Modeling, analysis and discretization of stochastic logistic equations
Henri Schurz 11 Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901-4408, USA
Received by the editors November 24, 2005 and, in revised form, January 10, 2006
The well-known logistic model has been extensively investigated in determinisc theory. There are numerous case studies where such type of non linearities occur in Ecology, Biology and Environmental Sciences. Due to the presence of environmental fluctuations and a lack of precision of measurements, one has to deal with erect so randomness on such model s. As a more realistic modeling, we suggest nonlinear stochastic direrential equations (SDEs) dX(t) = [(rho + lambda X(t))(K - X(t)) - mu X(t)]dt + sigma X(t)(alpha)vertical bar K - X(t)vertical bar(beta)dW(t) of Ito type to model the growth of populations or innovations X, driven by a Weiner process W and positive real constants rho, lambda, K, mu, alpha, beta >= 0. We discuss well-posedness, regularity (boundedness) and uniqueness of their solutions. However, explicit expressions for analytical solution of such random logistic equations are rearly known. Therefore one has to resort to numerical solution of SDEs for studying various aspects like the time-evolution of growth patterns, exit frequencies, mean passage times and impact of flostuating growth parameters. We present some basic aspects of adequate numerical analysis of these random extensions of these models such as numerical regularity and mean square convergence. The problem of keeping reasonable boundaries for analytic solutions under discretization plays an essential role for practically meaningful models, in particular the preservation of the continuous state space can be circumvented by appropriate methods. Balanced implicit methods (see Schurz, IJNAM 2 (2), p. 197-220, 2005) are used to construct stongly converging approximations with the desired monotone properties. Numerical studies can bring out salent features of the stochastic logistic models (e.g. almost sure monotonicity, almost sure uniform boundedness, delayed initial evolution or earlier points of inflection compared to deterministic model).
AMS subject classifications: 65C30, 65L20, 65D30, 34F05, 37H10, 60H10
Key words: logistic growth; stochastic logistic equation; properties of solutions; numerical methods; balanced implicit methods; boundedness; convergence; stability; monotonicity