# Mathematical Modeling of Evolution

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Biological evolution is a very complex process. Using mathematical modeling, one can try to clarify its features. But to what extent can that be done? For the case of evolution, it seems unrealistic to develop a detailed and fundamental description of phenomena as it is done in theoretical physics. Nevertheless, what can we do? Can mathematical models help us to systemize our knowledge about evolution? Can they provide us with more a profound understanding of the particularities of evolution? Can we imagine (using mathematical representation) some hypothetical stages of evolution? Can we use mathematical models to simulate some kind of artificial "evolution"?

In order to clarify such questions, it is natural to review the already developed models in a systematic manner. In evolutionary modeling one can distinguish the following branches:

• Models of molecular-genetic systems origin have been constructed in connection with the origin of life problem. Quasispecies and hypercycles by M. Eigen and P. Schuster and sysers by V.A. Ratner and V.V. Shamin are the best known. These models describe mathematically some hypothetical evolutionary stages of prebiological self-reproducing macromolecular systems.

• Artificial life evolutionary models are aimed at understanding of the formal laws of life and evolution. These models analyze the evolution of artificial “organisms”, living in computer-program worlds.

• Applied evolutionary models are computer algorithms, which use evolutionary methods of optimization to solve practical problems. The genetic algorithm by J.H. Holland and the evolutionary programming, initiated by L. Fogel et al., are well-known examples of these researches.

The analysis, accomplished in the child nodes, demonstrates that the relations between evolutionary models and experiments are rather abstract. The evolutionary models are mainly intended to describe general features of evolution process rather then concrete experiments. Only particular models (e.g. some models of mathematical genetics) are used to interpret certain experimental data. Moreover, some branches of evolutionary modeling (life origin models, artificial life evolutionary models) go to more abstract level and describe imaginary evolutionary processes: not the processes as-we-know-them, but the processes as-they-could-be.

This abstractness is understandable: because the biological world is very complex and diversified, we firstly try to generalize a lot of experiments and only then we interpret this generalized representation in mathematical models.

Historically, the profound experimental researches have stimulated the creation of evolutionary theories. For example, the mathematical theories of population genetics by R.A. Fisher, J.B.S. Haldane, and S. Wright were based on experimental genetic investigations, performed in the first half of the 20-th century. The outstanding achievements of the molecular biology, which were attained in the 1950-1960s, constituted the underlying background for the life origin models by M. Eigen et al and the models of regulatory genetic systems by S.A. Kauffman.

Currently, the evolutionary models are intensively developed in close connection with computer science researches, especially in Artificial Life investigations. There is an obvious tendency towards modeling of evolution of cybernetic, computer-like, intelligent features of biological organisms. Current evolutionary models are actively incorporating such notions as learning, neural networks, adaptive behavior. Nevertheless, a lot of problems, concerning the evolution of animal cognition abilities, have to be investigated. Let’s outline these problems briefly.

Biological evolution was able to create complex, harmonic, and very effective biocybernetic control systems, which govern the animal behavior. But how do these cybernetic systems operate? How did they emerge through evolution? What kinds of information processing and memory structures are used in animal control systems? How did animal cognitive abilities evolve? What kinds of "internal models" of the environment emerge in the animal "minds"? How are these "models" used in animal behavior? What were the transitional stages between animal cognitive abilities and human intelligence?

In order to investigate such a wide spectrum of problems, it is natural to use a certain evolutionary strategy and to analyze the animal control systems and emergence of animal "intelligent" features step by step, considering the biological evolutionary process as underlying background. Such a field of investigations can be called "Evolutionary biocybernetics". The conceptual background for the investigations of the evolution of animal cognition abilities was described in the first chapters of "The Phenomenon of Science" by V.F. Turchin [1]. Some approaches towards the developments of the evolutionary biocybernetics were outlined in the paper [2].

Conclusion. The mathematical modeling of evolution was profoundly elaborated in several directions: life origin models, mathematical population genetics, models of evolution of genetic regulatory systems, artificial life evolutionary models. These models provide us with better understanding of biological evolutionary phenomena; they also give generalized descriptions of biological experiments. Some models provide us with more abstract pictures – they describe artificial evolutionary processes: not the processes as-we-know-them, but the processes as-they-could-be. Thus, mathematical modeling of evolution is profound, well-elaborated, intensively developing field of theoretical investigations. Nevertheless, there are serious problems to be analyzed: the problems of evolution of cybernetic, computer-like, "intelligent" features of biological organisms. The theoretical investigations of these problems could constitute the subject of a future scientific discipline "Evolutionary biocybernetics".

References:

1. Turchin, V. F. The Phenomenon of Science. A Cybernetic Approach to Human Evolution. Columbia University Press, New York, 1977.

2. Red'ko, V.G. Towards the evolutionary biocybernetics // Proceedings of The Second International Symposium on Neuroinformatics and Neurocomputers, Rostov-on-Don, 1995, pp. 422-429.