**Population dynamics** is the branch of life sciences that studies the size and age composition of populations as dynamical systems, and the biological and environmental processes driving them (such as birth and death rates, and by immigration and emigration). Example scenarios are ageing populations, population growth, or population decline.

Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more recently the scope of mathematical biology has greatly expanded. The first principle of population dynamics is widely regarded as the exponential law of Malthus, as modeled by the Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.

A more general model formulation was proposed by F.J. Richards in 1959, further expanded by Simon Hopkins, in which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy are covered as special cases of the general formulation. The Lotka–Volterra predator-prey equations are another famous example, as well as the alternative Arditi–Ginzburg equations. The computer game *SimCity* and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.

In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analyzed, and provide important results that may be applied to health policy decisions.

The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the *intrinsic rate of increase*. It is

where the derivative is the rate of increase of the population, *N* is the population size, and *r* is the intrinsic rate of increase. Thus *r* is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate. The concept is commonly used in insect population biology to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.^{[2]}

Exponential growth describes unregulated reproduction. It is very unusual to see this in nature. In the last 100 years, human population growth has appeared to be exponential. In the long run, however, it is not likely. Thomas Malthus believed that human population growth would lead to overpopulation and starvation due to scarcity of resources. They believed that human population would grow at rate in which they exceed the ability at which humans can find food. In the future, humans would be unable to feed large populations. The biological assumptions of exponential growth is that the per capita growth rate is constant. Growth is not limited by resource scarcity or predation.^{[3]}

where *λ* is the discrete-time per capita growth rate. At *λ* = 1, we get a linear line and a discrete-time per capita growth rate of zero. At *λ* < 1, we get a decrease in per capita growth rate. At *λ* > 1, we get an increase in per capita growth rate. At *λ* = 0, we get extinction of the species.^{[3]}

Some species have continuous reproduction.

where is the rate of population growth per unit time, *r* is the maximum per capita growth rate, and *N* is the population size.

At *r* > 0, there is an increase in per capita growth rate. At *r* = 0, the per capita growth rate is zero. At *r* < 0, there is a decrease in per capita growth rate.

“Logistics” comes from the French word **logistique**, which means “to compute”. Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population. Consider an analogy with a thermostat. When the temperature is too hot, the thermostat turns on the air conditioning to decrease the temperature back to homeostasis. When the temperature is too cold, the thermostat turns on the heater to increase the temperature back to homeostasis. Likewise with density dependence, whether the population density is high or low, population dynamics returns the population density to homeostasis. Homeostasis is the set point, or carrying capacity, defined as K.^{[3]}

where is the density dependence, *N* is the number in the population, *K* is the set point for homeostasis and the carrying capacity. In this logistic model, population growth rate is highest at 1/2 *K* and the population growth rate is zero around *K*. The optimum harvesting rate is a close rate to 1/2 *K* where population will grow the fastest. Above *K*, the population growth rate is negative. The logistic models also show density dependence, meaning the per capita population growth rates decline as the population density increases. In the wild, you can't get these patterns to emerge without simplification. Negative density dependence allows for a population that overshoots the carrying capacity to decrease back to the carrying capacity, *K*.^{[3]}

According to r/K selection theory organisms may be specialised for rapid growth, or stability closer to carrying capacity.

This equation uses *r* instead of *λ* because per capita growth rate is zero when *r* = 0. As r gets very high, there are oscillations and deterministic chaos.^{[3]} Deterministic chaos is large changes in population dynamics when there is a very small change in *r*. This makes it hard to make predictions at high *r* values because a very small *r* error results in a massive error in population dynamics.

Population is always density dependent. Even a severe density independent event cannot regulate populate, although it may cause it to go extinct.

Not all population models are necessarily negative density dependent. The Allee effect allows for a positive correlation between population density and per capita growth rate in communities with very small populations. For example, a fish swimming on its own is more likely to be eaten than the same fish swimming among a school of fish, because the pattern of movement of the school of fish is more likely to confuse and stun the predator.^{[3]}

Cellular automata are used to investigate mechanisms of population dynamics. Here are relatively simple models with one and two species.

In fisheries and wildlife management, population is affected by three dynamic rate functions.

- Natality or birth rate, often recruitment, which means reaching a certain size or reproductive stage. Usually refers to the age a fish can be caught and counted in nets.
- Population growth rate, which measures the growth of individuals in size and length. More important in fisheries, where population is often measured in biomass.
- Mortality, which includes harvest mortality and natural mortality. Natural mortality includes non-human predation, disease and old age.

If *N*_{1} is the number of individuals at time 1 then

where *N*_{0} is the number of individuals at time 0, *B* is the number of individuals born, *D* the number that died, *I* the number that immigrated, and *E* the number that emigrated between time 0 and time 1.

If we measure these rates over many time intervals, we can determine how a population's density changes over time. Immigration and emigration are present, but are usually not measured.

All of these are measured to determine the harvestable surplus, which is the number of individuals that can be harvested from a population without affecting long-term population stability or average population size. The harvest within the harvestable surplus is termed "compensatory" mortality, where the harvest deaths are substituted for the deaths that would have occurred naturally. Harvest above that level is termed "additive" mortality, because it adds to the number of deaths that would have occurred naturally. These terms are not necessarily judged as "good" and "bad," respectively, in population management. For example, a fish & game agency might aim to reduce the size of a deer population through additive mortality. Bucks might be targeted to increase buck competition, or does might be targeted to reduce reproduction and thus overall population size.

For the management of many fish and other wildlife populations, the goal is often to achieve the largest possible long-run sustainable harvest, also known as maximum sustainable yield (or MSY). Given a population dynamic model, such as any of the ones above, it is possible to calculate the population size that produces the largest harvestable surplus at equilibrium.^{[4]} While the use of population dynamic models along with statistics and optimization to set harvest limits for fish and game is controversial among scientists,^{[5]} it has been shown to be more effective than the use of human judgment in computer experiments where both incorrect models and natural resource management students competed to maximize yield in two hypothetical fisheries.^{[6]}^{[7]} To give an example of a non-intuitive result, fisheries produce more fish when there is a nearby refuge from human predation in the form of a nature reserve, resulting in higher catches than if the whole area was open to fishing.^{[8]}^{[9]}

Population dynamics have been widely used in several control theory applications. With the use of evolutionary game theory, population games are broadly implemented for different industrial and daily-life contexts. Mostly used in multiple-input-multiple-output (MIMO) systems, although they can be adapted for use in single-input-single-output (SISO) systems. Some examples of applications are military campaigns, resource allocation for water distribution, dispatch of distributed generators, lab experiments, transport problems, communication problems, among others. Furthermore, with the adequate contextualization of industrial problems, population dynamics can be an efficient and easy-to-implement solution for control-related problems. Multiple academic research has been and is continuously carried out.

- Delayed density dependence
- Holocene extinction
- Lotka-Volterra equations
- Minimum viable population
- Maximum sustainable yield
- Nicholson–Bailey model
- Overshoot (population)
- Pest insect population dynamics
- Population cycle
- Population dynamics of fisheries
- Population ecology
- Population genetics
- Population modeling
- Ricker model
*r*/*K*selection theory- Sigmoid curve
- Societal collapse
- System dynamics

**^**Brotz, Lucas; Cheung, William W. L; Kleisner, Kristin; Pakhomov, Evgeny; Pauly, Daniel (2012). "Increasing jellyfish populations: Trends in Large Marine Ecosystems".*Hydrobiologia*.**690**(1): 3–20. doi:10.1007/s10750-012-1039-7.**^**Jahn, Gary C; Almazan, Liberty P; Pacia, Jocelyn B (2005). "Effect of Nitrogen Fertilizer on the Intrinsic Rate of Increase of*Hysteroneura*setariae(Thomas) (Homoptera: Aphididae) on Rice (Oryza sativaL.)".*Environmental Entomology*.**34**(4): 938–43. doi:10.1603/0046-225X-34.4.938.- ^
^{a}^{b}^{c}^{d}^{e}^{f}Yang, Louie. Population Dynamics. Davis: UC Davis, 2014. **^**Clark, Colin (1990).*Mathematical bioeconomics : the optimal management of renewable resources*. New York: Wiley. ISBN 978-0471508830.**^**Finley, C; Oreskes, N (2013). "Maximum sustained yield: A policy disguised as science".*ICES Journal of Marine Science*.**70**(2): 245–50. doi:10.1093/icesjms/fss192.**^**Holden, Matthew H; Ellner, Stephen P (2016). "Human judgment vs. Quantitative models for the management of ecological resources".*Ecological Applications*.**26**(5): 1553–1565. doi:10.1890/15-1295. PMID 27755756.**^**Standard, Pacific (2016-03-11). "Sometimes, Even Bad Models Make Better Decisions Than People".*Pacific Standard*. Retrieved 2017-01-28.**^**Chakraborty, Kunal; Das, Kunal; Kar, T. K (2013). "An ecological perspective on marine reserves in prey–predator dynamics".*Journal of Biological Physics*.**39**(4): 749–76. doi:10.1007/s10867-013-9329-5. PMC 3758828. PMID 23949368.**^**Lv, Yunfei; Yuan, Rong; Pei, Yongzhen (2013). "A prey-predator model with harvesting for fishery resource with reserve area".*Applied Mathematical Modelling*.**37**(5): 3048–62. doi:10.1016/j.apm.2012.07.030.

- Andrey Korotayev, Artemy Malkov, and Daria Khaltourina.
*Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth*. ISBN 5-484-00414-4 - Turchin, P. 2003.
*Complex Population Dynamics: a Theoretical/Empirical Synthesis*. Princeton, NJ: Princeton University Press. - Weiss, Volkmar (2007). "The Population Cycle Drives Human History - from a Eugenic Phase into a Dysgenic Phase and Eventual Collapse".
*The Journal of Social, Political and Economic Studies*.**32**(3): 327–58.

- The Virtual Handbook on Population Dynamics. An online compilation of state-ot-the-art basic tools for the analysis of population dynamics with emphasis on benthic invertebrates.

Agent-based models have many applications in biology, primarily due to the characteristics of the modeling method. Agent-based modeling is a rule-based, computational modeling methodology that focuses on rules and interactions among the individual components or the agents of the system. The goal of this modeling method is to generate populations of the system components of interest and simulate their interactions in a virtual world. Agent-based models start with rules for behavior and seek to reconstruct, through computational instantiation of those behavioral rules, the observed patterns of behavior. Several of the characteristics of agent-based models important to biological studies include:

Modular structure: The behavior of an agent-based model is defined by the rules of its agents. Existing agent rules can be modified or new agents can be added without having to modify the entire model.

Emergent properties: Through the use of the individual agents that interact locally with rules of behavior, agent-based models result in a synergy that leads to a higher level whole with much more intricate behavior than those of each individual agent.

Abstraction: Either by excluding non-essential details or when details are not available, agent-based models can be constructed in the absence of complete knowledge of the system under study. This allows the model to be as simple and verifiable as possible.

Stochasticity: Biological systems exhibit behavior that appears to be random. The probability of a particular behavior can be determined for a system as a whole and then be translated into rules for the individual agents. Before the agent-based model can be developed, one must choose the appropriate software or modeling toolkit to be used. Madey and Nikolai provide an extensive list of toolkits in their paper "Tools of the Trade: A Survey of Various Agent Based Modeling Platforms". The paper seeks to provide users with a method of choosing a suitable toolkit by examining five characteristics across the spectrum of toolkits: the programming language required to create the model, the required operating system, availability of user support, the software license type, and the intended toolkit domain. Some of the more commonly used toolkits include Swarm, NetLogo, RePast, and Mason. Listed below are summaries of several articles describing agent-based models that have been employed in biological studies. The summaries will provide a description of the problem space, an overview of the agent-based model and the agents involved, and a brief discussion of the model results.

DemographyDemography (from prefix demo- from Ancient Greek δῆμος dēmos meaning "the people", and -graphy from γράφω graphō, implies "writing, description or measurement") is the statistical study of populations, especially human beings. As a very general science, it can analyze any kind of dynamic living population, i.e., one that changes over time or space (see population dynamics). Demography encompasses the study of the size, structure, and distribution of these populations, and spatial or temporal changes in them in response to birth, migration, aging, and death. Based on the demographic research of the earth, earth's population up to the year 2050 and 2100 can be estimated by demographers. Demographics are quantifiable characteristics of a given population.

Demographic analysis can cover whole societies or groups defined by criteria such as education, nationality, religion, and ethnicity. Educational institutions usually treat demography as a field of sociology, though there are a number of independent demography departments.Formal demography limits its object of study to the measurement of population processes, while the broader field of social demography or population studies also analyses the relationships between economic, social, cultural, and biological processes influencing a population.

DepensationIn population dynamics, depensation is the effect on a population (such as a fish stock) whereby, due to certain causes, a decrease in the breeding population (mature individuals) leads to reduced production and survival of eggs or offspring. The causes may include predation levels rising per offspring (given the same level of overall predator pressure) and the allee effect, particularly the reduced likelihood of finding a mate.

FishBaseFishBase is a global species database of fish species (specifically finfish). It is the largest and most extensively accessed online database on adult finfish on the web. Over time it has "evolved into a dynamic and versatile ecological tool" that is widely cited in scholarly publications.FishBase provides comprehensive species data, including information on taxonomy, geographical distribution, biometrics and morphology, behaviour and habitats, ecology and population dynamics as well as reproductive, metabolic and genetic data. There is access to tools such as trophic pyramids, identification keys, biogeographical modelling and fishery statistics and there are direct species level links to information in other databases such as LarvalBase, GenBank, the IUCN Red List and the Catalog of Fishes.As of November 2018, FishBase included descriptions of 34,000 species and subspecies, 323,200 common names in almost 300 languages, 58,900 pictures, and references to 55,300 works in the scientific literature. The site has about 700,000 unique visitors per month.

Fish mortalityFish mortality is a parameter used in fisheries population dynamics to account for the loss of fish in a fish stock through death. The mortality can be divided into two types:

Natural mortality: the removal of fish from the stock due to causes not associated with fishing. Such causes can include disease, competition, cannibalism, old age, predation, pollution or any other natural factor that causes the death of fish. In fisheries models natural mortality is denoted by (M).

Fishing mortality: the removal of fish from the stock due to fishing activities using any fishing gear. It is denoted by (F) in fisheries models.(M) and (F) are additive instantaneous rates that sum up to (Z), the instantaneous total mortality coefficient; that is, Z=M+F. These rates are usually calculated on an annual basis. Estimates of fish mortality rates are often included in mathematical yield models to predict yield levels obtained under various exploitation scenarios. These are used as resource management indices or in bioeconomic studies of fisheries.

Fish stockFish stocks are subpopulations of a particular species of fish, for which intrinsic parameters (growth, recruitment, mortality and fishing mortality) are traditionally regarded as the significant factors determining the stock's population dynamics, while extrinsic factors (immigration and emigration) are traditionally ignored.

Fisheries managementFisheries management is the activity of protecting fishery resources so sustainable exploitation is possible, drawing on fisheries science, and including the precautionary principle. Modern fisheries management is often referred to as a governmental system of appropriate management rules based on defined objectives and a mix of management means to implement the rules, which are put in place by a system of monitoring control and surveillance. A popular approach is the ecosystem approach to fisheries management. According to the Food and Agriculture Organization of the United Nations (FAO), there are "no clear and generally accepted definitions of fisheries management". However, the working definition used by the FAO and much cited elsewhere is:

The integrated process of information gathering, analysis, planning, consultation, decision-making, allocation of resources and formulation and implementation, with enforcement as necessary, of regulations or rules which govern fisheries activities in order to ensure the continued productivity of the resources and the accomplishment of other fisheries objectives.

Fisheries scienceFisheries science is the academic discipline of managing and understanding fisheries. It is a multidisciplinary science, which draws on the disciplines of limnology, oceanography, freshwater biology, marine biology, conservation, ecology, population dynamics, economics and management to attempt to provide an integrated picture of fisheries. In some cases new disciplines have emerged, as in the case of bioeconomics and fisheries law.

Fisheries science is typically taught in a university setting, and can be the focus of an undergraduate, master's or Ph.D. program. Some universities offer fully integrated programs in fisheries science.

KaseseKasese is a town north of Lake George in the Western Region of Uganda. It originally grew around the copper mine at Kilembe, while attention later turned to cobalt mining. It is the chief town of Kasese District, and the district headquarters are located there. Kasese is also the largest town in the Rwenzururu region. Charles Mumbere, the Omusinga of Rwenzururu, maintains a palace in the town.

Little stintThe little stint (Calidris minuta) (or Erolia minuta), is a very small wader. It breeds in arctic Europe and Asia, and is a long-distance migrant, wintering south to Africa and south Asia. It occasionally is a vagrant to North America and to Australia. The genus name is from Ancient Greek kalidris or skalidris, a term used by Aristotle for some grey-coloured waterside birds. The specific minuta is Latin for "small.

Malthusian growth modelA **Malthusian growth model**, sometimes called a **simple exponential** growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows. The model is named after Thomas Robert Malthus, who wrote *An Essay on the Principle of Population* (1798), one of the earliest and most influential books on population.

Malthusian models have the following form:

where

The model can also been written in the form of a differential equation:

dP/dt = rP

with initial condition:
P(0)= P_{0}

This model is often referred to as the *exponential law*. It is widely regarded in the field of population ecology as the first principle of population dynamics, with Malthus as the founder. The exponential law is therefore also sometimes referred to as the *Malthusian Law*. By now, it is a widely accepted view to analogize Malthusian growth in Ecology to Newton's First Law of uniform motion in physics.

Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources:

A model of population growth bounded by resource limitations was developed by Pierre Francois Verhulst in 1838, after he had read Malthus' essay. Verhulst named the model a logistic function.

Mass diffusivityDiffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is encountered in Fick's law and numerous other equations of physical chemistry.

The diffusivity is generally prescribed for a given pair of species and pairwise for a multi-species system. The higher the diffusivity (of one substance with respect to another), the faster they diffuse into each other. Typically, a compound's diffusion coefficient is ~10,000× as great in air as in water. Carbon dioxide in air has a diffusion coefficient of 16 mm2/s, and in water its diffusion coefficient is 0.0016 mm2/s.Diffusivity has an SI unit of m2/s (length2 / time). In CGS units it is given

in cm2/s.

Mesopredator release hypothesisThe mesopredator release hypothesis is an ecological theory used to describe the interrelated population dynamics between apex predators and mesopredators within an ecosystem, such that a collapsing population of the former results in dramatically-increased populations of the latter. This hypothesis describes the phenomenon of trophic cascade in specific terrestrial communities.

A mesopredator is a medium-sized, middle trophic level predator, which both preys and is preyed upon. Examples are raccoons, skunks, snakes, cownose rays, and small sharks.

MetapopulationA metapopulation consists of a group of spatially separated populations of the same species which interact at some level. The term metapopulation was coined by Richard Levins in 1969 to describe a model of population dynamics of insect pests in agricultural fields, but the idea has been most broadly applied to species in naturally or artificially fragmented habitats. In Levins' own words, it consists of "a population of populations".A metapopulation is generally considered to consist of several distinct populations together with areas of suitable habitat which are currently unoccupied. In classical metapopulation theory, each population cycles in relative independence of the other populations and eventually goes extinct as a consequence of demographic stochasticity (fluctuations in population size due to random demographic events); the smaller the population, the more chances of inbreeding depression and prone to extinction.

Although individual populations have finite life-spans, the metapopulation as a whole is often stable because immigrants from one population (which may, for example, be experiencing a population boom) are likely to re-colonize habitat which has been left open by the extinction of another population. They may also emigrate to a small population and rescue that population from extinction (called the rescue effect). Such a rescue effect may occur because declining populations leave niche opportunities open to the "rescuers".

The development of metapopulation theory, in conjunction with the development of source-sink dynamics, emphasised the importance of connectivity between seemingly isolated populations. Although no single population may be able to guarantee the long-term survival of a given species, the combined effect of many populations may be able to do this.

Metapopulation theory was first developed for terrestrial ecosystems, and subsequently applied to the marine realm. In fisheries science, the term "sub-population" is equivalent to the metapopulation science term "local population". Most marine examples are provided by relatively sedentary species occupying discrete patches of habitat, with both local recruitment and recruitment from other local populations in the larger metapopulation. Kritzer & Sale have argued against strict application of the metapopulation definitional criteria that extinction risks to local populations must be non-negligible.Finnish biologist Ilkka Hanski of the University of Helsinki was an important contributor to metapopulation theory.

MortalityMortality is the state of being mortal, or susceptible to death; the opposite of immortality.

Mortality may also refer to:

Fish mortality, a parameter used in fisheries population dynamics to account for the loss of fish in a fish stock through death

Mortality (book), a 2012 collection of essays by Anglo-American writer Christopher Hitchens

Mortality (computability theory), a property of a Turing machine if it halts when run on any starting configuration

Mortality rate, a measure for the rate at which deaths occur in a given population

Mortality/differential attrition, an error in the internal validity of a scientific study

Population dynamics of fisheriesA fishery is an area with an associated fish or aquatic population which is harvested for its commercial or recreational value. Fisheries can be wild or farmed. Population dynamics describes the ways in which a given population grows and shrinks over time, as controlled by birth, death, and migration. It is the basis for understanding changing fishery patterns and issues such as habitat destruction, predation and optimal harvesting rates. The population dynamics of fisheries is used by fisheries scientists to determine sustainable yields.The basic accounting relation for population dynamics is the BIDE (Birth, Immigration, Death, Emigration) model, shown as:

N1 = N0 + B − D + I − Ewhere N1 is the number of individuals at time 1, N0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1. While immigration and emigration can be present in wild fisheries, they are usually not measured.

A fishery population is affected by three dynamic rate functions:

Birth rate or recruitment. Recruitment means reaching a certain size or reproductive stage. With fisheries, recruitment usually refers to the age a fish can be caught and counted in nets.

Growth rate. This measures the growth of individuals in size and length. This is important in fisheries where the population is often measured in terms of biomass.

Mortality. This includes harvest mortality and natural mortality. Natural mortality includes non-human predation, disease and old age.If these rates are measured over different time intervals, the harvestable surplus of a fishery can be determined. The harvestable surplus is the number of individuals that can be harvested from the population without affecting long term stability (average population size). The harvest within the harvestable surplus is called compensatory mortality, where the harvest deaths are substituting for the deaths that would otherwise occur naturally. Harvest beyond that is additive mortality, harvest in addition to all the animals that would have died naturally.

Care is needed when applying population dynamics to real world fisheries. Over-simplistic modelling of fisheries has resulted in the collapse of key stocks.

Population ecologyPopulation ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment. It is the study of how the population sizes of species change over time and space. The term population ecology is often used interchangeably with population biology or population dynamics.

The development of population ecology owes much to demography and actuarial life tables. Population ecology is important in conservation biology, especially in the development of population viability analysis (PVA) which makes it possible to predict the long-term probability of a species persisting in a given habitat patch. Although population ecology is a subfield of biology, it provides interesting problems for mathematicians and statisticians who work in population dynamics.

Population modelA population model is a type of mathematical model that is applied to the study of population dynamics.

Threatened speciesThreatened species are any species (including animals, plants, fungi, etc.) which are vulnerable to endangerment in the near future. Species that are threatened are sometimes characterised by the population dynamics measure of critical depensation, a mathematical measure of biomass related to population growth rate. This quantitative metric is one method of evaluating the degree of endangerment.

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