Evolution: an axiomatic perspective.

In this paper we present an axiomatic theory of evolution which is inspired by the reading of a paper written by G.V. Schiaparelli in 1898. Schiaparelli was a famous astronomer, but he also studied the Darwinian ideas. We propose five axioms which can characterize the theory of evolution. We have also written these axioms using the language of the logic of predicates of first order with some constant monadic and dyadic predicates and appropriate functionals. But we can use other types of logic. So we can examine the concepts of species and of speciation. Then we introduce the interesting notion of generation distance. Moreover we give a theorem which establishes a geometrical model of our theory. If we analyze further the fourth axiom (which concerns the notion of generation distance) we can propose an elementary dynamical model by which we can represent possible evolutionary dynamics. These dynamics partially depend on random quantities.     

Usefulness of information in bistable system and Dyson model for the origin of metabolism

Useful or meaningful information for the transition between disordered and ordered states occurring in a chemical system, which organizes itself out of molecular chaos, is stochastically discussed from the viewpoint of dynamical system theory. To make the discussion perspicuous it is helpful to assume that the time evolution of the chemical system proceeds by a univariate, bistable one-step process of birth-and-death type. The model for the evolution of primordial metabolism proposed by Dyson is a bistable system, and hence this model answers the purpose of illustrating the usefulness of information.     

The frame of non-canonical theory of heredity: from genes to epigenes

Particular theory of heredity that exceeds the limits of mendelian genetics is suggested. The model based on five sufficiently obvious assumptions (accepted as axioms) As consequence of these axioms the strict statements concerningfunctional heredity memory were formulated in mathematical terms. Molecular-genetic realization of the memory cells appears as new class of heredity units--epigenes. In the epigenes part f hereditary information is contained, encoded and transmitted beyond the primary structure of DNA molecules of genome. Epigenes capable to conserve sequences of genes functional states in the course of ontogenesis and provide transmission of information contained in this states throw consequent generations. It was shown that epigenes differ from genes at least by encoding method of heredity information. There are three functional-equivalent classes of really existing epigenes mechanisms: dynamic, modificational and transpositional; and there is one hypothetical class--invertional. It was shown that a lot of experimental data concerning epigenetic mechanism of heredity is in accord with theoretical conclusions concerning epigenes existence. Moreover, we constructed an artificial epigenes by genetic engineering methods. The existence of epigenes means that obtaining complete genome sequence, its physical and genetic maps, as well as distinguishing the rules of genes function encoding by its primary structure do not provide complete decoding of hereditary information. The role of epigenes in ontogenesis and phylogenesis was examined. It was shown that even elementary epigenetic systems could determine key ontogenesis events. Epigenetic system could serve as the basis of non-darwinian evolutionary strategies by means of "memorization of rather unsuccessfully steps of evolution" and conservation of alternative variants of ontogenesis. Teleonomic hypothesis on functional heredity memory was formulated. This theory provides explanation of phenomena of acquired features inheritance and molecular mechanisms of stress-induced evolution.     

A theoretical framework for defining some concepts in evolution.

We present a theoretical framework for biological evolution with the intention of giving precise mathematical definitions of some concepts in evolutionary biology such as fitness, evolutionary pressure, specialization and natural selection. In this framework, such concepts are identified with well-known mathematical terms within the theory of dynamical systems. We also discuss some more general implications in evolution; for instance, the fact that our model naturally exhibits a frequency spectrum of the type 1/f for low frequencies of evolutionary events.     

Chaos and language

Human language is a complex communication system with unlimited expressibility. Children spontaneously develop a native language by exposure to linguistic data from their speech community. Over historical time, languages change dramatically and unpredictably by accumulation of small changes and by interaction with other languages. We have previously developed a mathematical model for the acquisition and evolution of language in heterogeneous populations of speakers. This model is based on game dynamical equations with learning. Here, we show that simple examples of such equations can display complex limit cycles and chaos. Hence, language dynamical equations mimic complicated and unpredictable changes of languages over time. In terms of evolutionary game theory, we note that imperfect learning can induce chaotic switching among strict Nash equilibria.     

Modeling the topological organization of cellular processes

The cell as a dynamical system presents the characteristics of having a dynamical structure. That is, the exact phase space of the system cannot be fixed before the evolution and integrative cell models must state the evolution of the structure jointly with the evolution of the cell state. This kind of dynamical systems is very challenging to model and simulate. New programming concepts must be developed to ease their modeling and simulation. In this context, the goal of the MGS project is to develop an experimental programming language dedicated to the simulation of this kind of systems. MGS proposes a unified view on several computational mechanisms (CHAM, Lindenmayer systems, Paun systems, cellular automata) enabling the specification of spatially localized computations on heterogeneous entities. The evolution of a dynamical structure is handled through the concept of transformation which relies on the topological organization of the system components. An example based on the modeling of spatially distributed biochemical networks is used to illustrate how these notions can be used to model the spatial and temporal organization of intracellular processes.     

A possible scenario for spatial and temporal evolution of self-organizing biophysical structures

The stationary diffraction theory, the spectral theory of open systems, and the theory of Morse critical points of dispersion equations made it possible, when applied to bounded self-organizing biological media, to formulate the analytical dispersion laws and construct nonlinear evolution equations describing local space and time processes in a biomacromolecular continuum. By the simplest examples of the solution of the nonlinear evolution equations for planar medium interfaces, a variety of dynamical phenomena occurring in the self-organization of macromolecules are shown.     

Linking dynamical and population genetic models of persistent viral infection

This article develops a theoretical framework to link dynamical and population genetic models of persistent viral infection. This linkage is useful because, while the dynamical and population genetic theories have developed independently, the biological processes they describe are completely interrelated. Parameters of the dynamical models are important determinants of evolutionary processes such as natural selection and genetic drift. We develop analytical methods, based on coupled differential equations and Markov chain theory, to predict the accumulation of genetic diversity within the viral population as a function of dynamical parameters. These methods are first applied to the standard model of viral dynamics and then generalized to consider the infection of multiple host cell types by the viral population. Each cell type is characterized by specific parameter values. Inclusion of multiple cell types increases the likelihood of persistent infection and can increase the amount of genetic diversity within the viral population. However, the overall rate of gene sequence evolution may actually be reduced.     

On the stock estimation for some fishery systems

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called “observer” can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation of the unmeasurable stock state x(t). Moreover the convergence speed of towards x(t) can be chosen.     

Evolutionary stability in Lotka-Volterra systems

The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.     

Weak ergodicity of population evolution processes

The weak ergodic theorems of mathematical demography state that the age distribution of a closed population is asymptotically independent of the initial distribution. In this paper, we provide a new proof of the weak ergodic theorem of the multistate population model with continuous time. The main tool to attain this purpose is a theory of multiplicative processes, which was mainly developed by Garrett Birkhoff, who showed that ergodic properties generally hold for an appropriate class of multiplicative processes. First, we construct a general theory of multiplicative processes on a Banach lattice. Next, we formulate a dynamical model of a multistate population and show that its evolution operator forms a multiplicative process on the state space of the population. Subsequently, we investigate a sufficient condition that guarantees the weak ergodicity of the multiplicative process. Finally, we prove the weak and strong ergodic theorems for the multistate population and resolve the consistency problem.     

Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems

In the present paper we have first introduced a measure of dynamical entropy of an ecosystem on the basis of the dynamical model of the system. The dynamical entropy which depends on the eigenvalues of the community matrix of the system leads to a consistent measure of complexity of the ecosystem to characterize the dynamical behaviours such as the stability, instability and periodicity around the stationary states of the system. We have illustrated the theory with some model ecosystems.     

Consequences of plant-herbivore coevolution on the dynamics and functioning of ecosystems

The potential consequences of plant-herbivore coevolution for ecosystem functioning are investigated using a simple nutrient-limited ecosystem model in which plant and herbivore traits are subject to adaptive dynamics. Although the ecological model is very simple and always reaches a stable equilibrium in the absence of evolution, coevolution can generate a great diversity of dynamical behaviors. The evolutionary dynamics can lead to a stable equilibrium. If the evolution of plants is fast enough, certain values of the trade-off parameters lead to complex evolutionary cycles bounded by physiological constraints. The dynamical behavior of the model is very different when the dynamics of inorganic nutrient is ignored and plant competition is modeled by a logistic growth function. This emphasizes the importance of including explicit nutrient dynamics in studies of plant-herbivore coevolution.     

Abstract representations of biological systems in supercategories

An abstract representation of biological systems from the standpoint of the theory of supercategories is presented. The relevance of such representations forG-relational biologies is suggested. In section A the basic concepts of our representation, that is class, system, supercategory and measure are introduced. Section B is concerned with the mathematical representation starting with some axioms and principles which are natural extensions of the current abstract representations in biology. Likewise, some extensions of the principle of adequate design are introduced in section C. Two theorems which present the connection between categories and supercategories are proved. Two other theorems concerning the dynamical behavior of biological and biophysical systems are derived on the basis of the previous considerations. Section D is devoted to a general study of oscillatory behavior in enzymic systems, some general quantitative relations being derived from our representation. Finally, the relevance of these results for a quantum theoretic approach to biology is discussed.     

具脉冲扩散效应的单种群动力学模型

讨论了两斑块间脉冲扩散的单种群动力学模型,利用离散动力系统频闪映射理论,得到了种群持续生存的充分条件．结论31,1~了现实的生物种群动力学性质,也丰富了脉冲微分方程理论．     

On the Stock Estimation for a Harvested Fish Population

We consider a stage-structured model of a harvested fish population and we are interested in the problem of estimating the unknown stock state for each class. The model used in this work to describe the dynamical evolution of the population is a discrete time system including a nonlinear recruitment relationship. To estimate the stock state, we build an observer for the considered fish model. This observer is an auxiliary dynamical system that uses the catch data over each time interval and gives a dynamical estimate of the stock state for each stage class. The observer works well even if the recruitment function in the considered model is not well known. The same problem for an age-structured model has been addressed in a previous work (Ngom et al., Math. Biosci. Eng. 5(2):337–354, 2008).     

Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

旋转向量场中的阶二周期解

建立了一类具有状态脉冲控制的害虫控制模型以及一类特殊模型,根据半连续动力系统几何理论,研究了旋转向量场中阶二周期解的存在性和稳定性及变化规律.为现实的生物资源管理提供了可靠的策略依据,也丰富了半连续动力系统理论.