Here Rt is the number of sexually mature adult recruits, St is the number of surviving mature adults, and lt, pt, rt and st are the respective numbers observed at time t. The total number of mature adults is At = Rt + St and at = rt + st is the number of mature adults observed at time t. The symbol means 'is distributed as'. This "Poisson/binomial" LPA (or PBLPA) model is integer value and its dynamics occur on a lattice.
One way to construct a deterministic "skeleton" for the PBLPA model is by iterating the conditional expectation (so that the "most likely" data triple (Lt+i, Pt+i, At+i) to occur at time t +1, given the observed triple (lt,pt, at), is assumed to be the mean of the random variables in the PBLPA model). This results in the continuous state space LPA model (2).
On the other hand, we can obtain a deterministic skeleton that remains on the integer lattice (where real data is observed) by using another measure of central tendency, namely, the mode. By iterating the conditional mode we obtain a deterministic lattice mode described, as it turns out (assuming the unlikely event of a non-unique conditional mode), by the equations3
3 These equations result from formulas for the mode of a binomial random variable and the mode of a Poisson random variable. The following derivations are due to Michael Trosset and Shandelle Henson (private communication). The pdf for a binomial random variable binomial(n, p) is f (x) = x!(n-x)iPx (1 — p)n-x. If x = m is the mode, then f (m + 1) < f (m) and hence p (n + 1) — 1 < m. Also f (m — 1) < f (m) implies m < p (n + 1). Since m is an integer, and since p (n + 1) is almost always an integer, it follows that m = floor[p(n + 1)]. The pdf fo a Poisson random variable poisson(^) is f (x) = ^ ^ . For the mode m, we see that f (m +1) < f (m) implies ^ — 1 < m and f (m — 1) < f (m) implies m < [i. Since ^ is almost always not an integer, we have m = floor[^]. We also point out that the equation for At+i is different from that given in (Cushing et al. 2003) because of the nature of the experimental protocol involved in the study discussed in that book.
At+i = floor [(Pt + 1)exp( — ^A^l + floor [(1 — Ma)(At + 1)] .
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