Lectures 2 & 3 - Population ecology mathematics refresher

Transcription

1 Lcturs & - Poultio cology mthmtics rrshr To s th mov ito vloig oultio mols, th olloig mthmtics crisht is suli I i out r mthmtics ttook! Eots logrithms i i q q q q q q ( tims) / c c c c ) ( ) (

2 Clculus Dirtitio To clcult th rt-o-chg o uctio ith rsct to vril us irtitio This mouts to clcultig th slo o th tgt to th uctio Altrtivly c stimt th mimum vlu o uctio ith rsct to vril o itrst I its most simlst orm y y Eml : Dirtit th uctio y = -- y 8 Wh irtitig logrithms ots th olloig ruls ly y y y y (Not tht y ) Chi rul This is ossily th most imortt rul i irtil clculus Us th chi rul h you hv to irtit uctio y g tht is itsl uctio o th vril tht is ig irtit Eml uctios r, g g g Th rul c lso rss s tht thr hs trsormtio o th uctio u g y y u u giv

3 Eml : Dirtit th olloig to uctios usig th Chi Rul y y ( ) Th solutios to th olloig qutios r s ollos: lt u u u u Thror s y y y u u / y u u No ck-trsorm such tht y y ( ) ( ) Prouct quotit ruls Ot th uctio to irtit hs g g gg g g g g Eml : Clcult usig th rouct rul

4 Eml : Clcult usig th quotit rul 8 Emls Dirtit th uctio 9 9 Dirtit th uctio Dirtit th uctio

5 Dirtit th uctio s y 8 8 s y s y Fi i y First, lt y

6 Fi i Lt z y y z z y z z y Fi i 8

7 8 Fi y i y y Itgrtio Itgrtio is th oosit o irtitio I clculus thr is mtho us to th ck-clcult th rivtiv o uctio to its origil orm I th tirivtiv o is I its simlst orm C It is crucil to lys hv th th itgrtio costt C i your rssio solutio W c chck our solutio y tkig th rivtiv C Itgrtio, s ith irtitio, hs th olloig ruls: I ish to itgrt t to vlus sy t som lor, L, ur, U, ous th th Fumtl Thorm o th Clculus lis It is i s U L ' U U L L Th Fumtl Thorm o th Clculus is lso us to stimt th r ur curv or volum i t lst to-imsios r itgrt

8 8 Eml : Clcult t th ous 9 Emls 9 C C C C C C Itgrt

9 Fiig th mim/miim o uctio Th mimum or miimum o uctio c ou suig irtitio For ml, i hv cotiuous uctio (y) th mim/miim, ithi rg o vlus k th omi, r stimt sily y irtitig (y) ith rsct to, sttig th qutio to zro th solvig or Ths vlus r th rlc ito th origil qutios solv or th corrsoig y vlus I thr is mor th o mimum or miimum vlu i th cs o olyomil - th oul hv mor th o solutio I th cs o high school solutio i hv qurtic o th orm y c th th mimum/miimum vlu is y such tht Eml : A qurtic y olyomil y 9 r rst i th grh lo Wht r th mim/ miim? 8 8 y = y = 9 Th mimum miimum vlus r thror clcult y tkig th rivtiv o y ith rsct to, sttig th rivtiv qul to zro solvig or Th mimum miimum y stimts r clcult y sustitutig th mimum miimum stimts ck ito th origil qutio For th qurtic th miimum ovr th itrvl [-,] is y y or th olyomil ovr th itrvl [-,] 9

10 y 9 y ~ 9 Fiig th roots o uctio Th root(s) o uctio r thos vlus hr th uctio itrscts th -is Crti uctios hv roots tht r sy to clcult ths r oti y sttig solvig or Hovr, i o-lir this is ot trivil tsk A usul mthos is tht roos y Nto Rhso is ko s th Nto-Rhso s mthos o Roots or NRMR NRMR is ct or lir rolms roimt or o-lir rolms s o th ssumtio tht th solutio to uctio c stimt y Tylor s sris sio rou rsol stimt clos to th solutio, sy A Tylor s sris sio is o th orm ( ) ( ) I th NRMR oly th lir! trms r cosir such tht th solutio c ou rom stimt Itrtig ith th solutio ill solv th rolm quickly A Tylor sio rou th iitil stimt,, ill yil th roimt solutio such tht ( ) ( ) ( ) As ish to solv or th ( ) uctio ( ) th is th grl orm o ( ) hr is som corrctio ctor O ll-hv uctio this mtho covrgs to th solutio qurticlly, ith lir uctios oly o itrtio ill cssry NRMR c ict grhiclly or uctio tht hs irst rivtiv tht is itsl lir uctio I vlut th rivtiv t our irst stimt,, th c clcult th slo o th stright li tht ill itrsct th ojctiv uctio s tgt Th slo, or th rivtiv, is clcult s W c solv or such tht, th solutio rom th Tylor sris rivtio This rivtio c grhiclly illustrt i Figur

11 9 '( ( ) () ) (, ()) - - Figur : Th Nto-Rhso mtho o roots Th iitil guss,, is us to clcult (tmorry) solutio,, through th clcultio o th slo o lir rivtiv

12 Eml : Wht r th roots o Eml y ) ctorizig ) usig Nto- Rhso s mtho? For th qurtic th roots r y ( )( ) such tht = or = -, y 9 ( )( ) = or = or = - or th olyomil W ot tht or th qurtic th miimum is ou t (,-) thror th roots ill o ithr si o th miim cus th qurtic is symmtric Th miimum root is ou y sttig iitil guss o Th uctios r ( ) Thror ( ) 8 ( ), so our irst stimt is This is th rsustitut i such tht 9 9 Atr to mor itrtios root o is ou Similrly th rightmost root is ou t Ituitio: s th miim s (,-) th roots ill symmtric rou For th olyomil otic tht th mim/miim r ou t, Th iitil gusss r thr or t, th mi-oit t Th solutios r s ollos i (i) '(i) i+ i E E- i (i) '(i) i+ i E- 8 9E- i (i) '(i) i+ i Th roots r thror ou t =