Random-walk model to study cycles emerging from the exploration-exploitation trade-off

We present a model for a random walk with memory, phenomenologically inspired in a biological system. The walker has the capacity to remember the time of the last visit to each site and the step taken from there. This memory affects the behavior of the walker each time it reaches an already visited site modulating the probability of repeating previous moves. This probability increases with the time elapsed from the last visit. A biological analog of the walker is a frugivore, with the lattice sites representing plants. The memory effect can be associated with the time needed by plants to recover its fruit load. We propose two different strategies, conservative and explorative, as well as intermediate cases, leading to nonintuitive interesting results, such as the emergence of cycles.

Figure 1

Mean square displacement vs time for probability $\rho =0$ (black), $\rho =1$ (orange), and intermediate values and corresponding to the recovery time of the plants $\tau =20$. Simulations were performed on a square lattice of $5000\times 5000$ sites, ${10}^5$ time steps, and ${10}^3$ realizations.

Figure 2

Contour plot of the fraction of realizations that end in a cycle, as a function of parameters $\rho$ and $\tau$. Simulations were performed on a square lattice of $5000\times 5000$ sites, ${10}^5$ time steps, and ${10}^4$ realizations. The gray region corresponds to realizations that do not end in cycles due to finite observation time.

Figure 3

Contour plot of the mean cycle length as a function of $\rho$ and $\tau$. Simulations were performed on a square lattice of $5000\times 5000$ sites, ${10}^5$ time steps, and ${10}^4$ realizations.

Figure 4

Contour plot of the mean transient length as a function of parameters $\rho$ and $\tau$. The color scale is logarithmic. Simulations were performed on a square lattice of $5000\times 5000$ sites and ${10}^6$ time steps.

Figure 5

Diffusion coefficient of the $\rho =0$ case (slope of the average MSD curves for each value of $\tau$). Forty uniformly distributed values of $\tau$ were considered between 5 and 200.

Figure 6

Mean square displacement vs time considering a smooth function for the probability to repeat the step taken in the previous visit. Here $\omega =10$ (black), $\omega =2$ (red), and $\omega =0.5$ (blue) and $\tau =20$. Simulations were performed on a square lattice of $5000\times 5000$ sites, ${10}^5$ time steps, and ${10}^3$ realizations. The inset shows the functional expression and shape of the probability distribution.