Analysis of a population model with advection and an autocatalytic-type growth

Abstract

This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth. As opposed to a logistic growth where the rate of growth of the population decreases from the onset, an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend. Employing the idea of Painlevé property, we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term. In particular, due to situations in resource distribution or environmental variations, if the advection is represented as a decaying function in time or an oscillating function in time, we are able to find exact solutions with interesting behavior. We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.