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 ${\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3,\dots$ 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, $\alpha =\mathrm{\Delta }\ell /L_{\max }$. As expected, the probability of forward movement is $1/2$ when $\alpha \to 0$. However, when $\alpha$ is finite, this probability is less than $1/2$ with a minimum value when $\alpha =1$. 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 $0<\alpha ⩽1$ 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 $0<\alpha ⩽1$ 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.

我们提出了一个空模型，将其与生物学数据进行比较，以测试在具有不同几何形状和边界条件的有限空间中的间歇运动期间，站点之间运动的固有持久性。我们描述了停止点序列的时空特性${\mathbf{[R}}_{\mathrm{1个}}，{\mathbf{[R}}_2，{\mathbf{[R}}_3，\dots$在有限空间内被随机游走者访问。不考虑停止点之间的路径，仅考虑位移。由于停止点之间的位移没有内在的相关性，因此它们之间的移动没有​​内在的持久性。因此，这表示一个空模型，当由于有限空间而产生外部偏差时，可以根据该空模型比较经验数据以求停点之间运动的方向持久性。这种比较是检验关于在各种生物中插入间歇性运动的止挡功能的假设的必要的第一步。我们调查了向前移动的概率，定义为小于$90°$ 两个连续位移向量之间的距离，取决于随机助步器可以执行的停靠点之间最大位移与系统尺寸之间的比率， $\alpha =\mathrm{\Delta }\ell /{\mathrm{大号}}_{\mathrm{最高}}$。如预期的那样，向前移动的概率为$\mathrm{1个}/2$ 什么时候 $\alpha \to 0$。但是，当$\alpha$ 是有限的，此概率小于 $\mathrm{1个}/2$ 最小值时 $\alpha =\mathrm{1个}$。对于某些边界条件，最小值介于$\mathrm{1个}/3$ 和 $\mathrm{1个}/4$在1D模式下，甚至可以在2D模式下更低。精确计算所有值在1D中向前移动的概率$0<\alpha ⩽\mathrm{1个}$对于几个边界条件。对于具有一个边界条件的圆形和正方形有界区域，以2D形式进行向前运动概率的分析计算。所有值的数值结果$0<\alpha ⩽\mathrm{1个}$给出了几种边界条件。还考虑了矩形和椭圆的情况，并提供了向前移动概率与纵横比的依赖关系的近似模型。最后，提出了一些实用的观点，说明如何将这些结果用于二维有界空间内动物运动的经验分析。