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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
</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.
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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
According to [[Daniel Pauly]],<ref name="NYT"/><ref>{{cite
==Fishing effort==
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==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>∞</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 [[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> ''t''+1</sub> (or density) of individuals in generation ''t'' + 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'' = (''R''<sub>0</sub> − 1) ''M'' is the [[carrying capacity]] of the environment.▼
▲* 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''):
▲* 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}}</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'':
▲: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> ''t''+1</sub> (or density) of individuals in generation ''t'' + 1 as a function of the number of individuals in the previous generation,
==Predator–prey equations==
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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==
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* [[Depensation]]
* [[Huffaker's mite experiment]]
* [[Wild fisheries]]▼
* [[Overfishing]]
* [[Overexploitation]]
* [[Population modeling]]
* [[
* [[Tragedy of the commons]]{{Div col end}}
▲* [[Wild fisheries]]
==References==
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==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
* Haddon, Malcolm (2001) [https://books.google.com/books?id=TP_6Z4ukIZQC
* Hilborn, Ray and Walters, Carl J (1992) [https://books.google.com/books?id=WJg0OVEQHcQC
* McCallum, Hamish (2000) [https://books.google.com/books?id=r9KnI2kkQ30C
* Prevost E and Chaput G (2001) [https://books.google.com/books?id=4wdFFEMFupcC
* 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.
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[[Category:Fisheries science]]
[[Category:Population ecology]]
[[Category:Population dynamics]]
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