Estimation of Probability Distributions for Individual Parameters Using Aggregate Population Data

A comparison of approximation methods for the estimation of probability distributions on parameters

“…Based on data collected from rice fields, a reasonable mathematical model would have to predict two key features that are exhibited in the data: dispersion and bifurcation (a unimodal density becomes a bimodal density) of the population density over time [4,9,10]. The growth rate distribution model, developed in [4] and [9], captures both of these features in its solutions.…”

“…The SS model cannot be used as is to model the mosquitofish population because it does not predict dispersion or bifurcation of the population in time under biologically reasonable assumptions [4,9]. However, by modifying the SS model so that the individual growth rates of the mosquitofish vary across the population (instead of being the same for all individuals in the population), one obtains a model, known as the growth rate distribution (GRD) model, which does in fact exhibit both dispersal in time of the mosquitofish population and the development of a bimodal density from a unimodal density (see [9,10]). In the growth rate distribution (GRD) model, the population density u(t, x; P ), discussed in [4] and developed in [9], is actually given by…”

“…We also assume here that µ = 0 and K = 0 in order to focus on only the distribution of growth rates; however, distributions could just as well be placed on µ and K. We next introduce two different approaches that can be used in the inverse problem for the estimation of the distribution of growth rates of the mosquitofish. The first approach, which has been discussed and used in [9] and [10], involves the use of delta functions. We assume that the probability distributions P M placed on the growth rates are discrete corresponding to a collection G M with the form…”

“…Furthermore, we are able to note that this least squares inverse problem (3) becomes a quadratic programming problem [9,10]. Letting p be the vector that contains p k , 1 ≤ k ≤ M, when using DEL(M) or a k , 1 ≤ k ≤ M, when using SPL(M,N), we let A be the matrix with entries given by…”

Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model

“…As we discuss in Section 2, these probability-based methods can be used with preselected prior distributions as with Monte Carlo methods, or they may be used with weighted sums of Dirac delta measures that do not assume a fixed form for the probability distribution functions. A version of this method has been applied to a population model for mosquitofish in rice paddies, and was used to successfully describe fish population dynamics by estimating distributed growth rate functions using aggregate experimental data [35].…”