Lotka-Volterra equations

The simplest ecological system can be constructed from the two interacting species, e.g., 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.

Colors of Noise

What is white, pink, "brown" and even black? It is around us every day and usually is very useful. Yet from time to time it annoys us and sometimes we even call police to keep it in check? The answer to this quite complex, as it is oriented towards people with physics background, question is unbelievably simple. Noise possesses all of the aforementioned qualities!

Special cases of the stochastic differential equation reproducing 1/f noise

Considerable part of stochastic models available on Physics of Risk website (ex., Agent based herding model of financial markets or Long-range memory stochastic model of return) are related to the general class of stochastic differential equations derived by our group [1, 2]. The general form of this class is the following stochastic differential equation:

\begin{equation} \mathrm{d} x = \left(\eta - \frac{\lambda}{2} \right)x^{2 \eta -1} \mathrm{d} t + x^\eta \mathrm{d} W . \label{sde} \end{equation}

In our talks at various scientific events and on Physics of Risk itself we frequently say that this equation also encompasses other widely known stochastic processes. Thus further in this text we will show some of the relations between this class and some widely known stochastic processes.