Age (x) nx lx. Age (x) nx lx dx qx
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1 Life Tables Dynamic (horizonal) cohor= cohor followed hrough ime unil all members have died Saic (verical or curren) = one census period (day, season, ec.); only equivalen o dynamic if populaion does no change age disribuion; assumes consan survivorship. Life Hisory Tables Time () = ime inerval used for separaing age caegories n = number alive a he beginning a age L = proporion of individuals alive a age Freq Age () n l Life Hisory Tables d = proporion of original populaion dying during he age inerval o +1 q = proporion of eising populaion dying during age inerval o +1; q = d /l Life epecancy E = T / l T = average life epecancy from curren ime: e.g. how much living will be done by cohor from beginning of period : T =Σ(L ); summed from o las Age () n l d q Age () n l d q L T e
2 aaliy f = oal naaliy; number of ferilized eggs produced in a given year by individuals surviving o age m = age specific naaliy; average number of ferilized eggs produced per individual surviving o beginning of age Loka proved ha any pair of unchanging l and m values will evenually give rise o a populaion wih a sable age disribuion Reproducive Rae R = rae of change in he populaion. If below 1., populaion is shrinking R = (l m ) Sum of he number of ferilized eggs produced per original individual during each age Age () n l d q L T e m f lm R = 8.5 ime age fecund Survive ,234 1, sum ,24 1,529 2, pc pc pc pc ime age fecund Survive ,234 1, sum ,24 1,529 2, pc pc pc pc R=1.5 ime age fecund Survive ,562 3, 5,785 11,141 21,457 41, ,735 3,342 6, sum ,839 3,535 6,814 13,125 25,279 48, pc pc pc pc R=1.83 2
3 Reproducive Raes R = ne reproducive rae; e number of offspring produced per individual. Also, a muliplier allowing us o deermine populaion size in he ne generaion. For simpliciy, assume discree generaions. R = (l m ) R rae of populaion increase as a funcion of ime. Uniless and dimensionless. sage a l d q Log1a Log1l k M f lm lm -r e -r lme -r L T e R o l m 8.75 l m e -r =1 lm Defining R R is a muliplier allowing us o deermine populaion size a fuure generaion. If discree generaions, ime in generaions hen =T and + 1 = * R General formula for populaion size a ime is: = * R Predics eponenial growh Year Reproducive Raes r = inrinsic rae of increase; per capia rae of increase; also Malhusian Parameer. When r is > populaions will increase, when i is < populaions will decrease; a populaion ha is no increasing or decreasing will have an r of. Unis number of new individuals per uni ime. sage a l d q Log1a Log1l R o lm 8.75 lm T c r.876 Double.787 number of individuals 1 Eponenial Growh ime r =.1 r =.2 r =.3 3
4 Deermining r If generaion ime (T) is known, can deermine r from Loka s Equaion: r 1 = e l using ieraion m This is difficul, and he equaion is no biologically meaningful r can be esimaed by: r R = ln( ) T c where T c is generaion ime. Cohor Generaion Time T c is he average birh-o-birh ime for a generaion. lm Tc = R Recall ha Thus T c R = = l m l m l m sage a l d q Log1a Log1l R o lm 8.75 lm T c r.876 Double.787 Loka s Soluion o r, solve for Tc firs, hen r T c = (19.25/8.75) = r = ln(2.3839)/3.68 =.876 How good is our esimae of r? True value of r difficul o ge: 1 = e r l m sage a l d q Log 1a Log 1l k M f l m l m -r e -r l m e -r L Ro l m 8.75 l m e -r =1 l m T c r.876 Double.787 sage a l d q Log 1a Log 1l k M f l m l m -r e -r l m e -r L Ro l m 8.75 l m e -r =1 l m T c r.876 Double.787 Value > 1 means esimae of r is oo small Value < 1 means esimae is oo large In his case, acual r slighly greaer han.876 4
5 How do we use his? R is he finie rae of populaion increase. I + 1 = * R represens he proporional change in a However, r is much more useful as a coninuous measure of 1 = R * populaion from o +1. Is always posiive. populaion growh. I is a raio, dimensionless and wih no unis. d d = r = e r Differenial equaion of logisic growh. Can only ell you he rae (d/d) of growh, can projec populaion size. Inegrae he firs equaion, and i is wrien in a form where you can projec populaion size given a ime period (). = R * By eension r R = e r Assuming infiniely small ime seps, swich from he discree o coninuous form. = ln( R) r is hus he naural log of R R = l m R is he ne reproducive rae, correced for moraliy. The unis of R is number of offspring per individual per lifespan. If R >1. populaion is increasing. 1 = e r l m True value of r, difficul o solve. The esimae from R is usually close enough. R rae of populaion increase as a funcion of absolue ime. Uniless, dimensionless. R rae of populaion increase as a funcion of generaion ime. Thus R=R only if T=1 Esimaing r: r R = ln( ) T However, his is only An esimaion of r. 5
6 r More abou r How long does i ake a populaion o double in size?? r = e =2 and o =1 =? = e ln 2=r r ime =.69/r Doubling Time for Populaion inrinsic rae of increase Human Populaion Growh year r doubling ime Given curren growh raes, wha will he world populaion be in 3 years?? = e r =6,426,11,45 e.125(3) 9,349,922,439 Variabiliy in r The maimum rae of increase obainable by a populaion is r ma Difference beween r ma and r is due o environmenal resisance r ma - qualiy of food, space, ec. are opimum, no compeiion or predaion. An eample sage a M Given hese daa answer he following Wha proporion of individuals survive o age 2? (l 2 ) Wha proporion of 2 year olds die before reaching age 3? (q 2 ) Wha is he epeced lifespan? (e ) How many oal offspring are produced by he hird year class? (f 3 ) Is he populaion growing or shrinking? (R ) Wha is he generaion ime? (T) How large will he populaion be in 243 years? (r) How long before he populaion doubles? (r) 6
7 An eample Wha proporion of individuals survive o age 2? (l 2 ) l = a /a = 2/15 =13.3% An eample Wha proporion of 2 year olds die before reaching age 3? (q 2 ) q = a +1 /a = 1/2 =5% sage a l sage a l d q An eample Wha is he epeced lifespan? (e ) L = (a +a +1 )/2 e =Sum(L )/a e =( )/15 = 145/15 =.94 An eample How many oal offspring are produced by he hird year class? (f 3 ) f = m *a f 3 =m 3 *a 3 = 6*1 = 6 sage a l d q L e sage a l d q M f
8 An eample Is he populaion growing or shrinking? (R ) R =Sum(l m ) R = = 1.28 An eample Wha is he generaion ime? (T) T=sum(l m )/sum(l m ) T=2.7/1.27 = 2.12 sage a l d q Log 1 a Log 1 l k M f l m l m sage a l d q Log 1 a Log 1 l k M f l m l m An eample How large will he populaion be in 243 years? (r) r=lnr /T r=ln(1.27)/2.12 =.114 = e r 243 =15 e 114* = 15 e sage a l d q Log 1 a Log 1 l k M f l m l m An eample How long before he populaion doubles? (r) Double ime =.69/r =.69/.114 = 6.5 years sage a l d q Log 1 a Log 1 l k M f l m l m
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