New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations

This paper examines how one can use Riemann–Liouville fractional Brownian motion (in contrast to complex-valued fractional Brownian motion) to take account of randomness in some biological systems, and the kind of results one may expect to obtain from this. Loosely speaking, the Hurst parameter of the fractional Brownian motion appears to be quite suitable for describing the aggressiveness of some biological processes. After a summary on fractional calculus completed by a new derivation of Taylor’s series of fractional order, we propose a new solution for the stochastic differential equation of the Malthusian growth model with fractional random growth rate, and apply it to the stability analysis of some nonlinear systems (the logistic law of growth, for instance). Then we derive, as a discrete space model, the equation for the birth-and-death process of fractional order; then its companion Poissonian process (of fractional order) is considered in a fully detailed way, including the fractional partial differential equation of its generating function, which is solved by using a new technique. Lastly, we consider a model of optimal management of two species populations in the presence of fractal noises, which is an application of stochastic optimal control in the presence of fractional noises. We show how one can solve this problem by using the Lagrange variational approach applied to the dynamical equations of the state moments of the system.