Andrzej Pekalski

Percolation and jamming in random sequential adsorption of linear segments on a square lattice

We present the results of a study of random sequential adsorption of linear segments (needles) on sites of a square lattice. We show that the percolation threshold is a nonmonotonic function of the length of the adsorbed needle, showing a minimum for a certain length of the needles, while the jamming threshold decreases to a constant with a power law. The ratio of the two thresholds is also nonmonotonic and it remains constant only in a restricted range of the needles length. We determine the values of the correlation length exponent for percolation, jamming, and their ratio.

Formation of Liesegang patterns: Simulations using a kinetic Ising model

A kinetic Ising model description of Liesegang phenomena is studied using Monte Carlo simulations. The model takes into account thermal fluctuations, contains noise in the chemical reactions, and its control parameters are experimentally accessible. We find that noisy, irregular precipitation takes place in dimension d=2 while, depending on the values of the control parameters, either irregular patterns or precipitation bands satisfying the regular spacing law emerge in d=3.

Model of macroeconomic evolution in stable regionally dependent economic fields

We develop a model for the evolution of economic entities within a geographical type of framework. On a square symmetry lattice made of three (economic) regions, firms, described by a scalar fitness, are allowed to move, adapt, merge or create spin-offs under predetermined rules, in a space- and time-dependent economic environment. We only consider here one timely variation of the “external economic field condition”. For the firm fitness evolution, we take into account a constraint such that the disappearance of a firm modifies the fitness of nearest-neighboring ones, as in Bak–Sneppen population fitness evolution model. The concentration of firms, the averaged fitness, the regional distribution of firms, and fitness for different time moments, the number of collapsed, merged and new firms as a function of time have been recorded and are discussed. Also the asymptotic values of the number of firms present in the three regions together with their average fitness, as well as the number of respective births and collapses in the three regions are examined. It appears that a sort of critical selection pressure exists. A power-law dependence, signature of self-critical organization is seen in the birth and collapse asymptotic values for a high selection pressure only. A lack of self-organization is also seen at region borders.

Lattice gas model of gradual evolution

A simple dynamical model is presented for describing the gradual evolution of a variable number of species. The system is studied through Monte Carlo simulations using a lattice gas formalism. Each species is characterized by a single, scalar parameter (“adaptation”) which is changed, within limits depending on the adaptation itself, at each time step. There are two independent mechanisms for removing a species from the system and one for creating a new species. We find that, regardless of the initial concentration of species, the system always reaches the same final state, characterized by the same concentration and the same average adaptation. The system is not homogeneous and contains species with different values of adaptation. The better adapted ones are found to form more symmetrical spatial patterns. Behavior similar to the one determined in the present model has been found in the evolving ecological and biological systems.

Dynamics of Populations in Extended Systems

Two models of spatially extended population dynamics are investigated. Model A describes a lattice model of evolution of a predator - prey system. We compare four different strategies involving the problems of food resources, existence of cover against predators and birth. Properties of the steady states reached by the predator-prey system are analyzed. Model B concerns an individual-based model of a population which lives in a changing environment. The individuals forming the population are subject to mutations and selection pressure. We show that, depending on the values of the mutation rate and selection, the population may reach either an active phase (it will survive) or an absorbing phase (it will become extinct). The dependence of the mean time to extinction on the rate of mutations will also be discussed. These two problems illustrate the fact that cellular automata or Monte-Carlo simulations, which take completely the spatial fluctuations into account, are very useful tools to study population dynamics.

Monte Carlo investigation of surface self‐diffusion: The role of anisotropic next‐nearest neighbor interactions

The asymmetric next‐nearest neighbor (NNN) Ising model has been studied for describing the self‐diffusion of adsorbate atoms at low temperature, furthermore adding a relevant ingredient in such a regime, i.e., a finite value Q for the trapping potential barrier. The Monte Carlo technique leads to the components of the (tracer) diffusion coefficient. An interesting feature is seen: a ‘‘V–Λ transition’’ occurs at half‐coverage (c=0.5) when the temperature is lowered. The activation energy (obtained from an Arrhenius plot) is seen to be varying as a function of coverage, and is not symmetrical with respect to c=0.5. The anisotropy of the diffusion and the change in its pattern as a function of the temperature and coverage are presented and discussed. Monte Carlo snapshots are shown at various coverage and temperature values.

Species richness in a model with resource gradient

In order to study the dependence of the species richness on heterogeneity of the habitat, we introduce an extended model of annual plants which combines the features of the island model and of gradient heterogeneity resources. First, we consider a native population of plants living on a square lattice of linear size L. After equilibration of this native population, seeds of several different species j = 2, ... , k of annual plants invade the system; they compete among themselves and the native ones. The system is exposed to a one-dimensional water gradient, and each species is characterised by a tolerance to a surplus of water, τ(j). We study the influences of the properties of the gradient of the resource (GR) on the species richness (SR) present in the system. We have shown that the relation between GR and SR is not straightforward and that several cases could be distinguished: For a large class of control parameters, SR increases linearly with GR. However, when the values of the control parameters are such as to create wide inhabitable regions, the relation between SR and GR ceases to have a monotonic character. We have also demonstrated that the average species richness as a function of the resource availability has a hump shape. Our results can be simply explained within our model and are in agreement with several previous field and theoretical works.

COMPARISON OF TWO DESCRIPTIONS OF HETEROGENEITY USED IN MODELLING PLANTS DYNAMICS

We analyse the role played by two different approaches of spatial heterogeneity in theoretical models of annual plants dynamics. The first approach is called quasi-continuous gradient in which one type of resource is changing gradually along the gradient line. In the second one, called the patches approach, part of the habitat is covered by patches and the resource has a different value in each patch. We show that when the spatial heterogeneity of the habitat is small, the two approaches yield the same average number of surviving species, even if a small number of patches is used. In a strong heterogeneity it takes many patches to get similar results as in the gradient case. The difference between the gradient and patchy description of the spatial heterogeneity increases with the number of species present in the system. We have also shown that even when the average number of surviving species is the same, the abundances of species are ordered in a different way, like different species are the dominant ones. The conclusion of this paper is that modelling spatial heterogeneity in a system of plants is not a simple task. Special care is needed when the heterogeneity of the habitat is large, since then depending of the choice of a method, some predictions may differ significantly, making the model non-robust. Therefore the type of theoretical approach must closely match the modelled ecosystem.

Simulation model for dynamics of three types of annual plants

A Monte Carlo type model describing dynamics of three pairs of annual plants living in a homogeneous habitat is presented and discussed. Each plant follows its own history with growing, fecundity and survival chances determined individually as functions of the plant’s condition and environment. The three plants - Valerianella locusta, Mysotis ramosissima and Cerastium semidecandrum differ by the weight of their seeds, which in the model determines the competition preference. Heavier seeds have a better chance for germination from a site containing seeds of different plants. Better colonisers produce more seeds and disperse them over a larger distance. I show that without absolute asymmetry in the impact effects between better competitors and better colonisers and in a spatially and temporarily homogeneous habitat, coexistence of species is possible, however only in a limited time. This is different from statements coming from models using mean-field type methods. I demonstrate also that in a system of two species clustering of plants of the same type are more frequent. From the calculated survival chances of seedlings and adult plants it follows that elimination of plants occur mostly at the early stages of the plants life cycle, which agrees with the field data.I show that this competition/colonisation trade-off model is sufficient to maintain coexistence and I determine the conditions for dominance of one type of plants. I show that the time of extinction of the weaker species goes down with increasing observation time as a power function with the exponent independent of the type of plants.