Date of Award
Doctor of Philosophy (PhD)
Mathematics and Statistics
Computational and Applied Mathematics
John A. Adam
James L. Cox, Jr.
The population growth of a single species is modeled by a differential equation with initial condition(s) so that the number of organisms in the population is derived using some mechanism of growth, i.e. a growth rate function. However, such deterministic models are often highly unrealistic in population dynamics because population growth is basically a random event. There are a large number of chance factors influencing growth that might not be taken into account by deterministic models. The effect of other species (for example, in the chance meeting of a predator), population fluctuations due to weather changes that would alter food supply, immigrations and emigrations, in addition to the very processes of birth and death are chance occurrences. A stochastic approach to the modeling of complex biological systems, such as populations, allows us to include all these secondary effects for which a detailed knowledge is impossible.
Assuming that the time and state space are described by continuous variables and that the system is Markovian (i.e, for small changes in time, there are corresponding small changes in the state space) a stochastic differential equation (SDE) is constructed by adding a random function, incorporating all random fluctuations, to a deterministic equation. If this random function is Gaussian and of "white noise type", the SDE leads to a Fokker-Planck equation, a second order parabolic partial differential equation for the probability density function. Using separation of variables and simple transformations the Fokker-Planck equation reduces to a time independent ordinary differential equation, the Schrodinger equation of quantum mechanics. Solving the latter we obtain expressions for the probability density function, which, in turn gives us various moments or averages.
In this paper three specific rate transforms for the modeling of autocatalytic (self stimulated) growth processes are presented. The probability density functions are analytically solvable by Fokker-Planck methods and are generalizations of transforms heretofore unknown. Thus, the moment formulas obtained are also generalizations of previous results.
We also consider population growth in a random environment and obtain general formulae for the steady-state and time dependent probability density function in two regimes--thus generalizing formulae previously obtained elsewhere.
In addition, a large collection of known potentials that exactly solve the Schrodinger equation are collected. Hence an extensive set of known analytic solutions to the Fokker-Planck equation, and solutions to a special form of the Riccati equation are listed. As far as we know two new Schrodinger potentials are presented for the first time.
"The Fokker-Planck and Related Equations in Theoretical Population Dynamics"
(1990). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/hcr7-7m93