A Mechanistic Derivation of the Logistic Model

The purpose of this note is to mechanistically derive the logistic population growth models from the well tested and received Droop equation. 1. A mechanistic derivation of the logistic equation. Population growth involves and often is determined by the birth and death processes. Most of the existing studies focus on birth process on the combined birth and death processes (the growth process). In general, death mechanisms are more numerous and difficult to study then birth mechanisms in a lab or field setting. In a short time frame, growth dynamics can be approximated by a linear differential equation with the coefficient called growth rate. Longer term, this growth rate shall be regarded as time dependent, or density dependent. The so-called Droop equation provides a time and experiment tested simple mathematical expression for biomass growth rate. We shall show that it also provides a convenient base for deriving the classical logistic equation. This section is adapted from Kuang et al. (2004). In 1968, Droop reported some surprising findings based on his most ambitious and comprehensive chemostat experiment to date in terms of concept, technical difficulty and mathematical analysis, it was to surpass by far all that had gone before (Leadbeater, 2006). The experiment studied the kinetics of vitamin B12 limitation in Monochrysis lutheri in continuous and exponentially growing batch cultures and in washed cell suspensions. The aim of this experiments was to relate specific growth rate to substrate concentration (Droop 1968). Contrary to conventional belief, the specific growth rate (m) of Monochrysis in the chemostats was found not to depend directly on medium substrate concentration. However, the one relationship that did stand out was that growth depended on the intracellular concentration of vitamin B12 (cell quota Q). The relationship between specific growth rate (μ ) and cell quota (Q) took the following simple form (Droop 1973, 1974). μ = μm ( 1− q Q ) . (1.1) This equation is called the Droop equation. We call μm ( 1− q Q ) the Droop function. The parameter q is the minimum quota necessary for life (the subsistence quota) and represents the value of cell quote Q at zero growth rate. μm is the growth rate 2000 Mathematics Subject Classification. 92D25 and 34C60.