Appendix I Derivation and description of simulation model
To study these scaling rules in more generality, we require a model that meets the
following criteria
1. It must take as some limiting form, the logistic model of Lande et al., but have the
flexibility to represent a greater number of forms of density dependence
2. it should have environmental stochasticity
3. it should have demographic stochasticity
4. It must be comparable (in some sense, in some limit) to the diffusion model of
Lande et al.
We meet these criteria in the following way. We will use a discrete time model of the
form
Nt 1 ~ f Nt  ,
in which the left hand side, a random number representing population size, is distributed
according to probability mass function f that depends on the population size at time t. We
will choose model equations and parameters so that the expectation of is the well known
theta-Ricker equation, i.e.,

ENt 1   Nt exp  


  Nt   
1      .
  K  


Then, with =∞ we recapture the ceiling model of Lande et al., while for 0<<∞ we
represent a wide spectrum of possibilities for the form of density dependence.
   N   
We note that the term exp   1   t    can be viewed as a density-dependent
   K  

 
reproductive rate and denote this quantity by λN. To incorporate environmental
stochasticity, we consider λN to be a random variable, unique for each t. As a growth rate,
it must be positive and continuously distributed. The gamma distribution is a flexible
x 1e  x / 
distribution with these properties, with density function px; ,    
, where
  
0<=x, ,>0 and (x)=(x-1)! is the complete gamma function. One property of the
gamma distribution is that for gamma distributed random variable X~Gamma(,), the
distribution of the rescaled variable Y=cX is also distributed as a random variable,
Y~Gamma(,δ), where =c. Thus, multiplying the random variableby Nt, we have that
x 1e  x /  N t  
the population size is a gamma random variate with p.d.f. px  
. Standard
Nt    
parameterizations of the gamma distribution (such as this one) are in terms of shape ()
and scale () parameters, not the mean, as we would like. Therefore, noting that the mean
is given by the product of the shape and scale parameters, =Nt, we rearrange this

formula to give the shape parameter in terms of the mean  
, and re-parameterize
Nt 
x t /  Nt  1e  x /  Nt  
accordingly p  x; t , N t ,   

N t  t /  Nt    t
 Nt 
 exp N 

now have Nt 1 ~ Gamma
, Nt   .
 




, where t  Nt exp N  . Thus, we
Next also added demographic stochasticity. The standard definition of demographic
stochasticity is that it is randomness to results from the discreteness and independence of
the individuals that make up the population. If the (random with respect to time) expected
size of the population at time t is the random variable generated by the previous equation,
then the realized population size is some discrete random variable with this mean. We
assumed all reproductive events were independent, and thus used a Poisson distribution
with rate parameter equal to the gamma random variable from the previous equation. A
probability distribution in which the parameters themselves are random variables are
known as mixture distributions, and the gamma-mixed Poisson distribution is the wellknown negative binomial (Johnson et al. p. 308). The p.m.f. for a gamma-mixed Poisson
distribution with shape and scale parameters and  is
   x  1  

PrX  x  

   1    1 
x

 1 

.

 , with mean  and variance
1 /   1
  1
Thus, we have
 exp N   x  1 
exp  N 
x




    1 
PrX  x   
,




 exp N   1    1     1 






with mean Nt exp t   Nt exp  



exp  1  Nt / K 
.
1/ Nt   1
 

  Nt   
1      (as we require) and variance
  K  


Now we look at the change in population size N=Nt+1-Nt. We can view this as a random
variable which results from the subtraction of a constant (Nt) from random quantity
(Nt+1). For the mean we have


E N   M N   E N t 1   N t  N t  exp  




unchanged, i.e., VARN   V N  
  N t    
1       1 , while the variance is
  K   

 
 
exp  1  Nt / K 
1/ Nt   1

 .
We now have a stochastic model for density-dependent population dynamics with
demographic and environmental stochasticity. The model is fully specified in terms of the
familiar parameters 0, K, and , and one parameter, , which governs the degree of
environmental stochasticity. Further, the expected size of the population, conditioned on
some population size in the previous time step, is give by the familiar deterministic theta   N   
Ricker model Nt 1  Nt exp   1   t    . This model is not analytically tractable, but
   K  

 
is readily simulated. Finally, for comparison with Lande et al. (2003), we have
expressions for the mean and variance. By taking the time interval to zero, we can
interpret these formulas as the infinitesimal mean and variance of a continuously
changing stochastic process (a diffusion approximation).