Density Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).

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1 FW 662 Densiy-dependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Long-erm densiy independen populaion growh is unlikely and an unrealisic assumpion. Birh and deah raes are more likely a funcion of populaion densiy or abundance. Densiy Dependence birhs are a decreasing funcion of densiy b() and deahs are an increasing funcion of densiy d(). birhs deahs b or d This resuls in populaion growh being a declining funcion of Populaion Growh (f() Hence populaion growh will be zero a some populaion size. This poin is usually referred o as K (or carrying capaciy) bu le s develop he model firs considering he explici funcions for birh and deah. The approach is described in Donovan and Weldon (22) bu I have modified i o mach he noaion in Goelli (998).

2 FW 662 Densiy-dependen populaion models We need wo new erms o accoun for changes in per capia birh and deah raes a he amoun by which he per capia birh rae changes in response o an addiion of one individual o he populaion. c dio for deah rae Our densiy independen discree model was a difference uaion expressed as: + + b d We replace b and d (he densiy independen birh rae and deah rae) wih: b a and d + c ow our new densiy dependen model looks like + ( b a ) ( d c ) + + Populaion growh rae is no easy o visualize from his uaion. Bu you can see ha iniially he populaion will grow geomerically because () is small bu as () grows a and c have a greaer influence on he populaion. The quesion is will he populaion decrease in size? Increase? Or sabilize??? Use a common acic and assume ha here is an uilibrium (i.e. a poin where he populaion size isn changing). + ( b a ) ( d c ) + + Subrac () from boh sides and add ( d + c ) ( b a ) ( d + c ) Divide by () ( b a ) ( d + ) c This ells us ha he populaion is a uilibrium when per capia birh and deah raes are ual. THIS MAKES SESE.

3 FW 662 Densiy-dependen populaion models ow we can solve he uaion for () Rearrange he uaion by subracing d and adding a b d c + a Facor ou () b d + ( a c) Divide by (a+c) b d a + c K Using his uaion you can calculae K, if you know b, d, and he wo facors ha accoun for densiy dependence. Densiy Dependence (discree logisic) Typically, K is specified and no calculaed from he birh and deah raes. Le s sar wih our geomeric model R The exisence of a carrying capaciy (K) suggess ha he populaion canno exceed his level. R R + + R +

4 FW 662 Densiy-dependen populaion models The logisic model resuls in he following populaion dynamics. When () is small hen geomeric growh and when ()K hen populaion growh is zero. Pop Size Time We can also look a he change in populaion size as a funcion of populaion size. Iniially he populaion is growing quickly bu hen decline o zero. Dela () 8 Dela Looking a he per capia rae of change, he populaion begins growing a R and declines o zero a K. From his graph you can see ha he densiy dependence is linear. (Dela())/()

5 FW 662 Densiy-dependen populaion models Densiy Dependence (coninuous logisic) d d K r This uaion predics he rae of change in populaion size and you need o derive a predicive uaion. WHICH IS + K ( K ) r e This model resuls in he same dynamics ha were demonsraed for he discree version of he model. However, he discree version of he model is capable of demonsraing a wider variey of dynamics han he coninuous version, which we will explore laer. We have derived wo predicive uaions for exponenial populaion growh and wo uaions for densiy dependen populaion growh. Densiy Independen Densiy Dependen Discree λ + R K Coninuous ( ) + r + e r K K e

6 FW 662 Densiy-dependen populaion models Oher models wih Densiy Dependence PLEASE PRIT AD READ GARY S LECTURE OTES (5-7) Many possibiliies exis for describing densiy dependence and here are several oher models ha have been developed. The Ricker uaion Ricker was a fishery biologis ineresed in predicing recruimen o fishery socks and developed he following densiy dependen uaion. oe ha densiy dependence in his uaion is no linear and becomes sronger a higher densiies (due o he exponenial funcion). + e R The Hassel Equaion + K ( R K + + ) R

7 FW 662 Densiy-dependen populaion models Comparing populaion growh among he logisic, Ricker, and Hassel models for R.75 and K. 2 8 () Logisic Ricker Hassel Time () Comparing he form of densiy dependence among he logisic, Ricker, and Hassel models. dela () / () Logisic Ricker Hassel () Comparing he per capia growh rae among he logisic, Ricker, and Hassel models dela () Logisic Ricker Hassel ()

8 FW 662 Densiy-dependen populaion models Maximum Susainable Yield (MSY) I will develop a model based on he logisic difference (discree) uaion and follow her derivaion in Williams e al (22). Sar wih our discree logisic difference uaion + R + and include a erm for oal Harves H() + + R H The populaion is a uilibrium ((+)()) when R H and he per capia harves rae h H h h R A given per capia harves rae h() corresponds o a specific uilibrium populaion size ha can susain i and can be seen by rearranging he above uaion in erms of. h K R This uaion indicaes ha he populaion can be susained in uilibrium for any value h ha is less ha he populaion growh rae R. The quesion is, Wha value corresponds o he larges susainable harves? dh d R 2R ( ) K

9 FW 662 Densiy-dependen populaion models Se his ual o and solve for, which resuls in K 2 Subsiuing ino our uaion for uilibrium harves: R H ( R ) H 2 ( RK ) H 4 Then he per capia harves (h) is h H h R 2

10 FW 662 Densiy-dependen populaion models Dela () MSY () MSY (dela ())/() ()

11 FW 662 Densiy-dependen populaion models Goelli, A primer of ecology. Sinauer Associaes Inc. ew York. Hassel, M.P. Densiy-dependence in single-species populaions. The Journal of Animal Ecology. 44: Ricker, W. E Sock and recruimen. Journal Fisheries Research Board of Canada : Ricker, W. E Compuaion and inerpreaion of biological saisics of fish populaions. Fisheries Research Board of Canada, Bullein 9. Oawa, Canada. Williams, B.K., J.D. ichols, M.J. Conroy. 22. Analysis and managemen of animal populaions. Academic Press. ew York.

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