Figure 1.
Simulation output for systems of territorial central place foragers.
The dependence of the saturation mean square displacement (saturation MSD) (resp.
) of the dimensionless territory border position
(
) on the dimensionless parameters
and
(
) from stochastic simulation output. The notation
denotes an ensemble average over stochastic simulations. The border movement is non-dimensionalised by dividing by
, the average distance between central places of adjacent territories. Panel (a) shows output from 1D simulations and panel (b) from 2D simulations. The best-fit lines for the 2D plots are
for
,
for
,
for
,
for
,
for
, and
for
.
Figure 2.
Diagram of the reduced analytic 1D model of territorial dynamics.
The CPs are fixed at positions ,
and
(left to right). The territory borders are intrinsically subdiffusive and have positions
and
. Each animal moves diffusively with a constant drift towards the CP and constrained to move between the two territory borders to its immediate right and left. The position of the animal studied in the main text is denoted by
. The animals at
and
are drawn purely for illustrative purposes. In the Results section,
is assumed to be at 0 and
.
Table 1.
Notation glossary.
Figure 3.
Comparison of the many-bodied simulation system and the reduced analytic model.
Saturation marginal probability distributions from simulations of systems of territorial central place foragers are overlaid on the same distributions (equations 8 and 9) from the reduced analytic models. Panels (a–d) compare the two distributions for the 1D system. Dashed lines denote the simulation output and solid lines the analytic approximation. The animal's central place (CP) is at position 0, whereas CPs of conspecifics exist at positions −1 and 1. The distribution decays to 0 at the conspecific CPs, where the animal cannot tread. The values used were (a) ,
, (b)
,
(c)
,
and (d)
,
. Panels (e–g) compare the two distributions for the 2D system. The black contours show the deciles (i.e. 10%, 20%, 30% etc.) of the height of the probability distribution for the simulation system. The red contours show the same quantities for the analytic approximation. The values used were (e)
,
, (f)
,
, (g)
,
and (h)
,
. As we increase
or
, the effect of the adiabatic approximation becomes more apparent, since each red contour is further away from the respective black contour. This is due to the fluctuations of the territory border being more pronounced for higher
or
.
Figure 4.
Panel (a) shows how the radius of the normalised (by dividing by the mean distance between CPs) 95% minimum convex polygon home range depends on
and
in the 2D analytic model. The various shapes (circles, squares, crosses etc.) show the exact values and the solid lines show the least-squares best-fit sigmoidal curves. Notice that whenever
, a buffer zone appears between adjacent territories. The proportion of exclusive area
scales with mass [13] so this value is plotted in panel (b) against the dimensionless parameter
for various
. Again, solid lines are derived from the best-fit sigmoidal curves whilst the points denoted by various shapes show exact values.
Figure 5.
Comparison with a previous model of territory formation.
The parameter from the reaction-diffusion model introduced in [9] (see also main text) is compared with the parameters
and
from the 1D analytic model introduced here. Panel (a) shows the
-value that gives the best-fit animal marginal distribution curve for each given value of
and
. The insets compare the probability distributions for particular values of
and
, where the solid lines represent our model and the dashed lines the reaction-diffusion model. The values used are (i)
,
, (ii)
,
, (iii)
,
, (iv)
,
. Panel (b) shows the best fit
-value for a given
. The
-values used for the insets are (i)
, (ii)
, (iii)
. Low values of
always give a better fit to a given marginal distribution from the reaction-diffusion model than higher values and do not affect the value of
that gives the best fit. Therefore we set
when performing the fitting for panel (b). Low values of
and
together with high values of
tend to give rise to good fits, but outside this range the two models show quite different results.