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Haemig PD  (2012 Laws of Population Ecology.  ECOLOGY.INFO 23

Laws of Population Ecology

Note: This online review is updated and revised continuously, as soon as results of new scientific research become available.  It therefore presents state-of-the-art information on the topic it covers.

The discovery of laws in ecology has lagged behind those of physics and chemistry.  The main reasons seem to be that ecology is a much younger science and that research in all its branches is woefully underfunded and understaffed.

However, as Colyvan & Ginzburg (2003) point out, misunderstandings and unrealistic expectations of what laws are have also hindered the search, as have mistaken beliefs that ecology is just too complex a science to have laws or that explanatory power can be improved by building complex models (Ginzburg & Jensen 2004).  Nevertheless, over the years, resourceful and perceptive researchers have been able to identify some of the laws that exist in ecology.

While much remains to be learned, it now appears that laws of ecology resemble laws of physics (Ginzburg 1986).  They describe idealized situations, have many exceptions, and need not be explanatory or predictive (Colyvan & Ginzburg 2003, Ginzburg & Colyvan 2004).  Nevertheless, because they express important principles and relationships that are applicable to a wide set of real-world situations, they fully deserve to be called laws.

This review lists and summarizes the laws of population ecology, a branch of ecological science that studies populations of animals and plants.  Future reviews will cover the laws of other branches of ecology. 

What is a Law?

A scientific law is "a regularity which applies to all members of a broad class of phenomena (Parker 1989);" a "generalized description of how things behave in nature under a variety of circumstances (Krebs 2001b)."  In this review, we deal with two kinds of laws: principles and allometries.

A principle is "a scientific law which is highly general or fundamental, and from which other laws are derived (Parker 1989)."  An allometry is a power function relating different things to each other.  For example, the allometries we examine in this paper relate the body size of organisms to different population quantities such as density, rate of population growth and length of population cycle.  Like Kepler's Laws, which describe the motions of heavenly bodies in their orbits, ecological allometries are regressions and hence do not refer to mechanisms.

Laws of Population Ecology

Currently, nine laws of population ecology are recognized.  Each is listed below and categorized as either a principle or an allometry.  Also listed is one candidate principle.

Principles:
Malthusian Law
Allee's Law
Verhulst's Law
Lotka-Volterra's Law
Liebig's Law

Allometries:
Fenchel's Law
Calder's Law
Damuth's Law
Generation-Time Law

Candidate Principle:
Ginzburg's Law

Sources for this list: (Ginzburg 1986; Berryman 1999, 2003; Turchin 2001, 2003; Ginzburg & Colyvan 2004).

Each law or candidate law is described in detail in the paragraphs that follow.  To immediately read about a certain law, click on the name of that law in the list above to link directly to it.

Malthusian Law

This law says that when birth and death rates are constant, a population will grow (or decline) at an exponential rate.

The Malthusian Law thus describes how populations grow or decline when nothing else happens.  It "describes the default situation for populations - how they behave in the absence of any disturbing factors (Ginzburg & Colyvan 2004)." 

Ginzburg (1986) pointed out that the Malthusian Law plays a role in ecology similar to that of Newton's First Law in physics.  Before Galileo and Newton, Aristotle asserted that the default state of all objects was rest, and that motion occurred only when force was applied to an object.  Sir Isaac Newton, however, showed that the opposite was true: that uniform motion was the default state and that non-uniform movement and rest usually occurred only when force was applied to an object.  His first law incorporates the concept of inertia, which is "the tendency of a body to resist changing its velocity (Krebs 2001b)."

Like Newton's First Law, the Malthusian Law says that the default state of a population is not rest (i.e. a constant population) but motion (i.e. exponential growth or decline); and that when populations do not grow or decline exponentially it is because an external force (i.e. something in the environment) is altering the birth rates and/or death rates (Ginzburg 1986, Ginzburg & Colyvan 2004).  This external (environmental) force can be either an abiotic factor or a biotic factor, such as "the degree of inter-specific crowding, and densities of all other species in the community that could interact with the focal species (Turchin 2003)."

Etymology:  Named in honor of Thomas Robert Malthus (1766-1834) who first described this law (Malthus 1798).

Synonyms:  Exponential Law of Population Growth; Malthus's Law; Malthusian Principle; First Principle (Berryman 2003).

Allee's Law

Allee's Law says that there is a positive relationship between individual fitness and either the numbers or density of conspecifics (conspecifics are other individuals of the same species).  In other words, as the number of individuals in a population increases, or as population density increases, survival and reproduction also increase (Berryman 1999).  A good example occurs when animals aggregate in groups for protection and thereby dilute the threat that each individual faces from a predator:  For example, an individual sparrow in a flock of 4 sparrows that is attacked by a successful predator has a 75% chance of survival, while an individual sparrow in a flock of 100 has a 99% chance of survival. 

Increased numbers of conspecifics benefit a population because they increase predator dilution or saturation; increase antipredator vigilance or aggression; enhance cooperative defense of resources and cooperative defense against predators; increase social thermoregulation; increase collective modification or amelioration of the environment; increase the availability of mates; increase the success of pollination or fertilization success, enhance reproduction and reduce inbreeding, genetic drift, or loss of integrity by hybridization (Stephens et al. 1999)"  See also Courchamp et al. (1999) and Stephens & Sutherland (1999).

According to Allee's Law, there is reduced reproduction or reduced survival at low population densities or low population sizes.  For example, when the population size of an insect-pollinated plant becomes low, or if a low number of individuals flower during a year, fewer seeds will be produced per plant because insect pollinators will have a harder time finding few flowers than many flowers (Forsyth 2003).  Because small populations have lowered reproduction or survival, Allee's Law is of special interest to ecologists that work with endangered species.

Etymology:  Named by Berryman (2003) to honor Warder C. Allee (1885-1955) who first described this principle (Allee 1932).

Synonyms:  The Allee Effect; The Allee Principle; Second Principle (Berryman 2003), Cooperation (Allee 1932; Berryman 2003).

Verhulst's Law

Although individuals may benefit from the presence of conspecifics, population growth cannot go on forever without negative consequences.  Eventually, an upper bound is reached beyond which population density cannot increase.  Many different factors can limit a population, such as predators, disease, resource levels and competition with other species.  This law, however, deals with only one factor: intra-specific competition (i.e. competition between members of the same species).  Because the organisms limiting the population are also members of the population, this law is also called "population self-limitation" (Turchin 2001).

The Verhulst's Law says that at some point, the per-capita growth rate of a population is limited directly and immediately by its own density, through the process of intra-specific competition (Berryman 1999; Turchin 2001). 

The mechanisms of intra-specific competition that increase with rising population density and work to eventually limit the growth of the population include intra-specific aggression, territoriality, interference with search due to agonistic interactions with conspecifics, cannibalism, and competition for enemy-free space (Berryman 1999; Turchin 2003).  These mechanisms increase with rising population density because individuals struggle to occupy the insufficient amount of space now available - space needed to gather resources or to hide from or escape enemies (Berryman 2003).  Fortunately, other factors usually limit a population before it increases to the density where self-limitation by intra-specific competition begins.

Etymology:  Named by Berryman (2003) to honor Pierre-François Verhulst (1804-1849) who first described this law (Verhulst 1838).

Synonyms:  Third Principle (Berryman 2003); Intra-specific Competition; Population Self-limitation, Self-limitation (Turchin 2001).

Lotka-Volterra's Law

Organisms interact with other species and with the physical environment in many ways.  These interactions sometimes include "negative feedbacks."  An example of negative feedback is when an increase in the population of a prey species leads to an increase in the population of its predators (through increased reproduction), and this in turn feeds back to reduce the prey population through increased mortality from predation (Berryman 2002, 2003). 

Lotka-Volterra's law says that "when populations are involved in negative feedback with other species, or even components of their environments," oscillatory (cyclical) dynamics are likely to be seen (Berryman 2002, 2003).

Etymology:  Named in honor of Alfred James Lotka (1880-1949) and Vito Volterra (1860-1940) who independently described an early version of this law (Lotka 1925; Volterra 1926).  See also Hutchinson (1948).

Synonyms:  Fourth Principle (Berryman 2003); Hutchinson's Law (Berryman 2003); Law of Consumer-Resource Oscillations (Turchin 2001).

Liebig's Law

Many different environmental factors have the potential to control the growth of a population. These factors include the abundance of prey or nutrients that the population consumes and also the activities of predators.  A given population will usually interact with a multitude of different prey and predator species, and ecologists have described these many interactions by drawing food webs.  Yet, although a given population may interact with many different species in a food web, and also interact with many different abiotic factors outside the food web, not all of these interactions are of equal importance in controlling that population's growth.  Experience shows that "only one or two other species dominate the feedback structure of a population at any one time and place (Berryman 1993)."  The identity of these dominating species may change with time and location, but the number of species that limits a given population (i.e. actively controls its dynamics) is usually only one or two.

Liebig's Law, in its modern form, expresses this idea.  It says that of all the biotic or abiotic factors that control a given population, one has to be limiting (i.e. active, controlling the dynamics) (Berryman 1993, 2003).  Time delays produced by this limiting factor are usually one or two generations long (Berryman 1999).

Liebig's Law stresses the importance of limiting factors in ecology.  "A factor is defined as limiting if a change in the factor produces a change in average or equilibrium density (Krebs 2001a)."  One sometimes hears the remark that "everything is connected to everything else in nature," and that therefore a change in the abundance of one organism will affect the abundances of all others.  While it is true that everything in nature is connected to everything else by interactions, the conclusion just stated is exaggerated and misleading (Berryman 1993).  Research shows that only some of the many interactions are strong and important, and that relatively few of these limit the growth of the focal population at any given time and place (Berryman 1993, 2003).

Etymology:  Named in honor of Baron Justus von Leibig (1803-1873) who formulated an early version of this law (Liebig 1840).

Synomyms:  Liebig's Law of the Minimum; Law of the Minimum; Law of Feedback Dominance (Berryman 1993), Fifth Principle (Berryman 2003).

Fenchel's Law

Fenchel's law tells how exponential population growth is related to body size (mass).  It says that species with larger body sizes generally have lower rates of population growth.  More exactly, it states that the maximum rate of reproduction decreases with body size at a power of approximately 1/4 the body mass (Fenchel 1974).

Fenchel's law is expressed by the following allometric equation:

r = aW-1/4

Where r is the intrinsic rate of natural increase of the population, a is a constant that has 3 different values (one for unicellular organisms, one for heterotherms and one for homoiotherms), and W is the average body weight (mass) of the organism (Fenchel 1974).

Examples:  If species X has a body mass 10 times heavier than that of species Y, then X's maximum rate of reproduction will be approximately one-half that of Y's.  If X is 100 times heavier than Y, then X's maximum rate of reproduction will be approximately one-third that of Y's.  If X is 1,000 times heavier than Y, then X's maximum rate of reproduction will be one-fifth to one-sixth that of Y's.  If X is 10,000 times heavier than Y, then X's maximum rate of reproduction will be one-tenth that of Y's (Ginzburg & Colyvan 2004).

Etymology:  Named in honor of Tom Fenchel who first described this law (Fenchel 1974).

Synonyms: Fenchel Allometry (Ginzburg & Colyvan 2004).

Calder's Law

Calder's Law tells how the oscillation periods in populations of herbivorous mammals are related to body size (mass).  It says that species with larger body sizes generally have longer population cycles.  More exactly, Calder's Law states that the length of the population cycle increases with increasing body size at a power of approximately 1/4 the body mass (Calder 1983).

Calder's Law is expressed by the following allometric equation:

t = aW1/4

Where t is the average time of the population cycle, a is a constant, and W is the average body weight (mass) of the organism. 

Examples:  If species X has a body mass 10 times heavier than that of species Y, then X's populations will cycle (if they cycle) with a period 1.78 times as long as that of Y's populations.  If X is 100 times heavier than Y, then X's populations will cycle (if they cycle) with a period approximately three times longer than Y's.  If X is 1,000 times heavier than Y, then X's populations will cycle (if they cycle) with a period five to six times longer than Y's.  If X is 10,000 times heavier than Y, then X's populations will cycle (if they cycle) with a period ten times longer than Y's (Ginzburg & Colyvan 2004).

Prior to Calder's research, it was known that small mammalian herbivores such as lemmings (Lemmus) and voles (Microtus) had population cycles of 3 to 4 years, while larger-bodied varying hares (Lepus americanus) had population cycles of 8 to 10 years, and still larger-bodied moose (Alces alces) and reindeer (Rangifer tarandus) had population cycles of 20 to 40 years. However, no one before Calder had pointed out the correlation between larger body size and longer population cycles. 

Etymology:  Named in honor of William Alexander Calder III (1934-2002) who first described this law (Calder 1983).

Synonyms:  Calder Allometry (Ginzburg & Colyvan 2004).

Damuth's Law

Damuth's law tells how population density is related to body size (mass).  It says that species with larger body sizes generally have lower average population densities.  More exactly, it states that the average density of a population decreases with body size at a power of approximately 3/4 the body mass (Damuth 1981, 1987, 1991).

Damuth's Law is expressed by the following allometric equation:

d = aW-3/4

Where d is the average density of the population, a is a constant, and W is the average body weight (mass) of the organism. 

Example:  A mammal that is 16 times larger than a second mammal, will generally have an average population density 1/8 that of the second species (Ginzburg & Colyvan 2004).

Damuth's Law holds in most cases for terrestrial vertebrates and invertebrates (Damuth 1981, 1987, 1991).

Etymology:  Named in honor of John Damuth who first proposed this law (Damuth 1981).

Synonyms:  Damuth Allometry (Ginzburg & Colyvan 2004).

Generation-Time Law

This law tells how generation-time (the time period required for young to grow and mature to reproductive age) is related to body size.  It says that species with larger body sizes usually have longer generation-times.  More exactly, it states that the generation-time increases with increasing body size at a power of approximately 1/4 the body mass (Bonner 1965).  (The body mass used in this law is the body mass of the organism at the time of reproduction).

The Generation-Time Law is expressed by the following allometric equation:

g = aW1/4

Where g is the average generation-time of the population, a is a constant, and W is the average body weight (mass) of the organism. 

Examples:  If species X has a body mass 10 times heavier than that of species Y, then X's generation-time will be 1.76 times as long as that of Y's.  If X is 100 times heavier than Y, then X's generation-time will be  3.16 times longer than Y's.  If X is 1,000 times heavier than Y, then X's generation-time will be 5.62 times longer than Y's.  If X is 10,000 times heavier than Y, then X's generation-time will be 10.00 times longer than Y's (Ginzburg & Colyvan 2004).

All animals and plants appear to follow this law, and body length can usually be used in place of body mass (Bonner 1965).

Etymology:   John T. Bonner conducted extensive research that helped establish this law (Bonner 1965).  However, rumors abound that this law was known before Bonner (Ginzburg & Colyvan 2004).  I therefore follow Ginzburg & Colyvan (2004) in tentatively naming it the Generation-Time Law until the name of the discoverer of this law can be determined.

Synonyms:  Generation-Time Allometry (Ginzburg & Colyvan 2004)

A Candidate Law

The search for laws in population ecology is not over.  It continues, and some principles, allometries and rules not on our list may in fact be laws.  These "candidate laws" need more consideration, discussion, formulation and testing.  I will now discuss one such candidate: Ginzburg's Law.

Ginzburg's Law

This law asserts that the transfer of quality from mother to daughter (the maternal effect) influences population growth, and consequently that population growth at any one point in time is dependent not only on the current environment, but also on the environment of the previous generation (Ginzburg & Colyvan 2004).  Environmental alteration of the per-capita population growth rate is believed to occur by modifying the rate of change of this growth rate, rather than by directly altering the per-capita growth rate.  Thus, population growth is seen as being inertial, a second-order dynamic (Ginzburg & Colyvan 2004).

Ginzburg's Law says that the length of a population cycle (oscillation) is the result of the maternal effect and inertial population growth.  According to this law, the period lengths in the cycles of a population must be either two generations long or six or more generations long (Ginzburg & Colyvan 2004).  Predators or other environmental factors may be the cause of the population cycle, and they may also affect the amplitude and shape of the cycle, but the length of the cycle period is species-specific and not dependent on what causes the cycle.  This species-specific cycle period is called an eigenperiod.

Ginzburg's Law, with its eigenperiod concept, explains why similar species have similar cycle-periods, even though they may inhabit very different environments, are preyed upon by vastly different predators or, in some situations such as islands, are preyed upon by no predators (Ginzburg & Colyvan 2004).  It also explains why population cycle periods 3 to 5 generations long are unknown in nature.  I tentatively classify this candidate law as a principle because it provides an explanation for Calder's Law.

Etymology:  Named in honor of Lev Ginzburg who was the first to propose it (Ginzburg & Taneyhill 1994; Ginzburg & Colyvan 2004)

Synonyms:  The Maternal Effect (Ginzburg & Colyvan 2004).

References

Allee WC  (1932)  Animal aggregations: a study in general sociology.  University of Chicago Press, USA

Ashton K  (2001)  Are ecological and evolutionary rules being dismissed prematurely?  Diversity and Distributions 7: 289-295

Berryman AA  (1993)  Food web connectance and feedback dominance, or does everything really depend on everything else?  Oikos 68: 183-185

Berryman AA  (1999)  Principles of population dynamics and their application.  Cheltenham, UK

Berryman AA  (2002)  Population cycles: the case for trophic interactions.  Oxford University Press, New York

Berryman AA  (2003)  On principles, laws and theory in population ecology.  Oikos 103: 695-701

Bonner JT  (1965)  Size and Cycle. Princeton University Press, USA

Calder WA  (1983)  An allometric approach to population cycles of mammals.  Journal of Theoretical Biology 100: 275-282

Calder WA  (1996)  Size, function and life history.  Dover Publications, Mineola, NY, USA

Courchamp F, Clutton-Brock T, Grenfell B  (1999)  Inverse density dependence and the Allee effect.  Trends in Ecology and Evolution 14: 405-410

Colyvan M, Ginzburg LR  (2003)  Laws of nature and laws of ecology Oikos 101: 649-653

Damuth J  (1981)  Population density and body size in mammals.  Nature 290: 699-700

Damuth J  (1987)  Interspecific allometry of population density in mammals and other animals: the independence of body mass and population energy-use.  Biological Journal of the Linnean Society 31: 193-246

Damuth J  (1991)  Of size and abundance.  Nature 351: 268-269

Fenchel T  (1974)  Intrinsic rate of natural increase: the relationship with body size.  Oecologia 14: 317-326

Forsyth SA  (2003)  Density-dependent seed set in the Haleakala silversword: evidence for an Allee effect.  Oecologia 136: 551-557

Ginzburg LR  (1986)  The theory of population dynamics: 1. Back to first principles.  Journal of Theoretical Biology 122: 385-399

Ginzburg LR, Taneyhill DE  (1994)  Population cycles of forest Lepidoptera: a maternal effect hypothesis.  Journal of Animal Ecology 63: 79-92

Ginzburg LR, Colyvan M  (2004)  Ecological Orbits: how planets move and populations grow.  Oxford University Press, New York

Ginzburg LR, Jensen CXJ  (2004)  Rules of thumb for judging ecological theories.  Trends in Ecology and Evolution 19: 121-126

Hutchinson GE  (1948)  Circular causal systems in ecology.  Annuals of the New York Academy of Sciences 50: 221-246

Kleiber M  (1932)  Body Size and Metabolism.  Hilgardia 6 :315-353

Krebs CJ  (2001a)  Ecology.  Benjamin Cummings, San Francisco

Krebs RE  (2001b)  Scientific laws, principles and theories.  Greenwood Press, Westport, Connecticut

Liebig J  (1840)  Chemistry and its application to agriculture and physiology.  Taylor and Walton, London

Lotka AJ  (1925)  Elements of Physical Biology.  Williams & Wilkens, Baltimore, USA 

Malthus TR  (1798)  An Essay on the Principle of Population.  J Johnson, London

Parker SP  (1989)  McGraw-Hill Dictionary of Scientific and Technical Terms.  4th Edition.  McGraw-Hill, New York

Stephens PA, Sutherland WJ  (1999)  Consequences of the Allee effect for behaviour, ecology and conservation.  Trends in Ecology and Evolution 14: 401-405

Stephens PA, Sutherland WJ, Freckleton RP  (1999)  What is an Allee effect?  Oikos 87: 185-190

Turchin P  (2001)  Does population ecology have general laws?  Oikos 94: 17-26

Turchin P  (2003)  Complex Population Dynamics: A Theoretical/Empirical Synthesis.  Princeton University Press, USA

Verhulst P-F  (1838)  Notice sur la loi que la population suit dans son accrossement.  Corr. Math. Phys. 10:113-121

Volterra V  (1926)  Variazioni e fluttuazioni del numero d'individui in specie animali conviventi.  Mem. R. Accad. Naz. dei Lincei Ser. VI 2

West GB, Brown JH, Enquist BJ  (2000)  The origin of universal scaling laws in biology.  Pp. 87-112 in Scaling in Biology (Brown JH, West GB, Editors), Oxford University Press, NY

Information about this Review

This review is also available in the following languages:

Portuguese    Spanish

The author is:  Dr. Paul D. Haemig (PhD in Animal Ecology)

Photograph:  Wildlife at the Masai Mara Game Reserve in Kenya by Mark Karstad (United Arab Emerates)

The proper citation is:

Haemig PD  2012   Laws of Population Ecology.  ECOLOGY.INFO 23

If you are aware of any important scientific publications about the laws of population ecology that were omitted from this review, or have other suggestions for improving it, please write the author at the following e-mail address: 

director {at} ecology.info  

The author thanks Alan Berryman, Lev Ginzburg, Mark Colyvan, John Bonner and Peter Turchin, who generously shared their knowledge of population ecology with him.

© Copyright 2003-2012 Ecology Online Sweden.  All rights reserved.

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