View source for Population dynamics of fisheries

[[File:Moofushi Kandu fish.jpg|thumb|300px|right|Predator [[bluefin trevally]] sizing up [[Shoaling and schooling|schooling]] [[anchovy|anchovies]], in the [[Maldives]]]]
A [[fishery]] is an area with an associated [[fish]] or [[Aquatic animal|aquatic]] population which is harvested for its [[Commercial fishing|commercial]] or [[Recreational fishing|recreational]] value. Fisheries can be [[Wild fisheries of the world|wild]] or [[Fish farm|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 science|fisheries scientist]]s to determine [[Sustainable yield in fisheries|sustainable yields]].<ref>{{cite journal |last1=Wilderbuer |first1=Thomas K. |last2=Zhang |first2=Chang Ik |year=1999 |title=Evaluation of the population dynamics and yield characteristics of Alaska plaice, ''Pleuronectes quadrituberculatus'', in the eastern Bering Sea |journal=Fisheries Research |volume=41 |issue=2 |pages=183–200 |doi=10.1016/S0165-7836(99)00012-0 }}</ref><ref>{{cite journal |first1=Richard W. |last1=Zabel |first2=Chris J. |last2=Harvey |first3=Steven L. |last3=Katz |first4=Thomas P. |last4=Good |first5=Phillip S. |last5=Levin |display-authors=1 |year=2003 |title=Ecologically Sustainable Yield |journal=American Scientist |volume=91 |issue=2 |pages=150–157 |doi=10.1511/2003.2.150 |jstor=27858183 }}</ref>

The basic accounting relation for population dynamics is the [[Matrix population models|BIDE]] (Birth, Immigration, Death, Emigration) model, shown as:<ref>{{cite book |last=Caswell |first=H. |year=2001 |title=Matrix population models: Construction, analysis and interpretation |edition=2nd |publisher=Sinauer Associates |location=Sunderland, Massachusetts |isbn=0-87893-096-5 }}</ref>

: ''N''<sub>1</sub> = ''N''<sub>0</sub> + ''B'' &minus; ''D'' + ''I'' &minus; ''E''

where ''N''<sub>1</sub> is the number of individuals at time 1, ''N''<sub>0</sub> 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 of the world|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 (ecology)|biomass]].
*[[Fish mortality|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 [[Fish stocks|stocks]].<ref name=Epitaph /><ref name=Lessons />

==History==
The first principle of population dynamics is widely regarded as the exponential law of [[Malthus]], as modelled by the [[Malthusian growth model]]. The early period was dominated by [[demography|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, by which the models of Gompertz, Verhulst and also [[Ludwig von Bertalanffy]] are covered as special cases of the general formulation.<ref>{{cite journal |last=Richards |first=F. J. |year=1959 |title=A Flexible Growth Function for Empirical Use |journal=Journal of Experimental Botany |volume=10 |issue=2 |pages=290–301 |doi=10.1093/jxb/10.2.290 }}</ref>

==Population size==
The [[population size]] (usually denoted by ''N'') is the number of individual [[organism]]s in a [[population]].

The [[effective population size]] (''N''<sub>''e''</sub>) was defined by [[Sewall Wright]], 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 frequency|allele frequencies]] under random [[genetic drift]] or the same amount of [[inbreeding]] as the population under consideration". It is a basic parameter in many models in [[population genetics]]. ''N''<sub>''e''</sub> is usually less than ''N'' (the absolute population size).

[[Small population size]] results in increased [[genetic drift]].  [[Population bottleneck]]s are when population size reduces for a short period of time.

[[Overpopulation (biology)|Overpopulation]] may indicate any case in which the population of any species of animal may exceed the [[carrying capacity]] of its [[ecological niche]].

==Virtual population analysis==
{{Main|Virtual population analysis}}
'''Virtual population analysis (VPA)''' is a [[Cohort (statistics)|cohort]] modeling technique commonly used in [[fisheries science]] for reconstructing historical fish numbers at age using information on death of individuals each year. This death is usually partitioned into catch by fisheries and [[Fish mortality|natural mortality]]. VPA 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==
{{Main|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]]ity.<ref>{{cite web |last=Holsinger |first=Kent |date=2007 |archive-url=https://web.archive.org/web/20081120023317/http://darwin.eeb.uconn.edu/eeb310/lecture-notes/small-populations/node4.html |archive-date=November 20, 2008 |url=http://darwin.eeb.uconn.edu/eeb310/lecture-notes/small-populations/node4.html |title=Types of stochastic threats |access-date=2007-11-04 }}</ref> 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]]s known as [[population viability analysis|population viability analyses]] (PVA), where populations are modelled and future population dynamics are projected.

==Maximum sustainable yield==
{{main|Maximum sustainable yield}}
In [[population ecology]] and [[economics]], the [[maximum sustainable yield]] or '''MSY''' is, theoretically, the largest catch that can be taken from a fishery stock over an indefinite period.<ref>{{cite web |author=European Union |date=2006 |url=http://europa.eu/scadplus/leg/en/lvb/l66037.htm |title=Management based on maximum sustainable yield |archive-url=https://web.archive.org/web/20071223064801/http://europa.eu/scadplus/leg/en/lvb/l66037.htm |archive-date=2007-12-23 }}</ref><ref>{{cite web |author=European Union |date=2006 |url=http://europa.eu/rapid/pressReleasesAction.do?reference=MEMO/06/268&format=HTML&aged=0&language=EN&guiLanguage=en |title=Questions and Answers on Maximum Sustainable Yield (MSY) }}</ref> Under the assumption of logistic growth, the MSY will be exactly at half the [[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]].

This [[logistic function|logistic]] 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.<ref>{{cite web|author=WWF Publications|date=2007|title=The Maximum Sustainable Yield objective in Fisheries|url=https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/maximum-sustainable-yield|archive-url=|archive-date=|access-date=|website=}}{{dead link|date=February 2021|bot=Dlmarial|fix-attempted=yes}}
</ref><ref>{{cite web |author=New Zealand Ministry of Fisheries |url=http://www.fish.govt.nz/en-nz/SOF/Indicators.htm |title=MSY Harvest Strategies }}</ref>  Unlike the logistic (Schaefer) model, MSY in most modern fisheries models occurs at around 30-40% of the unexploited population size.<ref>See e.g. {{cite journal |last1=Thorpe |first1=Robert B. |first2=Will J. F. |last2=Le Quesne |first3=Fay |last3=Luxford |first4=Jeremy S. |last4=Collie |first5=Simon |last5=Jennings |display-authors=1 |year=2015 |title=Evaluation and management implications of uncertainty in a multispecies size-structured model of population and community responses to fishing |journal=Methods in Ecology and Evolution |volume=6 |issue=1 |pages=49–58 |doi=10.1111/2041-210X.12292 |pmid=25866615 |pmc=4390044 }}</ref> 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 biology|conservation biologists]] it is widely regarded as dangerous and misused.<ref name=Epitaph>{{cite journal |last=Larkin |first=P. A. |year=1977 |title=An Epitaph for the Concept of Maximum Sustained Yield |journal=Transactions of the American Fisheries Society |volume=106 |issue=1 |pages=1–11 |doi=10.1577/1548-8659(1977)106<1:AEFTCO>2.0.CO;2 }}</ref><ref name=Lessons>{{cite journal |author-link=Carl Walters |last1=Walters |first1=C. |last2=Maguire |first2=J. |year=1996 |title=Lessons for stock assessment from the northern cod collapse |journal=Reviews in Fish Biology and Fisheries |volume=6 |issue= 2|pages=125–137 |doi=10.1007/BF00182340 |s2cid=20224324 }}</ref>

==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]] 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.<ref>{{cite book |last=Houde |first=D. E. |year=2009 |chapter=Recruitment variability |editor-last=Jakobsen |editor-first=T. |editor2-last=Fogarty |editor2-first=M. J. |editor3-last=Megrey |editor3-first=B. A. |editor4-last=Moksness |editor4-first=E. |title=Fish Reproductive Biology |publisher=Wiley-Blackwell |isbn=978-1-4051-2126-2 }}</ref>

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"<ref>{{cite book |last=Beyer |first=J. E. |year=1981 |title=Aquatic ecosystems-an operational research approach |publisher=University of Washington Press |isbn=0-295-95719-0 }}
</ref> and “the major problem in fisheries science".<ref name="NYT">{{cite news |url=https://www.nytimes.com/2007/03/29/obituaries/29myers.html |title=Ransom A. Myers, 54, Dies; Expert on Loss of Fish Stocks |newspaper=The New York Times |date=29 March 2007 }}</ref> Fish produce huge volumes of larvae, but the volumes are very variable and mortality is high. This makes good predictions difficult.<ref>{{cite journal |last=Bakun |first=A. |year=1985 |url=http://www.calcofi.org/newhome/publications/CalCOFI_Reports/v26/pdfs/Vol_26_Bakun.pdf |title=Comparative studies and the recruitment problem: Searching for generalizations |archive-url=https://web.archive.org/web/20081201100227/http://www.calcofi.org/newhome/publications/CalCOFI_Reports/v26/pdfs/Vol_26_Bakun.pdf |archive-date=2008-12-01 |journal=CalCOFI Report |volume=26 }}</ref>

According to [[Daniel Pauly]],<ref name="NYT"/><ref>{{cite journal |url=http://seaaroundus.org/magazines/2007/Nature_RansomAldrichMyers.pdf |title=Ransom Aldrich Myers (1952-2007) |first=Daniel |last=Pauly |journal=Nature |date=10 May 2007 |volume=447 |issue=7141 |page=160 |doi=10.1038/447160a |pmid=17495917 |s2cid=4430142 }}</ref> the definitive study was made in 1999 by [[Ransom A. Myers|Ransom Myers]].<ref>{{cite journal |last=Myers |first=R. A. |date=1995 |archive-url=https://wayback.archive-it.org/all/20110425162945/http://www.fmap.ca/ramweb/papers-total/rec_marine_fish.pdf |archive-date=April 25, 2011 |url=http://www.fmap.ca/ramweb/papers-total/rec_marine_fish.pdf |title=Recruitment of marine fish: the relative roles of density-dependent and density-independent mortality in the egg, larval, and juvenile stages |journal=Marine Ecology Progress Series |volume=128 |pages=305–310 }}</ref> 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.”<ref name="NYT"/>

==Fishing effort==
Fishing effort is a measure of the anthropogenic work input used to catch fish. A good measure of the fishing effort will be approximately proportional to the amount of fish captured. Different measures are appropriate for different kinds of fisheries. For example, the fishing effort exerted by a [[fishing fleet]] in a trawl fishery might be measured by summing the products of the engine power for each boat and time it spent at sea (KW × days). For a gill-net fishery the effort might be measured by summing the products of the length of each set net and the time it was set in the water (Km × soak time). In fisheries where boats spend a lot of time looking for fish, a measure based on search time may be used.<ref>{{cite web |url=https://stats.oecd.org/glossary/detail.asp?ID=994OECD |title=Fishing effort |work=Glossary of Statistical Terms |publisher=Review of Fisheries in OECD Countries, 1998 |date=5 March 2003 |access-date=21 April 2019 }}</ref><ref>{{cite web |url=https://ec.europa.eu/fisheries/cfp/fishing_rules/fishing_effort_en |title=Fishing effort |publisher=European Commission |access-date=21 April 2019 }}</ref><ref>{{cite web |url=http://www.fao.org/gfcm/activities/fisheries/fleet/fishing-effort/en/ |title=Fishing effort measurement |publisher=Food and Agriculture Organization of the United Nations |access-date=21 April 2019 }}</ref>

==Overfishing==
[[File:Harvest Control Rule graph.gif|thumb|right|300px|The Traffic Light colour convention, showing the concept of Harvest Control Rule (HCR), specifying when a rebuilding plan is mandatory in terms of [[Precautionary principle|precautionary]] and limit reference points for [[Spawn (biology)|spawning]] [[biomass]] and fishing [[Fish mortality|mortality rate]].]]
{{Main|Overfishing}}
The notion of [[overfishing]] 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.<ref>{{cite web|last1=Coad|first1=Brian W.|last2=McAllister|first2=Don E.|title=Dictionary of Ichthyology|url=http://www.briancoad.com/dictionary/H.htm|access-date=2 January 2015|date=2008|url-status=dead|archive-url=https://web.archive.org/web/20110720124618/http://www.briancoad.com/dictionary/H.htm|archive-date=20 July 2011}}</ref>

* '''Biological overfishing''' occurs when fishing [[Mortality rate|mortality]] has reached a level where the stock [[biomass]] has negative [[marginalism|marginal growth]] (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]].  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==
{{Main|Metapopulation}}
A metapopulation is a group of spatially separated populations of the same [[species]] which interact at some level.  The term was coined by [[Richard Levins]] in 1969. The idea has been most broadly applied to species in naturally or artificially [[habitat fragmentation|fragmented habitats]]. In Levins' own words, it consists of "a population of populations".<ref>{{cite journal |last=Levins |first=R. |year=1969 |title=Some demographic and genetic consequences of environmental heterogeneity for biological control |journal=Bulletin of the Entomological Society of America |volume=15 |issue=3 |pages=237–240 |doi=10.1093/besa/15.3.237 }}</ref>

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]]'').  
<gallery>
Image:Thomas Malthus.jpg|Malthus
Image:gompertz.png|Gompertz
Image:Pierre Francois Verhulst.jpg|Verhulst
</gallery>

==Age class structure==
{{Main|Age class structure}}
Age can be determined by counting growth rings in [[fish]] scales, [[otoliths]], cross-sections of fin spines for species with thick spines such as [[triggerfish]], 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 [[normal distribution|bell curve]] for the population of 1-5 year-old fish, excepting a very low population for the 3-year-olds, implies a bad [[Spawn (biology)|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 (statistics)|sampling]] nets, and may in fact have a very healthy population.

==Population cycle==
{{Main|Population cycle}}
A population cycle occurs where [[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==
{{Main|Trophic cascade}}
Trophic cascades occur when [[predator]]s in a [[food chain]] suppress the abundance of their [[prey]], thereby releasing the next lower [[trophic level]] from [[predation]] (or [[herbivory]] if the intermediate trophic level is an [[herbivore]]). For example, if the abundance of large [[piscivore|piscivorous]] fish is increased in a [[lake]], the abundance of their prey, [[zooplankton|zooplanktivorous]] [[fish]], should decrease, large [[zooplankton]] abundance should increase, and [[phytoplankton]] [[biomass]] should decrease. This theory has stimulated new research in many areas of [[ecology]]. 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 [[lake]]s, [[piscivore|piscivorous]] fish can dramatically reduce populations of [[zooplankton|zooplanktivorous]] fish, [[zooplankton|zooplanktivorous]] fish can dramatically alter [[freshwater]] [[zooplankton]] communities, and [[zooplankton]] grazing can in turn have large impacts on [[phytoplankton]] communities. Removal of piscivorous fish can change lake water from clear to green by allowing phytoplankton to flourish.<ref>{{cite journal |last1=Carpenter |first1=S. R. |last2=Kitchell |first2=J. F. |last3=Hodgson |first3=J. R. |year=1985 |title=Cascading trophic interactions and lake productivity |journal=BioScience |volume=35 |issue=10 |pages=634–639 |doi=10.2307/1309989 |jstor=1309989 }}</ref>
# In the [[Eel River (California)|Eel River]], in Northern [[California]], fish ([[Rainbow trout|steelhead]] and [[California Roach|roach]]) consume fish larvae and predatory [[insects]]. These smaller [[predator]]s prey on [[midge]] larvae, which feed on [[algae]]. Removal of the larger fish increases the abundance of algae.<ref>{{cite journal |last=Power |first=M. E. |year=1990 |title=Effects of fish in river food webs |journal=Science |volume=250 |issue=4982 |pages=811–814 |doi=10.1126/science.250.4982.811 |pmid=17759974 |bibcode=1990Sci...250..811P |s2cid=24780727 }}</ref>
# In [[Pacific Ocean|Pacific]] [[kelp forest]]s, [[sea otters]] feed on [[sea urchin]]s. In areas where sea otters have been [[hunting|hunt]]ed to [[local extinction|extinction]], sea urchins increase in abundance and decimate [[kelp]]<ref>{{cite journal |last1=Estes |first1=J. A. |last2=Palmisano |first2=J. F. |year=1974 |title=Sea otters: their role in structuring nearshore communities |journal=Science |volume=185 |issue= 4156|pages=1058–1060 |doi=10.1126/science.185.4156.1058 |pmid=17738247 |bibcode=1974Sci...185.1058E |s2cid=35892592 }}</ref>

A recent theory, the [[mesopredator release hypothesis]], 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 [[Pierre François Verhulst|Verhulst's]] 1838 [[Logistic function#In ecology: modeling population growth|growth model]]: <math display="block"> \frac{dN}{dt} = r N \left(1 - \frac {N}{K} \right)</math> where ''N''(''t'') represents number of individuals at time ''t'', ''r'' the intrinsic growth rate and ''K'' is the [[carrying capacity]], or the maximum number of individuals that the environment can support.
* The individual growth model, published by [[Ludwig von Bertalanffy|von Bertalanffy]] 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]] of length (''L'') over time (''t''): <math display="block">L'(t) = r_B \left( L_\infty - L(t) \right)</math> where ''r''<sub>''B''</sub> is the von Bertalanffy growth rate and ''L''<sub>&infin;</sub> the ultimate length of the individual.
* [[Milner Baily Schaefer|Schaefer]] published a fishery equilibrium model based on the [[Pierre François Verhulst|Verhulst]] model with an assumption of a bi-linear catch equation, often referred to as the Schaefer short-term catch equation: <math display="block">H(E,X)=q E X\!</math> 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. {{pb}} Assuming the catch to equal the net natural growth in the population over the same period (<math>\dot{X}=0</math>), the equilibrium catch is a function of the long term fishing effort ''E'': <math display="block">H(E) = q K E \left(1 - \frac{qE}{r}\right)</math> ''r'' and ''K'' being biological parameters representing intrinsic growth rate and natural equilibrium biomass respectively.
* The [[Baranov catch equation]] of 1918 is perhaps the most used equation in fisheries modelling.<ref>{{Cite journal | last = Quinn | first = Terrance J. II| title = Ruminations on the development and future of population dynamics models in fisheries | doi = 10.1111/j.1939-7445.2003.tb00119.x | journal = Natural Resource Modeling | volume = 16 | issue = 4 | pages = 341–392| year = 2003| citeseerx = 10.1.1.473.3765| s2cid = 153420994}}</ref> It gives the catch in numbers as a function of initial population abundance ''N''<sub>''0''</sub> and fishing ''F'' and natural mortality ''M'': <math display="block">C = \frac{F}{F+M} \left(1-e^{-(F+M)T}\right) N_0</math> where ''T'' is the time period and is usually left out (i.e. ''T''=1 is assumed). The equation assumes that fishing and natural mortality occur simultaneously and thus "compete" with each other. The first term expresses the proportion of deaths that are caused by fishing, and the second and third term the total number of deaths.<ref>{{cite journal |last=Baranov |first=F. I. |year=1918 |title=On the question of the biological basis of fisheries |journal=Izvestiya |volume=1 |pages=81–128 }} (Translated from Russian by W.E. Ricker, 1945)</ref>
* The [[Ricker model]] is a classic discrete population model which gives the expected number (or density) of individuals ''N''<sub>''t''&nbsp;+&nbsp;1</sub> in generation ''t''&nbsp;+&nbsp;1 as a function of the number of individuals in the previous generation, <math display="block">N_{t+1} = N_t e^{r (1 - {N_t}/{k})}</math> Here ''r'' is interpreted as an intrinsic growth rate and ''k'' as the [[carrying capacity]] of the environment. The Ricker model was introduced in the context of the fisheries by [[Bill Ricker|Ricker]] (1954).<ref>Ricker, WE (1954). Stock and recruitment. Journal of the Fisheries Research Board of Canada.</ref>
* The [[Beverton–Holt model]], introduced in the context of fisheries in 1957, is a classic discrete-time population model which gives the [[expected value|expected]] number ''n''<sub>&nbsp;''t''+1</sub> (or density) of individuals in generation ''t''&nbsp;+&nbsp;1 as a function of the number of individuals in the previous generation, <math display="block">n_{t+1} = \frac{R_0 n_t}{1+ n_t/M}. </math> Here ''R''<sub>0</sub> is interpreted as the proliferation rate per generation and ''K''&nbsp;=&nbsp;(''R''<sub>0</sub>&nbsp;&minus;&nbsp;1)&nbsp;''M'' is the [[carrying capacity]] of the environment.

==Predator–prey equations==
{{See also|Lotka–Volterra equations|Competitive Lotka–Volterra equations|Nicholson–Bailey model}}
The classic predator–prey equations are a pair of first order, [[non-linear]], [[differential equation]]s used to describe the dynamics of [[Systems biology|biological systems]] in which two species interact, one a predator and one its prey. They were proposed independently by [[Alfred J. Lotka]] in 1925 and [[Vito Volterra]] in 1926.

An extension to these are the [[competitive Lotka–Volterra equations]], which provide a simple model of the population dynamics of species competing for some common resource.

In the 1930s [[Alexander John Nicholson|Alexander Nicholson]] and [[Victor Albert Bailey|Victor Bailey]] 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.<ref name="Hopper">{{cite journal |last=Hopper |first=J. L. |year=1987 |title=Opportunities and Handicaps of Antipodean Scientists: A. J. Nicholson and V. A. Bailey on the Balance of Animal Populations |journal=Historical Records of Australian Science |volume=7 |issue=2 |pages=179–188 |doi=10.1071/HR9880720179 |url=http://www.publish.csiro.au/paper/HR9880720179.htm }}</ref>

In the late 1980s, a credible, simple alternative to the Lotka–Volterra predator-prey model (and its common prey dependent generalizations) emerged, the ratio dependent or [[Arditi–Ginzburg equations|Arditi–Ginzburg model]].<ref>{{cite journal |last1=Arditi |first1=R. |last2=Ginzburg |first2=L. R. |year=1989 |title=Coupling in predator-prey dynamics: ratio dependence |journal=Journal of Theoretical Biology |volume=139 |issue=3 |pages=311–326 |doi=10.1016/S0022-5193(89)80211-5 |bibcode=1989JThBi.139..311A }}</ref> The two are the extremes of the spectrum of predator interference models. According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka–Volterra extreme on the interference spectrum that the model can simply be discounted as wrong. They are much closer to the ratio dependent extreme, so if a simple model is needed one can use the Arditi-Ginzburg model as the first approximation.<ref>{{cite book |last1=Arditi |first1=R. |last2=Ginzburg |first2=L. R. |year=2012 |title=How Species Interact: Altering the Standard View on Trophic Ecology |publisher=Oxford University Press |location=New York }}</ref>

==See also==
{{Div col|colwidth=20em}}
* [[Ecosystem model]]
* [[Depensation]]
* [[Huffaker's mite experiment]]
* [[Wild fisheries]]
* [[Overfishing]]
* [[Overexploitation]]
* [[Population modeling]]
* [[Fishing down the food web]]
* [[Tragedy of the commons]]{{Div col end}}

==References==
{{reflist|2}}

==Further reading==
* Berryman, Alan (2002) ''Population Cycles.'' Oxford University Press US. {{ISBN|0-19-514098-2}}
* [[Gerda de Vries|de Vries, Gerda]]; Hillen, Thomas; Lewis, Mark; Schonfisch, Birgitt and Muller, Johannes (2006)  [https://books.google.com/books?id=jJH17hiqHLUC&dq=Discrete+Dynamical+Systems:+The+Ricker+model&pg=PA212   ''A Course in Mathematical Biology''] SIAM. {{ISBN|978-0-89871-612-2}}
* Haddon, Malcolm (2001) [https://books.google.com/books?id=TP_6Z4ukIZQC&dq=9+recruitment+and+fisheries&pg=PA247 ''Modelling and quantitative methods in fisheries''] Chapman & Hall. {{ISBN|978-1-58488-177-3}}
* Hilborn, Ray and Walters, Carl J (1992) [https://books.google.com/books?id=WJg0OVEQHcQC&dq=%22Ricker+model%22&pg=RA3-PA278 ''Quantitative Fisheries Stock Assessment''] Springer. {{ISBN|978-0-412-02271-5}}
* McCallum, Hamish (2000) [https://books.google.com/books?id=r9KnI2kkQ30C&dq=%22Ricker+model%22&pg=PA175 ''Population Parameters''] Blackwell Publishing. {{ISBN|978-0-86542-740-2}}
* Prevost E and Chaput G (2001) [https://books.google.com/books?id=4wdFFEMFupcC ''Stock, recruitment and reference points''] Institute National de la Recherche Agronomique. {{ISBN|2-7380-0962-X}}.
* Plagányi, Éva, Models for an ecosystem approach to fisheries. FAO Fisheries Technical Paper. No. 477. Rome, FAO. 2007. 108p [http://www.fao.org/3/a1149e/a1149e.pdf]
* Quinn, Terrance J. II and Richard B. Deriso (1999) Quantitative Fish Dynamics.Oxford University Press {{ISBN | 0-19-507631-1}}
* Sparre, Per and Hart, Paul J B (2002) Handbook of Fish Biology and Fisheries,  Chapter13: ''Choosing the best model for fisheries assessment.'' Blackwell Publishing. {{ISBN|0-632-06482-X}}
* [[Peter Turchin|Turchin, P.]] 2003. Complex Population Dynamics: a Theoretical/Empirical Synthesis. Princeton, NJ: Princeton University Press.

{{fishery science topics|expanded=science}}
{{Fisheries and fishing}}

[[Category:Fisheries science]]
[[Category:Population ecology]]
[[Category:Population dynamics]]