of the species to the prevailing physical characteristics (such as temperature and moisture) of the environment that the population occupies.

To an extent, these factors can vary under their own dynamics, because the natural world, as we know it, is variable. But many of these factors will also be correlated with the size of the population, since they respond causally to the level of crowding in the population. As the population becomes more crowded, some resources will be in short supply, predators may be attracted to the concentration of prey, disease transmission may be more rapid, and some individuals of the population may be forced into less-preferred habitat. In this light, constant per capita birth and death rates appear quite unlikely over any period of time long enough for the population to grow or decline appreciably.

Constant conditions as experienced by each individual

Approximate constancy of birth and death rates is plausible in a stable environment for a limited period of time if, during that time, the population is so sparse that crowding effects are negligible and the population growth is limited only by basic physiological constraints on an individual's maximum rate of growth from birth to maturity and on an individual's maximum rate of producing offspring. It is implausible that individuals in a population should indefinitely experience constant conditions, even approximately, both because of background variation in the environment and because the exponential population growth or decline that occurs under constant conditions must eventually lead to enough change in population size that changes in the level of crowding will begin to have a substantial effect on the per capita birth and death rates.

Human populations

On a global scale, the human population over the past few centuries seems to have escaped the iron law that crowding will eventually terminate an episode of exponential population growth. The explanation for this apparent exception is that continuing progress in technology during this period, especially in agriculture and medicine, has compensated for the effects of numerical density, so that the effective biological crowding of humans has not changed substantially, even though the total human population has grown, and is continuing to grow, at a rate that in some respects is alarming.

Beneath the surface of this appearance, in aggregate, of constant effective human crowding allowing ongoing exponential population growth, there are many local and temporal inconstancies, many complicated details of social, biological, ecological, and economic processes, and unanswered questions about eventual limits to human knowledge and feasible technology. Our science does not have an adequate basis for predicting how far into the future the advances in technology will keep pace with the escalating growth of human population. Likewise, there is not an adequate basis for predicting whether the eventual end of this phase of approximately exponential growth in human population will be a gradual deceleration of growth until approximate population constancy is achieved, or whether there will be a catastrophic crash of some sort. Our inability to predict these outcomes with any certainty is not necessarily comforting.

Constant resource supply

If we imagine that the background environmental conditions for a population are constant, and the critical resources are in constant supply, then this constant total resource base will be divided among a varying number of individuals as the numbers of individuals change. If we imagine that the per capita birth and death rates change in proportion to the resources available per individual, with birth rates decreasing as population increases, and death rates increasing as population increases, we can readily construct a model in which the instantaneous per capita population growth rate, r, declines smoothly as the number of individuals in the population increases. This is a simple form of density-dependent population dynamics.

Such a model will exhibit very stable dynamics. For example, a population initiated at small numbers will initially grow exponentially, and then the growth rate will slow as crowding effects exert themselves in the larger population; and finally the population will cease to grow when the crowding just reaches the level where the birth rate (which has been declining as population increases) just balances the death rate (which has been increasing as population increases). If a population is initiated at larger numbers than this equilibrium level, the death rate will exceed the birth rate, so that the population will decline, and the rate of decline will diminish as the population approaches the equilibrium level (from above); and the population will cease declining when it reaches the same equilibrium level where birth rates equal death rates.

Carrying capacity

The dynamics described above, for a single population whose individuals share a constant (constant total, not constant per capita) resource supply, reveal an equilibrium population level to which the population will return, under its own dynamics, regardless of whether the population is temporarily displaced to larger or smaller numbers. This equilibrium population level is conventionally called the carrying capacity of this environment for this species. [See Carrying Capacity.]

Discussions of human population growth sometimes attempt to estimate the carrying capacity of the entire Earth for human beings, in an attempt to estimate the global population size at which the growth of the human population will cease. It must be understood that human carrying capacity in the context of those discussions is a moving target, since it depends on technology and culture.

Density-Dependent Growth in a Random Environment

If we imagine a population whose individuals share a resource supply that varies randomly over time, the variation in resource supply results effectively in a random variation in the carrying capacity of that environment for that population. At any moment, the population may be above or below the carrying capacity, and for the moment the population numbers will change in the direction that approaches the current carrying capacity. But the carrying capacity itself changes, randomly, so the effect is an appearance of the population numbers “chasing” after the fluctuating carrying capacity.

If, over time, the average carrying capacity is large relative to the range of variation in carrying capacity, the variation of population numbers will also be bounded by the “envelope” of the random carrying capacity. But if the fluctuations in carrying capacity are too wide, the variations in carrying capacity may occasionally carry the population to very low numbers. At very low numbers, additional chance factors begin to influence the population dynamics.

Population extinction

Recall that the per capita birth and death rates with which we began this discussion of population dynamics are averages over all the individuals in the population. When the number of individuals is large, differences between individuals will tend to cancel out in the average. But when numbers are small, the chance differences between individuals can dominate the dynamics of the population, as “luck of the draw” determines individual birth events and death events. For this reason, chance mechanisms operating in small populations can cause chance extinction following a run of bad luck.

Thus, on a particular time scale, there are two categories of population extinction. Chance extinction is driven primarily by the frequency with which random environmental variation brings the population to sufficiently low numbers that chance mechanisms operating on individuals can result in a string of deaths outstripping births until the last individual is dead. Deterministic extinction occurs when constant conditions (relative to the chosen time scale) are so unfavorable for the population that it declines approximately exponentially to extinction.

The forces of expanding human populations and land use changes are gradually driving many populations of many species to smaller and smaller numbers. This is resulting in an era of abnormally high extinction rates, which has motivated concerns for the preservation of biodiversity. The trends causing the persistent declines of these populations are deterministic, in the sense described above. In attempts to prevent some extinctions, protected reserves may be set aside for remnant populations. In designing and managing such reserves, it is important to allow a sufficient margin of safety to protect the small remnant population from chance extinction. The branch of population dynamics called population viability analysis deals with the prediction of extinction probabilities as a function of population size, environmental conditions, and conservation policies.

Other Populations

Far from being an inanimate and constant source of a flux of resources, the environment that real biological populations occupy includes other biological populations. The dynamics of these other populations, as they interact with the target population of interest, will dynamically revise the conditions defining the favorability of the environment for the target population and will dynamically revise the numerical value of the carrying capacity. Thus we need to understand the principal kinds of mechanism under lying the biological interactions among populations, and the implications of each interaction for the resulting dynamic stability and for the maintenance of biodiversity.


In the simplest model of two-species competition in an otherwise constant environment, individuals of both species compete in the consumption of the common resource, which is presumed to be in constant supply. This is an extension to two species of the model of population growth with a constant carrying capacity. Contrary to the stability manifested by the one-species version of this model, the two-species version tends to instability, because whichever species is most efficient at competing for the common resource tends to crowd out, and eliminate, the other (at which point the continuing dynamics are those of a one-species model, which is stable).

These unstable dynamics of systems of populations of two competitors are commonly encountered in real biological populations in simplified experimental laboratory conditions. But this degree of instability is obviously not characteristic of the dynamics of populations in natural communities of species.

In the simple two-species models, a stable equilibrium in which both species persist requires that each species depress its own growth, through self-crowding effects, more strongly than its growth is depressed by the competitive crowding effect of the other population. Ecologically, this implies that each species must be in some sense a “specialist,” so that the supply or availability of some aspect of the environment that is uniquely important to that species is more important than the supply of the common resource. This interpretation of the mathematical modeling result has given rise to the concept of an ecological niche.

According to niche theory, each species in an ecologically stable system plays a uniquely specialized role, and this specialization is at the heart of the dynamic stability of natural ecological systems, at least from the perspective of the persistence of the species in question. Whether the persistence of the species is instrumental to the stability of much of the rest of the system is less clear, but at least some other species will depend on this species’ persistence for its own stability.

Predator and prey

The second major mode of interaction between populations is one population using the other as a “resource,” as occurs in interactions between predator and prey or between parasite (or pathogen) and host. As with the simplest models of competition, the simplest models of predator–prey interaction tend to instability. The instability in the predator–prey models expresses itself as a tendency toward population cycles. The basic force behind the cycles is predator population buildup in the presence of ample prey, causing reduction of the prey population until the scarcity of prey leads to a decline in the predator population, which in turn allows the prey population to recover, and the cycle repeats. These cycles are dynamically similar to the “boom and bust” cycles of economic theory.

The predator–prey cycles, in the most elementary form of the models, can become so extreme that the predator exterminates the prey, or the predator itself becomes extinct when it has driven the prey to such sparseness that the predator cannot find enough prey to sustain itself. These extremely unstable dynamics are commonly encountered in actual predator–prey systems in simplified laboratory conditions, but obviously they are not characteristic of dynamics in natural communities of species.

As with the analysis of the dynamics of competition models, the stabilization of predator–prey dynamics seems to hinge on mechanisms whereby the predator population, or the prey population, or both, are to some degree self-limited in their dynamics.

The Whole and the Sum of Its Parts

While the stability of natural populations is far from absolute, there is no question that natural communities exhibit a greater degree of stability than would be predicted from the simplest models of population dynamics and two-species interaction. Natural communities also exhibit a greater degree of stability than is usually seen in attempts to maintain multispecies systems in a simple laboratory setting. In this sense, real natural communities are not just haphazard assemblages of species. Somehow or other, the species have to “fit” together, in part through mutual compatibility of their respective specializations. Thus we see that the dynamics of stable natural communities cannot be explained satisfactorily in terms of the most elementary dynamic models of their component species populations, or elementary dynamic models of pairwise interactions. The key to the system's relative dynamic stability seems to derive from particular properties that make each species different, and possibly also from a history of trial-and-error, or evolutionary, adjustment of species composition and species properties.

Niche theory, which was motivated at first by the discovery of the stabilizing effect of ecological specialization in competition models, seems to draw additional credibility from the discovery of the stabilizing effect of similar sorts of ecological specialization in predator–prey models. The theory leads to an image of ecological systems as intricately tuned mechanisms in which each species plays a functional, and possibly unique, role in the aggregate dynamics. This, in turn, suggests the possibility that a stable ecological system could be vulnerable to destabilization if some of its component species populations are eliminated, or if some new species that was not originally part of this system invades from the outside, or if physical conditions change in a way that disrupts the basis for some of the dynamic balance between species by eliminating one or more of the “specialties.”

This theorizing is behind the popular metaphors representing the global ecological system as a ship that might sink for the loss (or addition) of one rivet in the wrong place. The magnitude of ecologically significant global change, and the ongoing loss of biodiversity of at least some types of species, raises reasonable concerns about wholesale ecological collapse associated with mass extinctions. It is possible to construct multispecies computer models that display these sorts of catastrophic dynamics. But ecological reality is far too complicated, and these multispecies models themselves are too complicated, for us to be certain, before the fact, that any one such model is a dynamically accurate portrayal of reality.

Much important debate in environmental policy revolves around the prudent courses of action that should be recommended, given this uncertainty: Should socially and economically costly programs of intervention be undertaken as a precaution, or is it reasonable to allow an experiment with the ecology of an entire planet to proceed as in a science-fiction movie?

See also Biological Productivity; Growth, Limits to; Human Populations; Migrations; Urban Areas; and the biography of Malthus.


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              Daniel Goodman