Population dynamics of fisheries - AbsoluteAstronomy.com

Fishery

Generally, a fishery is an entity engaged in raising or harvesting fish which is determined by some authority to be a fishery. According to the FAO, a fishery is typically defined in terms of the "people involved, species or type of fish, area of water or seabed, method of fishing, class of boats,...

is an area with an associated fish

Fish

Fish are a paraphyletic group of organisms that consist of all gill-bearing aquatic vertebrate animals that lack limbs with digits. Included in this definition are the living hagfish, lampreys, and cartilaginous and bony fish, as well as various extinct related groups...

or aquatic

Aquatic animal

An aquatic animal is an animal, either vertebrate or invertebrate, which lives in water for most or all of its life. It may breathe air or extract its oxygen from that dissolved in water through specialised organs called gills, or directly through its skin. Natural environments and the animals that...

population which is harvested for its commercial

Commercial fishing

Commercial fishing is the activity of catching fish and other seafood for commercial profit, mostly from wild fisheries. It provides a large quantity of food to many countries around the world, but those who practice it as an industry must often pursue fish far into the ocean under adverse conditions...

or recreational

Recreational fishing

Recreational fishing, also called sport fishing, is fishing for pleasure or competition. It can be contrasted with commercial fishing, which is fishing for profit, or subsistence fishing, which is fishing for survival....

value. Fisheries can be wild

Wild fisheries of the world

A fishery is an area with an associated fish or aquatic population which is harvested for its commercial value. Fisheries can be marine or freshwater. They can also be wild or farmed. This article is an overview of the habitats occupied by the worlds' wild fisheries, and the human impacts on those...

or farmed. Population dynamics

Population dynamics

Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...

describes the ways in which a given population grows and shrinks over time, as controlled by birth, death, and emigration or immigration. 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 scientist

Fisheries science

Fisheries science is the academic discipline of managing and understanding fisheries. It is a multidisciplinary science, which draws on the disciplines of oceanography, marine biology, marine conservation, ecology, population dynamics, economics and management to attempt to provide an integrated...

s to determine sustainable yields

Sustainable yield in fisheries

The sustainable yield of natural capital is the ecological yield that can be extracted without reducing the base of capital itself, i.e. the surplus required to maintain ecosystem services at the same or increasing level over time. This yield usually varies over time with the needs of the...

.

The basic accounting relation for population dynamics is the BIDE

Matrix population models

Population models are used in population ecology to model the dynamics of wildlife or human populations. Matrix population models are a specific type of population model that uses matrix algebra...

model:

N₁ = N₀ + B − D + I − E

where 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

Wild fisheries of the world

, they are usually not measured.

A fishery population is affected by three dynamic rate functions:

Birth rate
Birth rate
Crude birth rate is the nativity or childbirths per 1,000 people per year . Another word used interchangeably with "birth rate" is "natality". When the crude birth rate is subtracted from the crude death rate, it reveals the rate of natural increase...

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
Biomass (ecology)
Biomass, in ecology, is the mass of living biological organisms in a given area or ecosystem at a given time. Biomass can refer to species biomass, which is the mass of one or more species, or to community biomass, which is the mass of all species in the community. It can include microorganisms,...

.

Mortality
Fish mortality
Fish mortality is a term widely used in fisheries science that denotes the loss of fish from a stock through death. The term is also commonly used in British English as a synonym for fish kill. Fish mortality can be divided into two types:...

. 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

Fish stock

Fish stocks are subpopulations of a particular species of fish, for which intrinsic parameters are the only significant factors in determining population dynamics, while extrinsic factors are considered to be insignificant.-The stock concept:All species have geographic limits to their...

History

The first principle of population dynamics is widely regarded as the exponential law of Malthus, as modelled by the Malthusian growth model

Malthusian growth model

The Malthusian growth model, sometimes called the simple exponential growth model, is essentially exponential growth based on a constant rate of compound interest...

. The early period was dominated by demographic

Demography

Demography is the statistical study of human population. It can be a very general science that can be applied to any kind of dynamic human population, that is, one that changes over time or space...

studies such as the work of Benjamin Gompertz

Benjamin Gompertz

Benjamin Gompertz was a British self educated mathematician and actuary, who became a Fellow of the Royal Society...

and Pierre François Verhulst

Pierre François Verhulst

Pierre François Verhulst was a mathematician and a doctor in number theory from the University of Ghent in 1825...

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, by which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy

Ludwig von Bertalanffy

Karl Ludwig von Bertalanffy was an Austrian-born biologist known as one of the founders of general systems theory . GST is an interdisciplinary practice that describes systems with interacting components, applicable to biology, cybernetics, and other fields...

are covered as special cases of the general formulation.

Population size

The population size

Population size

In population genetics and population ecology, population size is the number of individual organisms in a population.The effective population size is defined as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies under...

(usually denoted by N) is the number of individual organism

Organism

In biology, an organism is any contiguous living system . In at least some form, all organisms are capable of response to stimuli, reproduction, growth and development, and maintenance of homoeostasis as a stable whole.An organism may either be unicellular or, as in the case of humans, comprise...

s in a population

Population

A population is all the organisms that both belong to the same group or species and live in the same geographical area. The area that is used to define a sexual population is such that inter-breeding is possible between any pair within the area and more probable than cross-breeding with individuals...

.

The effective population size

Effective population size

In population genetics, the concept of effective population size Ne was introduced by the American geneticist Sewall Wright, who wrote two landmark papers on it...

(N_e) was defined by Sewall Wright

Sewall Wright

Sewall Green Wright was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. With R. A. Fisher and J.B.S. Haldane, he was a founder of theoretical population genetics. He is the discoverer of the inbreeding coefficient and of...

, who wrote two landmark papers on it (Wright 1931, 1938). He defined it as "the number of breeding individuals in an idealized population that would show the same amount of dispersion of allele frequencies

Allele frequency

Allele frequency or Gene frequency is the proportion of all copies of a gene that is made up of a particular gene variant . In other words, it is the number of copies of a particular allele divided by the number of copies of all alleles at the genetic place in a population. It can be expressed for...

under random genetic drift

Genetic drift

Genetic drift or allelic drift is the change in the frequency of a gene variant in a population due to random sampling.The alleles in the offspring are a sample of those in the parents, and chance has a role in determining whether a given individual survives and reproduces...

or the same amount of inbreeding

Inbreeding

Inbreeding is the reproduction from the mating of two genetically related parents. Inbreeding results in increased homozygosity, which can increase the chances of offspring being affected by recessive or deleterious traits. This generally leads to a decreased fitness of a population, which is...

as the population under consideration". It is a basic parameter in many models in population genetics

Population genetics

Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...

. N_e is usually less than N (the absolute population size).

Small population size

Small population size

Small populations behave differently from larger populations. They often result in population bottlenecks, which have harmful consequences for the survival of that population.-Demographic effects:...

results in increased genetic drift

Genetic drift

. Population bottleneck

Population bottleneck

A population bottleneck is an evolutionary event in which a significant percentage of a population or species is killed or otherwise prevented from reproducing....

s are when population size reduces for a short period of time.

Overpopulation

Overpopulation

Overpopulation is a condition where an organism's numbers exceed the carrying capacity of its habitat. The term often refers to the relationship between the human population and its environment, the Earth...

may indicate any case in which the population of any species of animal may exceed the carrying capacity

Carrying capacity

The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment...

of its ecological niche

Ecological niche

In ecology, a niche is a term describing the relational position of a species or population in its ecosystem to each other; e.g. a dolphin could potentially be in another ecological niche from one that travels in a different pod if the members of these pods utilize significantly different food...

Virtual population analysis

Virtual population analysis (VPA) is a modelling technique commonly used in fisheries science for reconstructing historical fish numbers using information on death of individuals each year. This death is usually partitioned into catch by fisheries and natural mortality.

VPA is the most commonly used term to refer to cohort

Cohort (statistics)

In statistics and demography, a cohort is a group of subjects who have shared a particular time together during a particular time span . Cohorts may be tracked over extended periods in a cohort study. The cohort can be modified by censoring, i.e...

reconstruction techniques used in fisheries. It is virtual in the sense that the population size is not observed or measured directly but is inferred or back-calculated to have been a certain size in the past in order to support the observed fish catches and an assumed death rate owing to non-fishery related causes.

Minimum viable population

The minimum viable population (MVP) is a lower bound on the population of a species, such that it can survive in the wild. More specifically MVP is the smallest possible size at which a biological population can exist without facing extinction from natural disasters or demographic, environmental, or genetic stochastic

Stochastic

Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

ity. The term "population" refers to the population of a species in the wild.

As a reference standard, MVP is usually given with a population survival probability of somewhere between ninety and ninety-five percent and calculated for between one hundred and one thousand years into the future.

The MVP can be calculated using computer simulation

Computer simulation

A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

s known as population viability analyses

Population viability analysis

Population viability analysis is a species-specific method of risk assessment frequently used in conservation biology.It is traditionally defined as the process that determines the probability that a population will go extinct within a given number of years.More recently, PVA has been described...

(PVA), where populations are modelled and future population dynamics are projected.

Maximum sustainable yield

In population ecology

Population ecology

Population 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 living together in groups change over time and space....

and economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, the maximum sustainable yield

Maximum sustainable yield

In population ecology and economics, maximum sustainable yield or MSY is, theoretically, the largest yield that can be taken from a species' stock over an indefinite period...

or MSY is, theoretically, the largest catch that can be taken from a fishery stock over an indefinite period. Under the assumption of logistic growth, the MSY will be exactly at half the carrying capacity

Carrying capacity

of a species, as this is the stage at when population growth is highest. The maximum sustainable yield is usually higher than the optimum sustainable yield

Optimum sustainable yield

In population ecology and economics, optimum sustainable yield is the level of effort that maximizes the difference between total revenue and total cost. Or, where marginal revenue equals marginal cost. This level of effort maximizes the economic profit, or rent, of the resource being utilized...

.

This logistic

Logistic function

A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...

model of growth is produced by a population introduced to a new habitat or with very poor numbers going through a lag phase of slow growth at first. Once it reaches a foothold population it will go through a rapid growth rate that will start to level off once the species approaches carrying capacity. The idea of maximum sustained yield is to decrease population density to the point of highest growth rate possible. This changes the number of the population, but the new number can be maintained indefinitely, ideally.

MSY is extensively used for fisheries management. Unlike the logistic (Schaefer) model, MSY in most modern fisheries models occurs at around 30% of the unexploited population size. This fraction differs among populations depending on the life history of the species and the age-specific selectivity of the fishing method.

However, the approach has been widely criticized as ignoring several key factors involved in fisheries management and has led to the devastating collapse of many fisheries. As a simple calculation, it ignores the size and age of the animal being taken, its reproductive status, and it focuses solely on the species in question, ignoring the damage to the ecosystem caused by the designated level of exploitation and the issue of bycatch. Among conservation biologists

Conservation biology

Conservation biology is the scientific study of the nature and status of Earth's biodiversity with the aim of protecting species, their habitats, and ecosystems from excessive rates of extinction...

it is widely regarded as dangerous and misused.

Recruitment

Recruitment is the number of new young fish that enter a population in a given year. The size of fish populations can fluctuate by orders of magnitude over time, and five to 10-fold variations in abundance are usual. This variability applies across time spans ranging from a year to hundreds of years. Year to year fluctuations in the abundance of short lived forage fish

Forage fish

Forage fish, also called prey fish or bait fish, are small fish which are preyed on by larger predators for food. Predators include other larger fish, seabirds and marine mammals. Typical ocean forage fish feed near the base of the food chain on plankton, often by filter feeding...

can be nearly as great as the fluctuations that occur over decades or centuries. This suggests that fluctuations in reproductive and recruitment success are prime factors behind fluctuations in abundance. Annual fluctuations often seem random, and recruitment success often has a poor relationship to adult stock levels and fishing effort. This makes prediction difficult.

The recruitment problem is the problem of predicting the number of fish larvae in one season that will survive and become juvenile fish in the next season. It has been called "the central problem of fish population dynamics" and “the major problem in fisheries science". Fish produce huge volumes of larvae, but the volumes are very variable and mortality is high. This makes good predictions difficult.

According to Daniel Pauly

Daniel Pauly

Daniel Pauly is a French-born marine biologist, well-known for his work in studying human impacts on global fisheries. He is a professor and the project leader of the Sea Around Us Project at the Fisheries Centre at the University of British Columbia. He also served as Director of the Fisheries...

, the definitive study was made in 1999 by Ransom Myers

Ransom A. Myers

Dr. Ransom Aldrich "Ram" Myers, Jr. was a world-renowned marine biologist and conservationist.He was the son of cotton planter, Ransom Aldrich Myers, Sr. and Fay A. Mitchell Myers...

. Myers solved the problem "by assembling a large base of stock data and developing a complex mathematical model to sort it out. Out of that came the conclusion that a female in general produced three to five recruits per year for most fish.”

Overfishing

The notion of overfishing

Overfishing

Overfishing occurs when fishing activities reduce fish stocks below an acceptable level. This can occur in any body of water from a pond to the oceans....

hinges on what is meant by an acceptable level of fishing.

A current operational model used by some fisheries for predicting acceptable levels is the Harvest Control Rule (HCR). This formalizes and summarizes a management strategy which can actively adapt to subsequent feedback. The HCR is a variable over which the management has some direct control and describes how the harvest is intended to be controlled by management in relation to the state of some indicator of stock status. For example, a harvest control rule can describe the various values of fishing mortality which will be aimed at for various values of the stock abundance. Constant catch and constant fishing mortality are two types of simple harvest control rules.

Biological overfishing occurs when fishing mortality
Mortality rate
Mortality rate is a measure of the number of deaths in a population, scaled to the size of that population, per unit time...

has reached a level where the stock biomass
Biomass
Biomass, as a renewable energy source, is biological material from living, or recently living organisms. As an energy source, biomass can either be used directly, or converted into other energy products such as biofuel....

has negative marginal growth
Marginalism
Marginalism refers to the use of marginal concepts in economic theory. Marginalism is associated with arguments concerning changes in the quantity used of a good or service, as opposed to some notion of the over-all significance of that class of good or service, or of some total quantity...

(slowing down biomass growth), as indicated by the red area in the figure. Fish are being taken out of the water so quickly that the replenishment of stock by breeding slows down. If the replenishment continues to slow down for long enough, replenishment will go into reverse and the population will decrease.

Economic or bioeconomic overfishing additionally considers the cost of fishing and defines overfishing as a situation of negative marginal growth of resource rent
Resource rent
In economics, rent is a surplus value after all costs and normal returns have been accounted for, i.e. the difference between the price at which an output from a resource can be sold and its respective extraction and production costs, including normal return...

. Fish are being taken out of the water so quickly that the growth in the profitability of fishing slows down. If this continues for long enough, profitability will decrease.

Metapopulation

A metapopulation is a group of spatially separated populations of the same species

Species

In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...

which interact at some level. The term was coined by Richard Levins

Richard Levins

Richard "Dick" Levins is a mathematical ecologist, and political activist. He is best known for his work on evolution in changing environments....

in 1969. The idea has been most broadly applied to species in naturally or artificially fragmented habitats

Habitat fragmentation

Habitat fragmentation as the name implies, describes the emergence of discontinuities in an organism's preferred environment , causing population fragmentation...

. In Levins' own words, it consists of "a population of populations".

A metapopulation generally consists of several distinct populations together with areas of suitable habitat which are currently unoccupied. 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 prone it is to extinction.

Although individual populations have finite life-spans, the population 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).

Age class structure

Age can be determined by counting growth rings in fish

Fish

scales, otoliths, cross-sections of fin spines for species with thick spines such as triggerfish

Triggerfish

Triggerfishes are about 40 species of often brightly colored fishes of the family Balistidae. Often marked by lines and spots, they inhabit tropical and subtropical oceans throughout the world, with the greatest species richness in the Indo-Pacific...

, or teeth for a few species. Each method has its merits and drawbacks. Fish scales are easiest to obtain, but may be unreliable if scales have fallen off of the fish and new ones grown in their places. Fin spines may be unreliable for the same reason, and most fish do not have spines of sufficient thickness for clear rings to be visible. Otoliths will have stayed with the fish throughout its life history, but obtaining them requires killing the fish. Also, otoliths often require more preparation before ageing can occur.

An age class structure with gaps in it, for instance a regular bell curve for the population of 1-5 year-old fish, excepting a very low population for the 3-year-olds, implies a bad spawning

Spawn (biology)

Spawn refers to the eggs and sperm released or deposited, usually into water, by aquatic animals. As a verb, spawn refers to the process of releasing the eggs and sperm, also called spawning...

year 3 years ago in that species.

Often fish in younger age class structures have very low numbers because they were small enough to slip through the sampling

Sampling (statistics)

In statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....

nets, and may in fact have a very healthy population.

Population cycle

A population cycle occurs where population

Population

s rise and fall over a predictable period of time. There are some species where population numbers have reasonably predictable patterns of change although the full reasons for population cycles is one of the major unsolved ecological problems. There are a number of factors which influence population change such as availability of food, predators, diseases and climate.

Trophic cascades

Trophic cascades occur when predators in a food chain

Food chain

A food web depicts feeding connections in an ecological community. Ecologists can broadly lump all life forms into one of two categories called trophic levels: 1) the autotrophs, and 2) the heterotrophs...

suppress the abundance of their prey, thereby releasing the next lower trophic level

Trophic level

The trophic level of an organism is the position it occupies in a food chain. The word trophic derives from the Greek τροφή referring to food or feeding. A food chain represents a succession of organisms that eat another organism and are, in turn, eaten themselves. The number of steps an organism...

from predation

Predation

In ecology, predation describes a biological interaction where a predator feeds on its prey . Predators may or may not kill their prey prior to feeding on them, but the act of predation always results in the death of its prey and the eventual absorption of the prey's tissue through consumption...

(or herbivory if the intermediate trophic level is an herbivore

Herbivore

Herbivores are organisms that are anatomically and physiologically adapted to eat plant-based foods. Herbivory is a form of consumption in which an organism principally eats autotrophs such as plants, algae and photosynthesizing bacteria. More generally, organisms that feed on autotrophs in...

). For example, if the abundance of large piscivorous

Piscivore

A piscivore is a carnivorous animal which eats primarily fish. Piscivory was the diet of early tetrapods , insectivory came next, then in time reptiles added herbivory....

fish is increased in a lake

Lake

A lake is a body of relatively still fresh or salt water of considerable size, localized in a basin, that is surrounded by land. Lakes are inland and not part of the ocean and therefore are distinct from lagoons, and are larger and deeper than ponds. Lakes can be contrasted with rivers or streams,...

, the abundance of their prey, zooplanktivorous

Zooplankton

Zooplankton are heterotrophic plankton. Plankton are organisms drifting in oceans, seas, and bodies of fresh water. The word "zooplankton" is derived from the Greek zoon , meaning "animal", and , meaning "wanderer" or "drifter"...

fish

Fish

, should decrease, large zooplankton

Zooplankton

abundance should increase, and phytoplankton

Phytoplankton

Phytoplankton are the autotrophic component of the plankton community. The name comes from the Greek words φυτόν , meaning "plant", and πλαγκτός , meaning "wanderer" or "drifter". Most phytoplankton are too small to be individually seen with the unaided eye...

biomass

Biomass

Biomass, as a renewable energy source, is biological material from living, or recently living organisms. As an energy source, biomass can either be used directly, or converted into other energy products such as biofuel....

should decrease. This theory has stimulated new research in many areas of ecology

Ecology

Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...

. Trophic cascades may also be important for understanding the effects of removing top predators from food webs, as humans have done in many places through hunting and fishing activities.

Classic examples

In lakes, piscivorous
Piscivore
A piscivore is a carnivorous animal which eats primarily fish. Piscivory was the diet of early tetrapods , insectivory came next, then in time reptiles added herbivory....

fish can dramatically reduce populations of zooplanktivorous
Zooplankton
Zooplankton are heterotrophic plankton. Plankton are organisms drifting in oceans, seas, and bodies of fresh water. The word "zooplankton" is derived from the Greek zoon , meaning "animal", and , meaning "wanderer" or "drifter"...

fish, zooplanktivorous
Zooplankton
Zooplankton are heterotrophic plankton. Plankton are organisms drifting in oceans, seas, and bodies of fresh water. The word "zooplankton" is derived from the Greek zoon , meaning "animal", and , meaning "wanderer" or "drifter"...

fish can dramatically alter freshwater
Freshwater
Fresh water is naturally occurring water on the Earth's surface in ice sheets, ice caps, glaciers, bogs, ponds, lakes, rivers and streams, and underground as groundwater in aquifers and underground streams. Fresh water is generally characterized by having low concentrations of dissolved salts and...

zooplankton
Zooplankton
Zooplankton are heterotrophic plankton. Plankton are organisms drifting in oceans, seas, and bodies of fresh water. The word "zooplankton" is derived from the Greek zoon , meaning "animal", and , meaning "wanderer" or "drifter"...

communities, and zooplankton
Zooplankton
Zooplankton are heterotrophic plankton. Plankton are organisms drifting in oceans, seas, and bodies of fresh water. The word "zooplankton" is derived from the Greek zoon , meaning "animal", and , meaning "wanderer" or "drifter"...

grazing can in turn have large impacts on phytoplankton
Phytoplankton
Phytoplankton are the autotrophic component of the plankton community. The name comes from the Greek words φυτόν , meaning "plant", and πλαγκτός , meaning "wanderer" or "drifter". Most phytoplankton are too small to be individually seen with the unaided eye...

communities. Removal of piscivorous fish can change lake water from clear to green by allowing phytoplankton to flourish.
In the Eel River
Eel River (California)
The Eel River is a major river system of the northern Pacific coast of the U.S. state of California. Approximately 200 miles long, it drains a rugged area in the California Coast Ranges between the Sacramento Valley and the ocean. For most of its course, the river flows northwest, parallel to the...

, in Northern California
California
California is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...

, fish (steelhead and roach
California roach
The California roach, Hesperoleucus symmetricus, is a cyprinid fish native to western North America and abundant in the intermittent streams throughout central California. It is the sole member of its genus....

) consume fish larvae and predatory insects. These smaller predators prey on midge
Midge
A midge is a very small, two-winged flying insect. "Midge" may also refer to:-Real:* Midge Costanza , American politician* Mildred Gillars , aka "Midge", American broadcaster of Nazi propaganda during World War II...

larvae, which feed on algae
Algae
Algae are a large and diverse group of simple, typically autotrophic organisms, ranging from unicellular to multicellular forms, such as the giant kelps that grow to 65 meters in length. They are photosynthetic like plants, and "simple" because their tissues are not organized into the many...

. Removal of the larger fish increases the abundance of algae.
In Pacific
Pacific Ocean
The Pacific Ocean is the largest of the Earth's oceanic divisions. It extends from the Arctic in the north to the Southern Ocean in the south, bounded by Asia and Australia in the west, and the Americas in the east.At 165.2 million square kilometres in area, this largest division of the World...

kelp forest
Kelp forest
Kelp forests are underwater areas with a high density of kelp. They are recognized as one of the most productive and dynamic ecosystems on Earth. Smaller areas of anchored kelp are called kelp beds....

s, sea otters feed on sea urchin
Sea urchin
Sea urchins or urchins are small, spiny, globular animals which, with their close kin, such as sand dollars, constitute the class Echinoidea of the echinoderm phylum. They inhabit all oceans. Their shell, or "test", is round and spiny, typically from across. Common colors include black and dull...

s. In areas where sea otters have been hunt
Hunting
Hunting is the practice of pursuing any living thing, usually wildlife, for food, recreation, or trade. In present-day use, the term refers to lawful hunting, as distinguished from poaching, which is the killing, trapping or capture of the hunted species contrary to applicable law...

ed to extinction
Local extinction
Local extinction, also known as extirpation, is the condition of a species which ceases to exist in the chosen geographic area of study, though it still exists elsewhere...

, sea urchins increase in abundance and decimate kelp
Kelp
Kelps are large seaweeds belonging to the brown algae in the order Laminariales. There are about 30 different genera....

A recent theory, the mesopredator release hypothesis

Mesopredator release hypothesis

The mesopredator release hypothesis is a relatively new hypothesis from 1988 which describes the phenomenon of trophic cascade in certain terrestrial communities. It states that as top predators decline in an ecosystem, an increase in the populations of mesopredators occurs...

, states that the decline of top predators in an ecosystem results in increased populations of medium-sized predators (mesopredators).

Basic models

The classic population equilibrium model is Verhulst's
Pierre François Verhulst
Pierre François Verhulst was a mathematician and a doctor in number theory from the University of Ghent in 1825...

1838 growth model:

where N(t) represents number of individuals at time t, r the intrinsic growth rate and K is the carrying capacity

Carrying capacity

, or the maximum number of individuals that the environment can support.

The individual growth model, published by von Bertalanffy
Ludwig von Bertalanffy
Karl Ludwig von Bertalanffy was an Austrian-born biologist known as one of the founders of general systems theory . GST is an interdisciplinary practice that describes systems with interacting components, applicable to biology, cybernetics, and other fields...

in 1934, can be used to model the rate at which fish grow. It exists in a number of versions, but in its simplest form it is expressed as a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

of length (L) over time (t):

where r_B is the von Bertalanffy growth rate and L_∞ the ultimate length of the individual.

Schaefer
Milner Baily Schaefer
Milner Baily Schaefer was born in Cheyenne, Wyoming, in 1912 and died at the age of 57 in San Diego, California in 1970. He is notable for his work on the population dynamics of fisheries....

published a fishery equilibrium model based on the Verhulst
Pierre François Verhulst
Pierre François Verhulst was a mathematician and a doctor in number theory from the University of Ghent in 1825...

model with an assumption of a bi-linear catch equation, often referred to as the Schaefer short-term catch equation:

where the variables are; H, referring to catch (harvest) over a given period of time (e.g. a year); E, the fishing effort over the given period; X, the fish stock biomass at the beginning of the period (or the average biomass), and the parameter q represents the catchability of the stock.

Assuming the catch to equal the net natural growth in the population over the same period (

), the equilibrium catch is a function of the long term fishing effort E:

r and K being biological parameters representing intrinsic growth rate and natural equilibrium biomass respectively.

The Ricker model
Ricker model
The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number a t+1 of individuals in generation t + 1 as a function of the number of individuals in the previous generation,Here r is interpreted as an intrinsic growth rate and k as...

is a classic discrete population model which gives the expected number (or density) of individuals N_t + 1 in generation t + 1 as a function of the number of individuals in the previous generation,

Here r is interpreted as an intrinsic growth rate and k as the carrying capacity

Carrying capacity

of the environment. The Ricker model was introduced in the context of the fisheries by Ricker

Bill Ricker

William Edwin Ricker, OC, FRSC is an important founder of fisheries science. He is best known for the Ricker model, which he developed in his studies of stock and recruitment in fisheries. The model can be used to predict the number of fish that will be present in a fishery...

(1954).

The Beverton–Holt model, introduced in the context of fisheries in 1957, is a classic discrete-time population model which gives the expected
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

number n_t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,

Here R₀ is interpreted as the proliferation rate per generation and K = (R₀ − 1) M is the carrying capacity

Carrying capacity

of the environment.

Nurgaliev's law
Nurgaliev's law
In population dynamics, Nurgaliev's law is an equation that describes the rate of change of the size of a population at a given time, in terms of the current population size...

says

where N is the size of a population, a is a half of the average probability of a birth of a male (the same for females) of a potential arbitrary parents pair within a year, and b is an average probability of a death of a fish within a year.

Predator-prey equations

The classic predator-prey equations are a pair of first order, non-linear, differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s used to describe the dynamics of biological systems

Systems biology

Systems biology is a term used to describe a number of trends in bioscience research, and a movement which draws on those trends. Proponents describe systems biology as a biology-based inter-disciplinary study field that focuses on complex interactions in biological systems, claiming that it uses...

in which two species interact, one a predator and one its prey. They were proposed independently by Alfred J. Lotka

Alfred J. Lotka

Alfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist best known for his proposal of the predator-prey model, developed simultaneously but independently of Vito Volterra...

in 1925 and Vito Volterra

Vito Volterra

Vito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....

in 1926.

An extension to these are the competitive Lotka-Volterra equations

Competitive Lotka-Volterra equations

The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to include trophic interactions.-Overview:...

, which provide a simple model of the population dynamics of species competing for some common resource.

In the 1930s Alexander Nicholson

Alexander John Nicholson

Alexander John Nicholson was an Irish Australian entomologist who specialised in insect population dynamics. He was Chief of the CSIR / CSIRO Division of Economic Entomology for 24 years and is credited with initiating the professional era in Australian entomology...

and Victor Bailey

Victor Albert Bailey

Victor Albert Bailey, 18 December 1895, Alexandria, Egypt - 7 December 1964, Geneva, Switzerland was a British-Australian physicist. He was the eldest of four surviving children of William Henry Bailey, a British Army engineer, and his wife Suzana, née Lazarus, an expatriate Romanian linguist.He...

developed a model to describe the population dynamics of a coupled predator-prey system. The model assumes that predators search for prey at random, and that both predators and prey are assumed to be distributed in a non-contiguous ("clumped") fashion in the environment.

External links

Predator-prey model – Scholarpedia
Growth models Mikael Höök, Uppsala University, 2009

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

History

Population size

Virtual population analysis

Minimum viable population

Maximum sustainable yield

Recruitment

Overfishing

Metapopulation

Age class structure

Population cycle

Trophic cascades

Basic models

Predator-prey equations

See also

Further reading

External links