A new governing equation for the bi-flux diffusion of a set of particles nested into subsets composing a cascade of two connected kinetic energy states is presented. The initial motivation was the problem of an invading population streaming along a given path such that a fraction of the individuals are retained to colonize the occupied territory while the complementary fraction keeps moving to conquer new sites. The solution to this problem was found with the help of a discrete approach. Following the clue suggested by the solution of the population dynamics problem considering retention a new formulation was developed for the continuum. It was possible to propose a new theory where the particles are divided into two distinct energy states. The primary state follows the classical Fick’s law and the secondary state which is closely related to the primary state obeys a new diffusion rule. This bi-flux theory is the most consistent assumption to interpret the presence of a fourth order differential term in the new governing equation. The new equation involves three parameters namely: The well-known diffusion coefficient D, a new coefficient associated with the perturbation of the classical diffusion process R, and also a new parameter β defining the fractions of the particles excited in the two afore mentioned energy states which may well be a stochastic variable. The new equation reads:

Beyond population dynamics, applications may be found in anomalous diffusion with multiple kinetic energy states, epidemics, capital flow and similar problems. With the present theory reaction effects may be dealt as an intrinsic property of the model controlled by the secondary diffusion coefficient R and a possibly stochastic variable β depending on the reactive properties of the medium.

Some clarifying examples are presented showing the most interesting features of the theory. Finally the extension of the theory for nonhomogeneous media and for a multi-energy or multi-flux diffusion state is presented.