Old models are the models previously featured on the old Physics of Risk website (which is still available at http://mokslasplius.itpa.lt/rizikos-fizika (note that all of its content is available only in Lithuanian)). Note that old models might not necessarily be identical to the corresponding models on the old website, they might be a completely new pieces of software, yet considering the topic tackled on the old website.
Belousov-Zhabotinsky reaction 1 is a chemical reaction, or more precisely a reaction family, known for exhibiting temporal and spatial oscillations.
This reaction is one of the classical examples of the natural non-linear oscillations. Another prominent example is the previously analyzed prey-predator interactions in the ecosystem. Interestingly enough despite being of a very different nature both of these example can be modeled using Lotka-Volterra equations.
Here, on the Physics of Risk, we frequently talk about two essential statistical features of the time series – probability and spectral densities. The probability density function should well known to our readers – it is related to the distribution of time series values. On the Physics of Risk we have also a Lithuanian-only article on this topic (see it here). So the time has come to discuss the power spectral density. Continue reading “Power spectral density (part 1)” »
Previously on Physics of Risk we have wrote about Lotka-Volterra equations. At that time we didn’t provide an interactive applet with the text. Only recently we have updated the text and provided an interactive Wolfram CDF applet. This applet was replaced by HTML5 app, yet it is still available for download.
In classical physics differential equations is the main tool to mathematically describe dynamical systems. Having obtained the mathematical descriptions of the system we should be able to predict the evolution of the system. It is noted that the evolution of the classical systems is pretty trivial – no matter what the initial condition is the system will “find” the stable state. Usually dissipative forces (such as frictions) are to be blamed for this. Though not all systems are so simple… Continue reading “Randomly generated strange attractors” »
The simplest ecological system can be constructed from the two interacting species, ex. prey and predator. This kind of system is very interesting in the terms of Physics of Risk primarily because it is nonlinear 1, and due to being real life example of competition (conflict). Also there are few known simple models for the prey-predator interaction. Among them there are both macroscopic, Lotka-Volterra equations, and microscopic, agent-based, models. In this text we continue the previous discussion by considering the agent-based model. Continue reading “Agent based prey-predator model” »
The simplest ecological system can be constructed from the two interacting species, ex. prey and predator. This kind of system is very interesting in the terms of Physics of Risk primarily because it is nonlinear 1, and due to being real life example of competition (conflict). Also there are few known simple models for the prey-predator interaction. Among them there are both macroscopic, Lotka-Volterra equations, and microscopic, agent-based, models. We will start our discussion from the macroscopic Lotka-Volterra model. Continue reading “Lotka-Volterra equations” »
In mathematics and computation theory there are a class of cellular automatons which are known as elementary automatons. This class of cellular automatons is restricted to the one dimensional grid (in the figures below the second dimension, ordinate (vertical) axis, is time) with cells either on or off. Another important simplification is that the actual state of the cell at given time, , depends only on the previous state of the same cell and the previous states of its immediate neighbors, i.e. on . Due to these restrictions and simplifications, generally speaking cellular automatons might evolve in the infinite dimensions, have infinite neighborhoods and have limitless number of possible cell states, these cellular automatons appear to be very simple, though as we show bellow they can replicate very complex and even chaotic behavior. Continue reading “Wolfram’s elementary automatons” »
Newton-Raphson, sometimes just Newton or Newton-Fourier, method is an approximate method in mathematical analysis for finding local roots of very complex functions (such as polynomials with large powers). Recall that root of the function is defined as a solution of . The essence of this method is to linearize function at the guessing point. The point where linearized function passes the abscissa axis is assumed to be a more precise estimate of the actual root. Continue reading “Newton-Raphson method” »