The classical rate equations for the concentration p(x,t) or the probability density in the diffusion-limited regime are extended by including non-Markovian terms. We present analytical and numerical results for a whole class of evolution models with conserved p, where the underlying equations are of convolution type
with temporally and spatially varying memory kernels. Based on our recent studies in the reaction-limited case with memory, we study now the influence of time and spatial couplings. Due to the balance between the conventional diffusive current and the additional force, originated by the feedback, the system exhibits a non-trivial stationary solution which depends on both the initial distribution and the memory strength. For a non-linear memory kernel of KPZ-type we get an asymptotic exact solution. Although the mean square displacement offers ultimately diffusion, the distribution function is determined by the memory strength, too. Differences to diffusion are observed in higher order cumulants. For an arbitrary memory kernel we find a criteria which enables us to get a non-trivial stationary solution.
Epidemiological processes are studied within a recently proposed social network model using the susceptible-infected-refractory dynamics (SIR) of an epidemic. Within the network model, a population of individuals may be characterized by H independent hierarchies or dimensions, each of which consists of groupings of individuals into layers of subgroups. Detailed numerical
simulations reveals that for H > 1, the global spreading results regardless of the degree of homophily α of the individuals forming a social circle. For H = 1, a transition from a global to a local spread occurs as the population becomes decomposed into increasingly homophilous groups. Multiple dimensions in classifying individuals (nodes) thus make a society (computer network) highly susceptible to large scale outbreaks of infectious diseases (viruses). The SIR-model can be extended by the inclusion of waiting times resulting in modified distribution function of the
Different evolution models are considered with feedback-couplings. In particular, we study the Lotka-Volterra system under the influence of a cumulative term, the Ginzburg-Landau model with a convolution memory term and chemical rate equations with time delay. The memory leads to a modified dynamical behavior. In case of a positive coupling the generalized Lotka-Volterra system exhibits a maximum gain achieved after a finite time, but the population will die out in the long time limit. In the opposite case, the time evolution is terminated in a crash. Due to the nonlinear feedback coupling the two branches of a bistable model are controlled by the the strength and the sign of the memory. For a negative coupling the system is able to switch over between both branches of the stationary solution. The
dynamics of the system is further controlled by the initial condition. The diffusion-limited reaction is likewise studied in case the reacting entities are not available simultaneously. Whereas for an external feedback the dynamics is altered, but the stationary solution remain unchanged, a self-organized internal feedback leads to a time persistent solution.