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Fig. 6. Analysis of the mathematical model. (A) Worm trajectories calculated using
the model represented by Eqn 2 and Eqn 3 with and without a spatial thermal
gradient. The black arrow indicates the direction of worm movement, the thick
grey arrow indicates the direction of the thermal gradient, and the dotted
line indicates an isotherm. The black line is the calculated trajectory
without the gradient. The grayed trajectory shows the calculated trajectory in
response to the perturbation caused by the gradient. The two labeled points in
the magnified region indicate the extrema of both 
(t) and
. At point x,
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|2 is greater than at
point y, so the worm curves more vigorously at x, leading to
a slight right turn that corrects isothermal alignment. (B) Functional forms
of f(), all of which can generate
calculated trajectories consistent with the movements of real C.
elegans responding to spatiotemporal thermal gradients. The requisite
feature for the qualitative behavior is that
f() is monotonic in
||. The thick line
corresponds to the function
f()=g||2,
the functional form we use in the analytical discussion in Results. (C) Plot
of
as a function of 
and
. The values of
are listed to the right of
their curves. Black dots represent stable fixed points, the white dot
represents an unstable point, and the gray circle switches from unstable to
stable when 
(t)>1. (D)
Plots of the fixed point
as a function of
. When
undergoes small oscillations
about 0, so does .
When approaches 1, the fixed
point is able to move to different branches of the graph, corresponding to the
looped trajectories exhibited by C. elegans responding to steep
superposed temporal gradients.
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