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Logistics equations were studied in the context of complex systems initially for the wrong reasons: their deterministic yet unpredictable (chaotic) solutions for more then 3 nonlinear coupled equations and their fractal behavior in the discrete time-step version seemed for a while as a preferred (if not royal) road to complexity.

Even after those hopes were realistically re-assessed one cannot ignore the ubiquitness of the logistic systems: Montroll 'all the social phenomena obey logistic growth', Robert May: 'Montroll introduced in this context the concept of "sociological force" which induces deviations from the default "universal" logistic behavior he considers generic to all social systems' [5]

Moreover, as described later in this report, when the spatial distributed and stochastic character of the equations are appropriately taken into account, the logistic systems turn out to lead naturally and generically to collective Macro objects with adaptive and highly resilient properties and to scaling laws of the Pareto type.

In 1798, T.R. Malthus [1] wrote the first equation describing the dynamics of a set of autocatalitically proliferating individuals:

dx / dt = (birth rate - death rate)x

(malthus autocatalytic equation)

with its obvious exponential solution:

x ~ e^rt

where r = (birth rate- death rate)

According to the contemporary estimations the coefficients were such as to insure the doubling of the population every 30 years or so.

The impact of the prediction of an exponential increase in the population was so great, that everybody breathed with relief when P.F. Verhulst [2] offered a way out of it:

dx /dt = rx - cx²

(Verhulst logistic equation)

where the c coefficient represents the effect of competition and other limiting factors (finite resources, finite space, finite population etc). The solution of this equation starts exponentially but saturates asymptotically to the carrying capacity

x = r / competition

The solution was verified on animal population: the population of sheep in Tasmania (after a couple was lost by British sailors on this island with no sheep predators [3]), pheasants, turtledoves, bees colony growth, e. coli cultures, drosofillas in bottles, water fleas, lemmings, etc.

Applications of the logistic curve in technological change and new product diffusion were considered on [8] The fit was found excellent it for 17 productss (e.g. detergents displacing of soap in US, Japan).

The application of the Logistic Equation has been used to describe social change diffusion: the rate of adoption is proportional [12] to the number of people that have adopted the change times the number of the agents that still haven't.

Unfortunately detailed data on the spatio-temporal patterns of propagation were collected only for a few instances [DATA] of novelty propagation (hybrid corn among farmers in Iowa, antibiotics among physicians in US family planning among rural population in Korea).

The modeling of the aggregate penetration of new products in the marketing literature generally follows the Bass model [7] which in turn is based on the theory of Rogers [7] for the diffusion of innovations.

This theory postulates in addition to the internal ("word of mouth") influences the role of communication methods - external influence (e.g., advertising, mass media). The equations turn out to be of the same generic logistic for,

Further developments of the equations addressed the problems of:

1. interaction between different products sharing a market,
2. competition between producers
3. effects of repeat purchase
4. the dynamics of substitution of an old product (technology) by a new one.

In epidemiology, Sir Ronald Ross [14] wrote a system of differential coupled equations to describe the course of malaria in humans and mosquito.

This model was taken up by Lotka in a series of papers [15] and in particular in [16] where the system of equations generalizing (to vectors and matrices) the logistic equation :

dx_i / dt = r_ij x_j - c_ij x_i x_j

(diff equation logistic system)

were introduced.

The interpretation given by Lotka to this logistic system in the malaria context was:

-x_i represented the number of malaria infected individuals in various populations indexed by i (e.g. i=1 humans and i= 2 mosquitos).

r_ij represent probability by an individual from the i population of catching the disease upon interacting with an individual from the j species.

c_ij represent the saturation corrections due to the fact that the newly infected individual can be already ill.

To this date, the best field data for these systems occurs in the context of epidemics (superior antigenpropagation). Bayley [34] reviews the applications in epidemiology (applications to influenza, measels and rabies) and in particular "multiple site" models [35] generalizing the original malaria problem studied by Ross and by Lotka.

Vito Voltera became involved independently with the use of differential equations in biology and social sciences [17] (the English translations of the references [17-??] together with further usefull information can be found in the collection [18].

In particular, Volterra re-deduced the logistic curve by reducing the verhulst-pearl equation to a variational principle (maximixing a function that he named "quantity of life" ). V.A. Kostitzin generalized the logistic equation [18] to the case of time and variable dependent coefficients and applied it to species genetic dynamics [19]. This was further generalized by A Kolmogoroff [22]. May showed that logistic systems [24] are almost certainly stable for a wide range of parameters.

The extension of the relevance of these models to many other subjects continued for the rest of the century e.g. [23]. See also [24] for a very mathematical study of the equations of the logistic form.

Another generalizationwas proposed by Eigen [36] and Eigen and Shuster [37] [38] was in the context of Darwinian selection and evolution in prebiotic environments. One assumes a species of autocatalytic molecules which can undergo mutations. The various mutant "quasi-species" have various proliferation rates and can also mutate one into the other.

In the resulting equations describing the system r_ij represent hen the increase in the population of species i due to mutations that transform the other species j into it. This dynamics is the crucial ingredient in the study of molecular evolution in terms of the autocatalytic dynamics of polynucleotide replicators.

Eigen and Schuster showed that the system reaches a "stationary mutant distribution of quasispecies". The selection of the fittest is not completely irrelevant but it refers now to the selection of the highest eigenvalue eigenvector. The importance of the spreading of the mutants over an entire genomic space neighbourhood of the current fitness maximum was emphasized in the context of a very hostile and changing environment in by many authors.

The spread of the population and the discreteness of the genetic space lead to a situation in which populations which would naively disappear (in the hypothesis of continuum genetic space) in fact survive, adapt and proliferate.

As remarked in Mikhailov and Mikhailov and Loskutov [46][47] the Eigen equations are relevant to market economics: If one denotes by i the agents that produce a certain kind of commodity and compete on the market one may denote by x_i the amount of commodity the agent i produces per unit time (production of i).Then (diff eq logistic system) is obtained by assuming that a certain amount of the profit is reinvested in production and by taking into account various competition and redistribution (and eventual cooperation) effects.

One may apply the same equations to the situation in which x_i represent the investment in a company i or the value of its shares. Marsili Maslov and Zhang have shown that the equation (diff eq logistic system) characterizes the optimal investment strategy.

The ecology-market analogy was postulated already in [Schumpeter] and [Alchian] See also [Nelson and Winter] [Jimenez and Ebeling] [Silverberg] [Ebeling and Feistel] [Jerne].

The extension to spatially extended logistic systems in terms of partial differential equations was first formulated in [25] in the context of the propagation of a mutant superior gene in a population.

dx_i /dt =( r_ij + d_j ? ) x_j - c_i,j x_i x_j

(spatially distributed logistic equation)

where the coefficient d_j and the Lagrangian operator ? ??_r?_r represent the spatial diffusion due to the jumping of individuals between neighboring locations.

The mathematical study of these "Fisher fronts" was taken up by [26] folowed up by a large literature in physics journals. [quote anomalous diffusion works].

For the further development of this direction, see [28-33].

A large body of mathematical literature [36-40] (much of it quite illegible to non-mathematicians) has addressed in the past the stochastic generalizations of the logistic/ lotka-volterra equations in which the coefficients and the unknowns are stochastic variables. One of the difficulties is that the continuum (differential equation ? like) notation of a stochastic process is not unambiguous. In general the interpretation is along the Ito lines [ref Ito] and becomes unambiguous when the process is specified at the discrete time level. [41]. Of particular importance for survaivability, resilience and sustainability are also the random space-time fluctuations in the coefficients r_ij due to the stochastic / discrete character of the substrate / environment. This randomness is also responsible for random variations in the sub-species fitness which in turn can be shown to be responsible for the emergence of power laws in the distribution of individuals between the various sub-populations. Similar effects are seen in the other applications of the logistic systems where microscopic discreteness and insuing randomness lead quite universally to stable power laws (even in arbitrary and dramatically non-stationary global conditions).

This multiscale distribution of the sub-populations constitutes an additional link betiween the microscopic stage of the natural selection and the macroscopic dynamics of the populations. Most of the effort was in the direction of extending the stability / cycles theorems existing in the deterministic case.

Moreover Horsthemke and Lefever applied it to the particular case of the stochastic extension of the logistic Verhulst equation.

Rather than having a self-averaging effect, this multiplicative noise leads to scaling variations in the sub-population sizes and consequently ([London][Biham et al][Paris]) to power tailed (Levy) fluctuations in the global sub-species populations.

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