Animal intermittent locomotion: A null model for the probability of moving forward in bounded space

We present a null model to be compared with biological data to test for intrinsic persistence in movement between stops during intermittent locomotion in bounded space with different geometries and boundary conditions. We describe spatio-temporal properties of the sequence of stopping points r₁,r₂,r₃,… visited by a Random Walker within a bounded space. The path between stopping points is not considered, only the displacement. Since there are no intrinsic correlations in the displacements between stopping points, there is no intrinsic persistence in the movement between them. Hence, this represents a null-model against which to compare empirical data for directional persistence in the movement between stopping points when there is external bias due to the bounded space. This comparison is a necessary first step in testing hypotheses about the function of the stops that punctuate intermittent locomotion in diverse organisms. We investigate the probability of forward movement, defined as a deviation of less than 90° between two successive displacement vectors, as a function of the ratio between the largest displacement between stops that could be performed by the random walker and the system size, α=Δℓ/L_max. As expected, the probability of forward movement is 1/2 when α→0. However, when α is finite, this probability is less than 1/2 with a minimum value when α=1. For certain boundary conditions, the minimum value is between 1/3 and 1/4 in 1D while it can be even lower in 2D. The probability of forward movement in 1D is calculated exactly for all values 0<α⩽1 for several boundary conditions. Analytical calculations for the probability of forward movement are performed in 2D for circular and square bounded regions with one boundary condition. Numerical results for all values 0<α⩽1 are presented for several boundary conditions. The cases of rectangle and ellipse are also considered and an approximate model of the dependence of the forward movement probability on the aspect ratio is provided. Finally, some practical points are presented on how these results can be utilised in the empirical analysis of animal movement in two-dimensional bounded space.