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, ||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.