Replicators in Fine-grained Environment: Adaptation and Polymorphism

Selection in a time-periodic environment is modeled via the two-player replicator dynamics. For sufficiently fast environmental changes, this is reduced to a multi-player replicator dynamics in a constant environment. The two-player terms correspond to the time-averaged payoffs, while the three and four-player terms arise from the adaptation of the morphs to their varying environment. Such multi-player (adaptive) terms can induce a stable polymorphism. The establishment of the polymorphism in partnership games [genetic selection] is accompanied by decreasing mean fitness of the population.Comment: 4 pages, 2 figure

Le Chatelier principle in replicator dynamics

The Le Chatelier principle states that physical equilibria are not only stable, but they also resist external perturbations via short-time negative-feedback mechanisms: a perturbation induces processes tending to diminish its results. The principle has deep roots, e.g., in thermodynamics it is closely related to the second law and the positivity of the entropy production. Here we study the applicability of the Le Chatelier principle to evolutionary game theory, i.e., to perturbations of a Nash equilibrium within the replicator dynamics. We show that the principle can be reformulated as a majorization relation. This defines a stability notion that generalizes the concept of evolutionary stability. We determine criteria for a Nash equilibrium to satisfy the Le Chatelier principle and relate them to mutualistic interactions (game-theoretical anticoordination) showing in which sense mutualistic replicators can be more stable than (say) competing ones. There are globally stable Nash equilibria, where the Le Chatelier principle is violated even locally: in contrast to the thermodynamic equilibrium a Nash equilibrium can amplify small perturbations, though both this type of equilibria satisfy the detailed balance condition.Comment: 12 pages, 3 figure

Modelling carbon dynamics from urban land conversion: fundamental model of city in relation to a local carbon cycle

<p>Abstract</p> <p>Background</p> <p>The main task is to estimate the qualitative and quantitative contribution of urban territories and precisely of the process of urbanization to the Global Carbon Cycle (GCC). Note that, on the contrary to many investigations that have considered direct anthropogenic emission of CO₂(urbanized territories produce ca. 96–98% of it), we are interested in more subtle, and up until the present time, weaker processes associated with the conversion of the surrounding natural ecosystems and landscapes into urban lands. Such conversion inevitably takes place when cities are sprawling and additional "natural" lands are becoming "urbanized".</p> <p>Results</p> <p>In order to fulfil this task, we first develop a fundamental model of urban space, since the type of land cover within a city makes a difference for a local carbon cycle. Hence, a city is sub-divided by built-up, „green" (parks, etc.) and informal settlements (<it>favelas</it>) fractions. Another aspect is a sub-division of the additional two regions, which makes the total number reaching eight regions, while the UN divides the world by six. Next, the basic model of the local carbon cycle for urbanized territories is built. We consider two processes: carbon emissions as a result of conversion of natural lands caused by urbanization; and the transformation of carbon flows by "urbanized" ecosystems; when carbon, accumulated by urban vegetation, is exported to the neighbouring territories. The total carbon flow in the model depends, in general, on two groups of parameters. The first includes the NPP, and the sum of living biomass and dead organic matter of ecosystems involved in the process of urbanization, and namely them we calculate here, using a new more realistic approach and taking into account the difference in regional cities' evolution.</p> <p>Conclusion</p> <p>There is also another group of parameters, dealing with the areas of urban territories, and their annual increments. A method of dynamic forecasting of these parameters, based on the statistical regression model, was already suggested; nevertheless we shall further develop a new technique based on one idea to use the gamma-distribution. This will allow us to calculate the total carbon balance and to show how urbanization shifts it.</p