Nonlinear model of the firefly flash

Abstract

A low dimensional nonlinear model based on the basic lighting mechanism of a firefly is proposed (Saikia and Bora in Nonlinear model of the firefly flash. http://export.arxiv.org/pdf/2002.01183). The basic assumption is that the firefly lighting cycle can be thought to be a nonlinear oscillator with a robust periodic cycle. We base our hypothesis on the well-known light producing reactions involving enzymes, common to many insect species, including the fireflies. We compare our numerical findings with the available experimental results which correctly predicts the reaction rates of the underlying chemical reactions. Toward the end, a time-delay effect is introduced for possible explanation of appearance of multiple-peak light pulses, especially when the ambient temperature becomes low.

Appendix

1.1 Approximation of a DDE system

As we know that a dynamical system with time delay is basically an infinite dimensional system, the delay system can be reduced to a set of N ordinary differential equations, where N is a large positive integer [26].

Consider our DDE with two delays $\tau _{1,2}$,

$$\begin{aligned} \frac{\mathrm{d}x}{\mathrm{d}t}= & {} f[x(t),x(t-\tau _{1}),x(t-\tau _{2}),y(t)], \end{aligned}$$

(69)

$$\begin{aligned} \frac{\mathrm{d}y}{\mathrm{d}t}= & {} g[x(t),y(t)], \end{aligned}$$

(70)

assuming $\tau _{1}>\tau _{2}$ numerically. We now divide the larger of the delay into N different discrete divisions, each of size ${\varDelta }t$ such that ${\varDelta }t=\tau _{1}/N$ or equivalently $N=\tau _{1}/{\varDelta }t$. As $N\gg 1$, ${\varDelta }t$ is very small so that we have,

$$\begin{aligned}&x(t+{\varDelta }t) \nonumber \\&\quad \simeq x(t)+{\varDelta }t \nonumber \\&\qquad f[x(t),x(t-N{\varDelta }t),x(t-M{\varDelta }t),y(t)], \end{aligned}$$

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where $M\,(<N)$ is an integer such that $M=\tau _{2}/{\varDelta }t$ (please see explanation about M at the end of this section). Denoting now the values of x(t) at a discrete time $t_{j}$ as $x_{j}(t)$, we have,

$$\begin{aligned} x_{j+1}(t)=x_{j}(t-{\varDelta }t), \end{aligned}$$

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which can be written as

$$\begin{aligned} x_{j+N}(t)= & {} x_{j}(t-N{\varDelta }t),\nonumber \\ \Rightarrow x_{N}(t)= & {} x_{0}(t-N{\varDelta }t), \end{aligned}$$

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so that we can have now

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}x_{0}(t)=f[x_{0}(t),x_{N}(t),x_{M}(t),y(t)]. \end{aligned}$$

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Following this prescription, we can now write the original DDE system as,

$$\begin{aligned} \frac{\mathrm{d}x_{0}}{\mathrm{d}t}= & {} f(x_{0},x_{N},x_{M},y), \end{aligned}$$

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$$\begin{aligned} \frac{\mathrm{d}x_{j}}{\mathrm{d}t}= & {} \frac{N}{2\tau _{1}}(x_{j-1}-x_{j+1}),\quad 1\le j\le N-1, \end{aligned}$$

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$$\begin{aligned} \frac{\mathrm{d}x_{N}}{\mathrm{d}t}= & {} \frac{N}{\tau _{1}}(x_{N-1}-x_{N}), \end{aligned}$$

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$$\begin{aligned} \frac{\mathrm{d}y}{\mathrm{d}t}= & {} g(x_{0},y), \end{aligned}$$

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for $N\gg 1$, which can now be solved with any standard method. For $N\rightarrow \infty$, the above system will reduce to the original DDEs.

One important point to be noted here about the value of N. While any large positive integer as a value of N will do for a single delay DDE system, for a system with double delays such as in our case, it is important that the difference between the two delays $(\tau _{1}-\tau _{2})$, assuming $\tau _{1}>\tau _{2}$, is exactly divisible by ${\varDelta }\text {t}$ for the chosen value of N. This is required as we have now divided the entire time interval $[0,\tau _{1}]$ into discrete divisions, the smaller of the delays, which in this case is $\tau _{2}$ falls exactly at one node, otherwise the discrete set of equations will not maintain the analytical difference between the delays for small values of N. In our chosen set of values of $\tau _{1}=2.53$ and $\tau _{2}=2.3$, N should be a multiple of 11. If the difference between the delays is not exactly divisible by ${\varDelta }t$, N should be large enough to make the this difference equivalent to the analytical value, otherwise even a small value of N gives quite good results.