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Let 0e(t) be the phase (of the angle in radians) in the flashing of the external stimulus at time t (e.g., an attractive male neighbor). The frequency We of the flashing is assume to be given and fixed with day=wo. dt Let A(t) be the phase of a particular firefly under consideration at the same time. If A(t) is ahead of te (t), it will try to slow down. If (A(t) is behind, it will try to speed up. So a model for 1/4 is given by d0 - The size of the external stimulus is measured by the parameter a, and w is the natural frequency of this firefly in the absence of an external stimulus. Let o(t) = Be - A be the relative phase, T = at, and s = we-w. The above equation becomes dt = involving only one parameter 8, which can be positive or negative. Consider the positive case: (a) Plot the phase diagram (of do do VS. b) schematically for three different cases (8 = 0,0 < 1, 8 > 1) and for the range -TT <0Th. (b) Describe the equilibria in the three cases and their stability (using arrows to indicate the direction of increasing 6). (c) Describe what happens to A(t) eventually in the three cases: i) for the very strong stimulus case, & = 0; ii) for the weak stimulus case, s > 1; and iii) for the moderate and realistic case, 0 < s < 1. (We call the case of non-synchronous change in phase a phase drift, and we call the establishment of a steady phase relative to the external phase a phase locking. When phase locking occurs, the fireflies flash at the same frequency.) (d) Derive the condition (an inequality among W, O and We) for phase locking.

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