Random walks with asymmetric time delays

TL;DR

It is observed that by shifting appropriately fitness functions the mean position of a walker depends on time delays, which is a joint effect of both stochasticity and time delays present in the system.

Abstract

We studied simple random-walk models with asymmetric time delays. Stochastic simulations were performed for hyperbolic-tangent fitness functions and to obtain analytical results we approximated them by step functions. A novel behavior has been observed. Namely, the mean position of a walker depends on time delays. This is a joint effect of both stochasticity and time delays present in the system. We also observed that by shifting appropriately fitness functions we may reverse the effect of time delays - the mean position of the walker changes the sign.

Figures (7)

• Figure 1: Phase portrait in the phase space $(x_{t-\tau}, x_{t})$ for the deterministic dynamics. Blue arrows show directions in which the system can move. The state space $\Omega$ for $\tau=3$ is shown in green, the invariant set $T$ is shown in red.
• Figure 2: Hyperbolic tangent fitness functions; $d=-20$ on the left and $d=20$ on the right, $\rho=0.3.$
• Figure 3: Histograms for the hyperbolic-tangent models with parameters $\tau =20$ on the left and right, $\tau_A=0$ in the middle. $d=-20$ on the left and $d=20$ on the right. $\rho=0.99$ for all simulations. Simulation were run for $10^8$ steps.
• Figure 4: The dependence of the expected value of the walker's position $\mu$ on the time delay $\tau$ with hyperbolic-tangent fitness functions with $\rho=0.99$, $\tau=0, 2, 4, \ldots 68$. $d=-20$ on the left and $d=20$ on the right. The equation of the blue line (linear regression) is $0.14\tau + 0.12$ on the left and $-0.13\tau - 0.01$ on the right.
• Figure 5: The dependence of the expected value of the walker's position $\mu$ on $d$ with hyperbolic-tangent fitness functions with $\rho=0.99$, $\tau=20$, $d=-40, \ldots, -2, 0, 2, \ldots, 40$.