Andrzej Pȩkalski

Simple model for the dynamics of correlations in the evolution of economic entities under varying economic conditions

From some observations on economic behaviors, in particular changing economic conditions with time and space, we develop a very simple model for the evolution of economic entities within a geographical type of framework. We raise a few questions and attempt to investigate whether some of them can be tackled by our model. Several cases of interest are reported. It is found that the model even in its simple form can lead to a large variety of situations, including: delocalization and cycles, but also pre-chaotic behavior.

Transfer matrix methods in the Blume-Emery-Griffiths model

Abstract The critical properties of the plane Blume-Emery-Griffiths (BEG) model are analyzed using two transfer matrix approaches. The two methods and the domains of their applicability are discussed. The phase diagram is derived and compared with the one obtained by the position-space renormalization group (PSRG). The critical indices η i and conformal anomaly c are computed at Ising-like and Potts-like critical points and a good agreement with the conformal invariance predictions is found. A new, very effective method of estimating critical points is introduced and an attempt to estimate critical end points is also made.

Mutations and changes of the environment in a model of biological evolution

Time evolution of a model of population of diploid organisms is studied using the Monte Carlo technique. Reproduction of the individuals follows the Mendelian rules, including the recombination. We compare the effects of harmful mutations and changes of the environment on the population size, its average adaptation and the average age of the individuals. We show that at short time scale the mutations change the dynamics more deeply. If however the environment was subject to several changes, the effect is much more drastic and may even lead to the extinction of the population.

A model of population dynamics

A model describing the evolution of a sexual population in a selective environment is presented. The population is composed of individuals each characterized by its phenotype and age. Within the standard Monte Carlo simulation technique, we calculate the time dependence of the average fitness, average adaptation to the environment and the distribution of the phenotypes. We show that the former quantities exhibit damped oscillations, meaning that in the absence of mutations the population is driven by the selection to a homogeneous one. The distribution of the phenotypes, as well as the adaptation is Gaussian at each time step of the process. The role of the maximum age and length of the genetic strings on the dynamics of the population is also discussed.

Monte Carlo simulation of oxygen diffusion in planar model of 123 YBCO Low-temperature regime and effect of trapping barrier

Abstract The asymmetric next-nearest neighbour lattice gas model is used for describing the oxygen-vacancy phase diffusion in the basal plane of 123 YBaCuO superconducting ceramics at low temperatures. A finite value Q for the trapping potential barrier is added. The Monte Carlo technique has been used to obtain the components of the (tracer) diffusion coefficient. The character of its dependence on the coverage is shown to vary, from convex at high temperatures to concave at lower ones. The temperature of the transition depends on the trapping barrier height. The activation energy (obtained from an Arrhenius plot) is seen to be varying as a function of coverage and the trapping barrier, and is not symmetrical with respect to c =0.5 if Q is finite. The same of course is true for the diffusion coefficient. The value of the “final-state-energy” is reported and compared to the ground-state energy. This indicates the likely occurence of fine structure at specific coverage values corresponding to a recently predicted devils staircase distribution of superstructure phases. The time evolution of twins is also reported. A discussion of the limited validity of the model for describing the 123 structural phase diagram is presented.

Model of annual plants dynamics with facilitation and competition

An individual-based model describing the dynamics of one type of annual plants is presented. We use Monte Carlo simulations where each plant has its own history and the interactions among plants are between nearest neighbours. The character of the interaction (positive or negative) depends on local conditions. The plants compete for two external resources-water and light. The amount of water and/or light a plant receives depends on the external factor but also on local arrangement. Survival, growth and seed production of plants are determined by how well their demands for the resources are met. The survival and seeds production tests have a probabilistic character, which makes the dynamics more realistic than by using a deterministic approach. There is a non-linear coupling between the external supplies. Water evaporates from the soil at a rate depending on constant evaporation rate, local conditions and the amount of light. We examine the dynamics of the plant population along two environmental gradients, allowing also for surplus of water and/or light. We show that the largest number of plants is when the demands for both resources are equal to the supplies. We estimate also the role of evaporation and we find that it depends on the situation. It could be negative, but sometimes it has a positive character. We show that the link between the type of interaction (positive or negative) and external conditions has a complex character. In general in favourable environment plants have a stronger tendency for competitive interactions, leading to mostly isolated plants. When the conditions are getting more difficult, cooperation becomes the dominant type of interactions and the plants grow in clusters. The type of plants-sun-loving or shade tolerating, plays also an important role.

Statistical physics model of an evolving population

There are many possible approaches by a theoretical physicist to problems of biological evolution. Some focus on physically interesting features, like the self-organized criticality (P. Bak, K. Sneppen, Phys. Rev. Lett 71 (1993); N. Vadewalle, M. Ausloos, Physica D 90 (1996) 262). Others put on more effort taking into account factors considered by biologists to be important in determining one or another aspect of biological evolution (D. Derrida, P.G. Higgs, J. Phys. A 24 (1991) L985; I. Mroz, A. Pekalski, K. Sznajd-Weron, Phys. Rev. Lett. 76 (1996) 3025; A. Pekalski, Physica A 265 (1999) 255). The intrinsic complexity of the problem enforces nevertheless drastic simplifications. Certain consolation may come from the fact that the mathematical models used by biologists themselves are quite often even more “coarse grained”.

A simple model of colonization

Presented is a simple model, based on computer simulations, of a population colonizing an adjacent habitat with different living conditions. Members of the population are characterized (apart from their positions on the lattice) by a single feature—a qualitative trait (phenotype). Its agreement with the optimum, determining the external conditions, relates to the survival probability. The process is influenced by three factors—selection pressure, fecundity, gene flow and mutation rate. The last is less important and only a combination of the first three factors determines whether an expansion is successful or not.

Evolution of populations in a changing environment

A model of a system of populations (metapopulation) is presented in which a population is characterized by a mean phenotype (or equivalently a continuous character) averaged over all individuals belonging to the population. The populations interact by exchanging genetic information (gene flow) and competing in the natural selection. We show, via Monte Carlo simulations, that the extinction rate of the populations depends crucially on two factors – interactions between the populations and the changes of the external conditions. Apart from big cataclysms, increased rate of extinctions in the system of interacting populations may be also provoked by small, but correlated, changes in the environment. As in the case of a constant habitat, which leads to a steady extinction rate, there are only three possible scenarios for the fate of the metapopulation.

Evolution under stabilizing selection through gene flow

A simple dynamic model is constructed to describe evolutionary changes in populations characterized by a single, quantitative character. The populations interact via gene flow under stabilizing selection. Using standard Monte Carlo simulations we find, in agreement with biological data, that reunion of dispersed populations will lead to one of the three different scenarios, total union of the genetic pools, their partial mixing or complete isolation of the populations. Spatial gradients of the average value of the character were found. The biologically most interesting case, that of partial exchange of the genetic pools, shows some aspects of critical behavior.