Supplemenary maerial o: Princeon Universiy Press, all righs reserved From: Chaper 3: Deriving Classic Models in Ecology and Evoluionary Biology A Biologis s Guide o Mahemaical Modeling in Ecology and Evoluion S. P. Oo and T. Day (2005) Princeon Universiy Press Supplemenary Maerial 3.1: Translaing beween numbers and frequencies ihe diploid model of selecion Here, we examine how selecion alers he numbers of individuals wihin a diploid populaion and show ha hese changes are consisen wih he allele frequency recursion, (3.13a). Le s begin a he gamee pool sage and assume ha he frequency of allele A among gamees is p(). Immediaely afer gamees unie randomly o form zygoes, le he number of each genoype be n AA (), n Aa (), and n aa (), and le he oal populaion size be n() = n AA () + n Aa () + n aa (). The probabiliy ha genoype AA survives o adulhood imes is feriliy (he number of individuals ha i conribues o he nex generaion) is measured by W AA. This represens he absolue finess of AA. Similarly, he absolue finesses of Aa and aa are W Aa and W aa, respecively. According o hese definiions, he oal number of individuals conribued o he nex generaion is n(+1) = W AA n AA () + W Aa n Aa () + W aa n aa (). We can also find he average value of he absolue finess by summing he absolue finess of each genoype imes he frequency of ha genoype: n W = W AA AA Equaion (S3.1.1a) can be rewrien as: n + W Aa Aa + W aa ( ) n aa (S3.1.1a) W = +1, (S3.1.1b) 1
which demonsraes ha he populaion changes in size by a facor equal o he average value of he absolue finess. Assuming ha each genoype conribues o he gamee pool in proporioo is absolue finess, he frequency of allele A ihe gamee pool a he nex generaion will equal: p( +1) = W AA n AA ( ) + 1 2 W Aan Aa W AA n AA ( ) + W Aa n Aa W AAn + W aa n aa ( ) = AA ( ) + 1 2 W Aan Aa ( ) W n( ). (S3.1.2a) Assuming ha he zygoes a ime were formed by random union of gamees, hey will be in Hardy-Weinberg proporions (Table 3.2), and he numbers of each genoype will equal he Hardy-Weinberg proporions imes he populaion size: n AA () = p() 2 n(), n Aa () = 2 p() q() n(), and n aa () = q() 2 n(). This allows us o rewrie he mean finess as W = W AA p( ) 2 + 2W Aa p( ) q( ) + W aa q( ) 2 and equaion (S3.1.2a) as: p( +1) = W AA p( ) 2 + W Aa p( ) q( ) Equaion (S3.1.2b) is he same as equaion (3.13a) of he ex. W, (S3.1.2b) We can also describe he dynamics ierms of he change in allele frequency over he course of a generaion: Δp = p( +1) p( ) = p ( ) q( ) p( ) ( W AA W Aa ) + q( ) ( W Aa W aa ) W. (S3.1.2c) This is similar in form o he haploid difference equaion; in boh cases, he change in allele frequency is proporional o p() q(), indicaing ha evoluionary change slows whenever one of he alleles is rare (p() or q() near 0). As was he case wih he haploid model, i is relaive finess (measured as he raio of W ij o some sandard) ha deermines he evoluionary dynamics of allele frequency. Thus, we can muliply or divide all of he finesses (S3.1.2b) by any common facor and his will no affec he evoluionary dynamics. Muliplying or dividing he finesses by a common facor will, however, aler he equaion for he populaion dynamics. In paricular, equaion (S3.1.1b) reveals ha 2
n( +1) = W n(). (S3.1.3) Therefore muliplying he finesses by a common facor σ, will cause he dynamics of he populaion size o be alered by a facor σ. In summary, he recursion equaion (S3.1.3) for he populaion size always depends on he geneic composiion of he populaion, unless he genoypes are equally fi. Bu he recursion equaion (S3.1.2b) for he allele frequency does no depend ohe size of he populaion as long as he relaive finess of each genoype is independen of he populaion size. If, however, he genoypes differ iheir sensiiviy o compeiion and o populaion size (e.g., if he finess of AA declines exponenially wih populaion size, W AA = e α n( ) ) hen boh equaions (S3.1.3) and (S3.1.2b) are necessary o predic he oucome of selecion (e.g., Problem 3.17). 3
Supplemenary Maerial 3.2: A discree-ime version of he Loka-Volerra predaor-prey model. Here we derive he discree-ime Loka-Volerra predaor-prey model; he equivalen coninuous-ime model is described ihe ex by equaions (3.18). The key difference ihe discree-ime model is ha we mus specify an order o he evens ha occur wihin a ime uni. Arbirarily, we assume ha each ime uni consiss of a census, followed by prey birhs, predaion, predaor birhs, and finally predaor deahs. As ihe exponenial model, we assume ha he prey have a per capia reproducive oupu of R individuals per ime uni ihe absence of he predaor. Following prey reproducion, a predaor has a conac probabiliy of c of finding any one of he prey per ime uni, so ha he oal expeced number of conacs beween predaors and prey wihihe communiy is c n 1 () n 2 (). A each conac, he probabiliy ha he predaor successfully aacks he prey is a. Nex, we assume ha reproducion of he predaor is enirely dependen ohe number of prey i consumes and ha one prey iem is he resource equivalen of ε predaor offspring. Finally, we assume ha a fracion, d, of predaors die per ime uni. These assumpions are described ihe form of a flow diagram in Figure S3.2.1. We now derive he recursion equaion by applying Recipe 2.1 o boh species afer every even ihe life cycle: n 1 '() = n 1 () + (R 1) n 1 () n 2 '() = n 2 () afer prey birhs n 1 ''() = n 1 '() a c n 1 '() n 2 '() n 2 ''() = n 2 '() + ε a c n 1 '() n 2 '() afer predaion n 1 '''() = n 1 ''() n 2 '''() = n 2 ''() d n 2 ''() afer predaor deahs Subsiuing each line ino he nex, we ge a discree-ime version of he predaor-prey model: n 1 ( +1) = R n 1 () a c R n 1 () n 2 () (S3.2.1a) n 2 ( +1) = (1 d) ( n 2 () + ε a c R n 1 () n 2 ()) (S3.2.1b) 4
As a check, if here were no conacs beween predaors and prey (c = 0), he prey should grow according o he exponenial growh model (3.1b), and he predaors should die off by a facor of (1 d) each ime uni, which is indeed rue for equaions (S3.2.1). Figure S3.2.1: A flow diagram for he Loka-Volerra predaor-prey model in discreeime. 5