The von Mises and Fisher distributions are spherical analogues to theNormal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth.
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Figure 4.
Comparison between simulations of the stochastic velocity-jump model and the fundamental solution of (6), equation (9) in two dimensions. Results shown at times
Figure 5.
Comparison between simulations of the stochastic velocity-jump model and the fundamental solution of (6) in three dimensions. All plots are generated at time
Figure 6.
Simulations of green sea turtle navigation towards Ascension island: we refer to [31] for full details. We consider (a) weak navigators and (b) strong navigators, showing in each case (top row) the individual-based model and (bottom row) the macroscopic model. For the IBM turtle positions (white circles) are shown at the indicated times, superimposed on ocean currents as illustrated by its direction (arrows) and magnitude (arrow length/density map), while for the macroscopic model (6) the population density is indicated by the density map, where the scale indicates the number/km
Figure 7.
Population density distributions describing the different responses of a population to linear landscape features such as seismic lines. In each case we solve (6), where
Figure 8.
Simulations of brain tumour growth using real patient data: see [34] for details. These are test cases to show the effect of changing the concentration parameter. (a) shows the fractional anisotropy for a two-dimensional axial slice of a real patient brain. Yellow indicates high fractional anisotropy, and thus high alignment, while blue indicates isotropic tissue. The initial condition for the brain tumour simulation is indicated by a black dot. (b)-(e) show two artificial tumours generated using real patient DTI data for two different values of
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