The well-known logistic model has been extensively
investigated in deterministic theory. There are numerous case studies
where such type of nonlinearities 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 effects of randomness on such models. As a more realistic modeling,
we suggest nonlinear stochastic differential equations (SDEs) $$dX(t) = [(\rho + \lambda X(t))(K - X(t)) - \mu X(t)]dt + \sigma
X(t)^{\alpha}| K - X(t)|^{\beta}dW(t)$$ of Itô type to model the growth of populations or innovations $X$, driven
by a Wiener process $W$ and positive real constants $\rho$, $\lambda$, $K$, $\mu$, $\alpha$, $\beta \geq 0$. We discuss well-posedness, regularity (boundedness)
and uniqueness of their solutions. However, explicit expressions for
analytical solution of such random logistic equations are rarely 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 fluctuating 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 intervals with reflecting or absorbing
barriers. A discretization of the continuous state space can be circumvented by
appropriate methods. Balanced implicit methods (see Schurz, IJNAM 2 (2),
pp. 197-220, 2005) are used to construct strongly converging
approximations with the desired monotone properties. Numerical studies
can bring out salient 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).