Editing Predation (section)

=== Population dynamics ===
{{further|Population dynamics|Lotka–Volterra equations}}

[[File:Milliers fourrures vendues en environ 90 ans odum 1953 en.jpg|right|upright=1.5|thumb|Numbers of [[snowshoe hare]] (''Lepus americanus'') (yellow background) and Canada lynx (black line, foreground) furs sold to the [[Hudson's Bay Company]] from 1845 to 1935|alt=A line graph of the number of Canada lynx furs sold to the Hudson's Bay Company on the vertical axis against the numbers of snowshoe hare on the horizontal axis for the period 1845 to 1935]]

In the absence of predators, the population of a species can grow exponentially until it approaches the [[carrying capacity]] of the environment.<ref>{{cite book |last1=Neal |first1=Dick |title=Introduction to population biology |date=2004 |publisher=Cambridge University Press |isbn=9780521532235 |pages=68–69}}</ref> Predators limit the growth of prey both by consuming them and by changing their behavior.<ref>{{cite journal |last1=Nelson |first1=Erik H. |last2=Matthews |first2=Christopher E. |last3=Rosenheim |first3=Jay A. |title=Predators Reduce Prey Population Growth by Inducing Changes in Prey Behavior |journal=Ecology |date=July 2004 |volume=85 |issue=7 |pages=1853–1858 |jstor=3450359 |doi=10.1890/03-3109 |bibcode=2004Ecol...85.1853N |url=https://rosenheim.faculty.ucdavis.edu/wp-content/uploads/sites/137/2014/09/Ecology2004ms.pdf }}</ref> Increases or decreases in the prey population can also lead to increases or decreases in the number of predators, for example, through an increase in the number of young they bear.

Cyclical fluctuations have been seen in populations of predator and prey, often with offsets between the predator and prey cycles. A well-known example is that of the [[snowshoe hare]] and [[lynx]]. Over a broad span of [[boreal forest]]s in Alaska and Canada, the hare populations fluctuate in near synchrony with a 10-year period, and the lynx populations fluctuate in response. This was first seen in historical records of animals caught by [[Fur trade|fur hunter]]s for the [[Hudson's Bay Company]] over more than a century.<ref name=Krebs>{{cite journal |last1=Krebs |first1=Charles J. |last2=Boonstra |first2=Rudy |last3=Boutin |first3=Stan |last4=Sinclair |first4=A.R.E. |title=What Drives the 10-year Cycle of Snowshoe Hares? |journal=BioScience |date=2001 |volume=51 |issue=1 |pages=25 |doi=10.1641/0006-3568(2001)051[0025:WDTYCO]2.0.CO;2|doi-access=free |hdl=1807/359 |hdl-access=free }}</ref><ref name=classics>{{cite journal |last1=Peckarsky |first1=Barbara L. |last2=Abrams |first2=Peter A. |last3=Bolnick |first3=Daniel I. |last4=Dill |first4=Lawrence M. |last5=Grabowski |first5=Jonathan H. |last6=Luttbeg |first6=Barney |last7=Orrock |first7=John L. |last8=Peacor |first8=Scott D. |last9=Preisser |first9=Evan L. |last10=Schmitz |first10=Oswald J. |last11=Trussell |first11=Geoffrey C. |title=Revisiting the classics: considering nonconsumptive effects in textbook examples of predator–prey interactions |journal=Ecology |date=September 2008 |volume=89 |issue=9 |pages=2416–2425 |doi=10.1890/07-1131.1 |pmid=18831163 |bibcode=2008Ecol...89.2416P }}</ref><ref name="The Snowshoe Hare 10-year Cycle – A Cautionary Tale">{{cite web |last1=Krebs |first1=Charley |last2=Myers |first2=Judy |title=The Snowshoe Hare 10-year Cycle – A Cautionary Tale |url=https://www.zoology.ubc.ca/~krebs/ecological_rants/?p=786 |website=Ecological rants |publisher=University of British Columbia |access-date=2 October 2018|date=12 July 2014 }}</ref><ref name="BBC Bitesize - Predators and their prey">{{cite web |title=Predators and their prey |url=https://www.bbc.co.uk/schools/gcsebitesize/science/ocr_gateway/understanding_environment/interdependencerev2.shtml |website=BBC Bitesize |publisher=[[BBC]] |access-date=7 October 2015}}</ref>

[[File:Lotka-Volterra model (1.1, 0.4, 0.4, 0.1).png|thumb|upright=1.35|left|[[Predator]]-prey population cycles in a [[Lotka–Volterra equations|Lotka–Volterra model]]<!-- with the parameters set to (1.1, 0.4, 0.4, 0.1)-->]]

A simple model of a system with one species each of predator and prey, the [[Lotka–Volterra equations]], predicts population cycles.<ref name="Goel1971">{{cite book |last1=Goel |first1=Narendra S. |first2=S. C. |last2=Maitra |first3=E. W. |last3=Montroll |title=On the Volterra and Other Non-Linear Models of Interacting Populations |publisher=Academic Press |year=1971 |isbn=978-0122874505}}</ref> However, attempts to reproduce the predictions of this model in the laboratory have often failed; for example, when the protozoan ''[[Didinium#Didinium nasutum|Didinium nasutum]]'' is added to a culture containing its prey, ''[[Paramecium caudatum]]'', the latter is often driven to extinction.<ref name=Princeton>{{cite book |last1=Levin |first1=Simon A. |last2=Carpenter |first2=Stephen R. |last3=Godfray |first3=H. Charles J. |last4=Kinzig |first4=Ann P. |last5=Loreau |first5=Michel |last6=Losos |first6=Jonathan B. |last7=Walker |first7=Brian |last8=Wilcove |first8=David S. |title=The Princeton guide to ecology |url=https://archive.org/details/princetonguideto00levi |url-access=limited |date=2009 |publisher=Princeton University Press |isbn=9781400833023 |pages=[https://archive.org/details/princetonguideto00levi/page/n218 204]–209}}</ref>

The Lotka–Volterra equations rely on several simplifying assumptions, and they are [[structural stability|structurally unstable]], meaning that any change in the equations can stabilize or destabilize the dynamics.<ref name=Murdoch>{{cite book |last1=Murdoch |first1=William W. |last2=Briggs |first2=Cheryl J. |last3=Nisbet |first3=Roger M. |title=Consumer-resource dynamics |date=2013 |publisher=Princeton University Press |isbn=9781400847259|page=39}}</ref><ref>{{cite book |last1=Nowak |first1=Martin |last2=May |first2=Robert M. |title=Virus Dynamics : Mathematical Principles of Immunology and Virology |date=2000 |publisher=Oxford University Press |isbn=9780191588518 |page=8}}</ref> For example, one assumption is that predators have a linear [[functional response]] to prey: the rate of kills increases in proportion to the rate of encounters. If this rate is limited by time spent handling each catch, then prey populations can reach densities above which predators cannot control them.<ref name=Princeton/> Another assumption is that all prey individuals are identical. In reality, predators tend to select young, weak, and ill individuals, leaving prey populations able to regrow.<ref name="Genovart2010">{{cite journal |last1=Genovart |first1=M. |last2=Negre |first2=N. |last3=Tavecchia |first3=G. |last4=Bistuer |first4=A. |last5=Parpal |first5=L. |last6=Oro |first6=D. |year=2010 |title=The young, the weak and the sick: evidence of natural selection by predation |journal=PLOS ONE|volume=5 |issue=3 | page=e9774 |doi=10.1371/journal.pone.0009774 |pmid=20333305 |pmc=2841644|bibcode=2010PLoSO...5.9774G |doi-access=free }}</ref>

Many factors can stabilize predator and prey populations.<ref>{{harvnb|Rockwood|2009|page=281}}</ref> One example is the presence of multiple predators, particularly generalists that are attracted to a given prey species if it is abundant and look elsewhere if it is not.<ref>{{harvnb|Rockwood|2009|page=246}}</ref> As a result, population cycles tend to be found in northern temperate and [[subarctic]] ecosystems because the food webs are simpler.<ref>{{harvnb|Rockwood|2009|pages=271–272}}</ref> The snowshoe hare-lynx system is subarctic, but even this involves other predators, including coyotes, [[goshawk]]s and [[great horned owl]]s, and the cycle is reinforced by variations in the food available to the hares.<ref>{{harvnb|Rockwood|2009|page=272–273}}</ref>

A range of mathematical models have been developed by relaxing the assumptions made in the Lotka–Volterra model; these variously allow animals to have [[spatial distribution|geographic distribution]]s, or to [[animal migration|migrate]]; to have differences between individuals, such as [[sex]]es and an [[age structure]], so that only some individuals reproduce; to live in a varying environment, such as with changing [[season]]s;<ref name="Cushing2005">{{cite journal |last1=Cushing |first1=J. M. |title=Book Review: Mathematics in population biology |journal=Bulletin of the American Mathematical Society |date=30 March 2005 |volume=42 |issue=4 |pages=501–506 |doi=10.1090/S0273-0979-05-01055-4 |doi-access=free }}</ref><ref>{{cite book|last=Thieme|first=Horst R. |title=Mathematics in Population Biology|url=https://books.google.com/books?id=cHcjnkrMweYC |year=2003 |publisher=Princeton University Press |isbn=978-0-691-09291-1}}</ref> and analysing the interactions of more than just two species at once. Such models predict widely differing and often [[chaos theory|chaotic]] predator-prey population dynamics.<ref name="Cushing2005"/><ref name="KozlovVakulenko2013">{{cite journal |last1=Kozlov |first1=Vladimir |last2=Vakulenko |first2=Sergey |title=On chaos in Lotka–Volterra systems: an analytical approach |journal=Nonlinearity |volume=26 |issue=8 | date=3 July 2013 |doi=10.1088/0951-7715/26/8/2299 |pages=2299–2314|bibcode=2013Nonli..26.2299K | s2cid=121559550 }}</ref> The presence of [[refuge (ecology)|refuge areas]], where prey are safe from predators, may enable prey to maintain larger populations but may also destabilize the dynamics.<ref>{{cite journal |doi=10.1016/0040-5809(87)90019-0 |title=Prey refuges and predator-prey stability |journal=Theoretical Population Biology |volume=31 |pages=1–12 |year=1987 |last1=Sih |first1=Andrew |issue=1 |bibcode=1987TPBio..31....1S }}</ref><ref>{{cite journal |doi=10.1016/0040-5809(86)90004-3 |pmid=3961711 |title=The effects of refuges on predator-prey interactions: A reconsideration |journal=Theoretical Population Biology |volume=29 |issue=1 |pages=38–63 |year=1986 |last1=McNair |first1=James N |bibcode=1986TPBio..29...38M }}</ref><ref name="BerrymanHawkins2006">{{cite journal |last1=Berryman |first1=Alan A. |last2=Hawkins |first2=Bradford A. |last3=Hawkins |first3=Bradford A. |title=The refuge as an integrating concept in ecology and evolution |journal=Oikos |volume=115 |issue=1 |year=2006 |pages=192–196 |doi=10.1111/j.0030-1299.2006.15188.x|bibcode=2006Oikos.115..192B }}</ref><ref>{{cite journal |doi=10.1016/j.tpb.2009.08.005 |pmid=19751753 |title=A predator–prey refuge system: Evolutionary stability in ecological systems |journal=Theoretical Population Biology |volume=76 |issue=4 |pages=248–57 |year=2009 |last1=Cressman |first1=Ross |last2=Garay |first2=József |bibcode=2009TPBio..76..248C }}</ref>