Population dynamics models based on cumulative density dependent feedback: A link to the logistic growth curve and a test for symmetry using aphid data

• Published in: Ecological modelling 2009-08, Vol.220 (15), p.1745-1751
• Main Authors: Matis, James H., Kiffe, Thomas R., van der Werf, Wopke, Costamagna, Alejandro C., Matis, Timothy I., Grant, William E.
• Format: Article
• Language: English
• Subjects: Animal and plant ecology; Animal, plant and microbial ecology; Aphididae; Aphis; Biological and medical sciences; Cumulative size dependency; Demecology; Density dependent feedback; fluctuations; Fundamental and applied biological sciences. Psychology; General aspects; General aspects. Techniques; Glycine max; Logistic growth model; Logistic probability density; Methods and techniques (sampling, tagging, trapping, modelling...); Minimal mechanistic model; Population eruption and decline; Population growth laws; size; Skew
• ISSN: 0304-3800; 1872-7026
• DOI: 10.1016/j.ecolmodel.2009.04.026

Density dependent feedback, based on cumulative population size, has been advocated to explain and mathematically characterize “boom and bust” population dynamics. Such feedback results in a bell-shaped population trajectory of the population density. Here, we note that this trajectory is mathematically described by the logistic probability density function. Consequently, the cumulative population follows a time trajectory that has the same shape as the cumulative logistic function. Thus, the Pearl–Verhulst logistic equation, widely used as a phenomenological model for density dependent population growth, can be interpreted as a model for cumulative rather than instantaneous population. We extend the cumulative density dependent differential equation model to allow skew in the bell-shaped population trajectory and present a simple statistical test for skewness. Model properties are exemplified by fitting population trajectories of the soybean aphid, Aphis glycines. The linkage between the mechanistic underpinnings of the logistic probability density function and cumulative distribution function models could open up new avenues for analyzing population data.