probabilities defines a probability distribution that is the fundamental 

 building block of a stochastic process model. 



A simple example can illustrate the differences among the different 

 types of models. Consider the survival of a group of individuals from time 

 t to t+1. Let N(£) denote the number of individuals alive at time t, and 

 N(£+l) the number alive at time t+1. A simple deterministic relationship is 



mt+i) = N(t) P (i) 



where p = the proportion surviving through one time unit. If N is re- 

 stricted to integer values, then similar restrictions must be placed on p as 

 well. For example, if N(t) = 10, then p = 0.5 is reasonable. But if N(t) = 9, 

 then p cannot be 0.5 because that will result in Nft+1) = 4.5 individuals, 

 an unrealistic outcome. Simply rounding N(£+l) to the nearest integer will 

 not work. As long as p > 0.5, the population will never die out — as t = 

 0,1,2, ...co, N(£) = 10,5,3,2, 1,1,.. .,1. In this model, the difference between 

 p = 0.5 and p = 0.4999 is the difference between immortality and certain 

 extinction. 



This simple deterministic model could be made stochastic by defining a 

 probability distribution for p. We would have to define at least one more 

 parameter to describe the variance of p and also consider many of the same 

 integer-counting problems that are associated with the deterministic 

 model. While there are practical solutions for this simple, one-step ex- 

 ample, the problems mount as more steps and more parameters are added. 



The more elegant and parsimonious way to address this problem is to 

 view it as a stochastic binomial process. In the binomial model, each indi- 

 vidual is assumed to have the same probability of surviving (p). N(£+l) is 

 viewed as the sum of N(£) independent trials with possible outcomes or 1 

 (0 = individual dies, 1 = individual survives). This gives rise to the familiar 

 binomial probability distribution 



Prob{Ntf+l) = X}= N(f)! p x ^ _ p] wuy-v (2 ) 



[N(£)-X]!X! 



which is defined for < X < N(t) and has mean = ~N(t)p and variance = 

 N(£)p[l- p]. The binomial model is widely applicable as a model of death 

 processes in ecology (Pielou 1977) and is used extensively within the 

 SLCM. In the following discussion, the independent variables N and p 

 used in the binomial distribution are assigned to a variety of state vari- 

 ables and probabilities, but their interpretation relative to the binomial 

 distribution is the same. 



Two extensions to the binomial, the multinomial and the binomial-beta 

 distributions, are also used within the SLCM. The multinomial is appro- 

 priate when there are more than two possible outcomes, all outcomes are 

 mutually exclusive, and all outcomes have fixed probabilities of occurring 

 that sum to one. For example, the allocation of subbasin escapement 

 among returns to the hatchery, subbasin harvest, natural spawning, and 

 natural mortality follows a multinomial. 



The binomial-beta distribution is used where the probability of survival 

 (p) is not assumed to be constant, but rather is itself a random variable. 

 The variation or uncertainty in p can result from three sources: (1) tempo- 

 ral changes in the environment or environmental stochasticity (Gilpin and 

 Soule 1986), (2) intrapopulation variability among individuals arising from 



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