On estimating the parameters of generalized logistic model from census data: Drawback of classical approach and reliable inference using Bayesian framework

Abstract

In ecology, a nonconstant functional relationship between per capita growth rate and population size is referred as density dependence and mathematical models are utilized to detect its presence in natural populations. The theta-logistic model (parameterized by rm: intrinsic growth rate; θ: shape parameter; and K: carrying capacity) has been extensively discussed through various generalizations in the literature due to its flexibility and sound ecological interpretations which can be generalized for many natural populations. In this article, we show that nonlinear least squares approach is not an appropriate choice for estimating the model using real data. Using simulation, we show that the unknown parameters are better estimated under a Bayesian framework. We utilize the Gibbs algorithm for simulating samples from the posterior density, which is approximated by grid approximation and Bayesian credible intervals are obtained. Reliability of the estimation process is shown by using simulated data sets. Rules for choosing the prior distributions are discussed. We also apply the proposed strategy to estimate parameters using real data sets from the global population dynamics database. The robustness of the method with respect to prior distribution of the parameters is investigated by taking different choice of priors. We also establish its effectiveness in estimating parameters in a predator-prey system. The discussed method is computationally intensive, but its simple implementation will be useful for fitting complex models to study growth patterns of natural populations.