Title: Estimation of trend in population time series data using growth curve models                

Abstract: Estimating the trend in population time series data using growth curve models is a central idea in population ecology. We provide a new framework to analyze ecological time series data by fitting mathematical models governed by fractional differential equations (FDE) that are used to incorporate memory in population processes. We show that how the FDE models can be utilized to estimate trend in population time series data and is shown to have better performance than the ordinary differential equation models. The application of FDE is exemplified by analyzing time series data on two bird species Phalacrocorax carbo (Great cormorant) and Parus bicolor (Tufted titmouse) and two mammal species Castor canadensis (Beaver) and Ursus americanus (American black bear) extracted from global population dynamics database. We fit five population growth models to these data; density-independent exponential, negative density-dependent logistic and θ-logistic model, positive density-dependent exponential Allee and strong Allee model. Both ordinary and fractional counterparts of these models are fitted to the population abundance data over time. Closed form equations of both the ordinary and fractional order models are calibrated on the real time series data. Markov chain Monte Carlo (MCMC) framework is used to estimate the model parameters and Akaike Information Criterion is used to select the best model. Estimating the return rate for each population we show that, populations, governed by FDE, return to the stable equilibrium faster than ordinary differential equation model. This demonstrates a synergistic interplay between memory and stability in natural populations.