Random walks with time delays

Project Description

Research project in Probability Calculus.

It is usually expected that time delays cause oscillations in dynamical systems - there might appear cycles around stationary points. Here we study random walks with asymmetric time delays. In our models, the probability of a walker to move to the right or to the left depends on the difference between two state-dependent tness functions evaluated at two different times. We observe a novel behavior - a dependence of the mean position of the walker on time delays. Moreover, the expect of time delays is reversed when one shifts (in a symmetric way) fitness functions. This is a joint effect of both stochasticity and time delays present in the system. Position of a random walker may be interpreted as a frequency of a given strategy in discrete replicator dynamics of evolutionary games. Then our results show that the effect of asymmetric time delays on the equilibrium structure of a population depends not only on time delays but also on details of fitness functions.

Initially I started with Python simulations using numpy and we experimented with different settings. We found a novel effect when using two time delays. I read prior publications when a continuous model was used instead of a discrete one or where there was only one time delay and in both cases the expected value was 0. However, using a discrete model with two time delays we found that the expected value is unequal to 0. At first, it was computer aided and the mean was only an approximation and to get a decent simulation result I had to choose 10⁹ steps, but with Python program it lasted almost 8 hours so I let it run in the night. But then we wanted test for different parameter combinations, so I rewrote the whole simulation program into C++ to get it faster and it was only 45min afterwards. I also wrote it with multiple threads, but in reality it was easier to just run separate terminal applications. I created the diagrams for the pulbication and analysis with pyplot package.

Publication

Łopuszański, Kamil, and Jacek Miękisz. "Random walks with asymmetric time delays." Physical Review E 105.6 (2022): 064131.