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Reaction diffusion systems are multi-agent systems where the agents may move freely / randomly (diffuse) in space as long as they do not encounter one another and react when they eventually meet.

The usual approach to reaction-diffusion processes in their field of origin (chemistry) was to express them in terms of density fields D(r,t) representing the average density of the different reactants (agents of a given type) as continuos functions of time t and spatial location r (in certain cases the dependence on r is neglected and the system is represented as a single object).

This approach stands in contrast to the Multi-Agent approach which consists of tracking in time the individual location of each and the individual transformations which the agents undergo upon meeting.

Whatever approach is taken the interest in a reaction-diffusion system is usually its spatio-temporal evolution.

In the density field approach the spatio-temporal distribution is explicitly expressed by the variables D (r,t) and the partial-differential equations governing them (see below) while in Multi-Agent approach the spatio-temporal distribution emerges only upon averaging the agents positions over finite space-time ranges (i.e. over many stochastic system instances ='configurations').

In the complex applications one often needs to represent space by a discrete regular mesh and record the number of agents of each kind on each mesh site.

The Multi-Agent approach is closer to the real system when there are only trace densities of the different reactants. Indeed, the partial differential equations approach describes the system with less accuracy when the discreteness of the reactants is apparent.

We will see later in the report, that for auto-catalytic reactionsthe microscopic discreteness influences the macroscopic long range behavior of the system in generating power laws, localized spatio-temporal structures and collective self-organized behavior. The continuous approach may miss occasionally, or even systematically, such effects as we will see below.

These special Reaction-Diffusion Multi-Agent effects are not restricted to describing chemical systems. Indeed reaction-diffusion processes of the type (spatially distributed logistic) have even been used extensively in population biology [Maynard] (where auto-catalysis is called "reproduction"), marketing (see section. ..) immunology, finance, social science, etc )).  Discretization is crucial in the behavior of such auto-catalytic models thus showing that the Multi-Agent is indispensable to researchers who need to model real-life emerging systems. Indeed, due to the spatio-temporal fluctuations in the autocatalytic coefficients r_ii even if the growth rate is in average negative < r_ii ><< 0,the system presents rare singular points where there is momentary growth. One can show that in a very wide range of conditions the islands of growth generated by one such singular fluctuation survive the actual life of the original and are able to keep growing due to the occurrence of another such fluctuation on their territory. Thus, an entire infinite chain of singular fluctuations (not too far one from the preceding one) might insure the survival forever of the x’s as a population. Note that while the collective islands looks like searching and opportunistically taking advantage of the environment fluctuations, the actual individuals are completely naive (zero intelligence /rationality). This mechanism explains the role of collective emergent objects such as cells, species, institutions, herds in insuring the sustainability and resilience of adaptive activity in situations which otherwiselook hopeless and doomed to decay.

Moreover, the multiplicative stochastic character of the x_i r_ii term can be shown to imply the emergence of robust scaling even in conditions in which the probability distribution r_ii and the nonlinear saturation terms are highly non stationary and lead to chaotic global dynamics.

This explains why inspecting the list of Scaling systems and the list of Logistic systems one sees a very strong overlap between them.

Next: Autocatalytic Growth
Up: Table of contents
Previous: Power laws and dynamics of their emergence