Appendix S2
Assume individual i’s size through time follows: Si (t + d t ) = Si (t ) + d t g + d t e i + e i ,t , where
2 2
e i has mean 0 and among-individual variance r sg, and e i , t has mean 0 and variance
(
)
2
2
through time 1- r s gd t. Let
denote the expected value across individuals.
s ( t + dt ) = ( Si ( t + dt )) - Si ( t + dt )
2
2
= ( Si ( t ) + gdt + eidt + ei,t )
2
2
(
- Si ( t ) + gdt
)
2
= Si ( t ) + 2 Si ( t ) gdt + 2 Si ( t ) eidt + g2dt2 + (eidt ) + (ei,t )
2
2
= Si ( t ) - Si ( t ) + 2dt Si ( t ) ei + (eidt ) + (ei,t )
2
= Si ( t ) - Si ( t )
2
2
2
2
2
- Si ( t ) - 2 Si ( t ) gdt - g2dt2
2
t
æ
ö
2
2
+ 2dt ç Si ( 0) + gt + ei t + åei,t ÷ei + (eidt ) + (ei,t )
è
ø
t =0
= Si ( t ) - Si ( t ) + 2dt ei2 t + (eidt ) + (ei,t )
2
2
2
= s 2 ( t ) + 2dt r 2s g2 t + dt2 r 2s g2 + (1- r 2 )s g2dt
2
2