Firefly Blinking

The flash of a firefly ($\theta =0$) in response to mimic an external stimulus flashing ($S$ = 0) is modeled by

$$\dot{\varphi }=\omega -A\sin (\varphi )\text{with}\dot{\varphi }=\dot{S}-\dot{\theta }$$

where $\varphi$ is the phase difference between the two flashes. The model shows how $\theta$ changes to converge itself to the stimulus $S$. By nondimensionalization the model can be simplified to

$${\varphi }^{\prime }=\mu -\sin \varphi$$

with exclusively $\mu$ imposing bifurcation on the system. The analysis concludes as follows

• When a fixed point ${\varphi }^{\text{*}}=0$ exists, the phase difference is able to be mitigated and the two frequencies $S,\theta$ arrange into phase
• With fixed ${\varphi }^{\text{*}}>0$ a constant phase difference will develop and the stimuli will be phase locked.
• With no ${\varphi }^{\text{*}}$ the following frequency won’t be able to entrain

Entrainment Range

Entrainment can be fixated to a certain range of the constant parameters in the system, namely $\omega$ = the natural frequency of the firefly, $A$ = the firefly’s ability to spontaneously change frequency (the higher, the faster it can do such), $S$ the frequency of the external stimulus. The range can be found by looking up the range $a\le \mu \le b$ in which fixed points exist. The bounding parameters can be substituted with the variables of the system and a certain value of i.e. $S$ can be fixated, for which entrainment can occur in the system.