Pulsed Interaction Signals as a Route to Biological Pattern Formation

We identify a mechanism for biological spatial pattern formation arising when the signals that mediate interactions between individuals in a population have pulsed character. Our general population-signal framework shows that while for a slow signal-dynamics limit no pattern formation is observed for any values of the model parameters, for a fast limit, on the contrary, pattern formation can occur. Furthermore, at these limits, our framework reduces, respectively, to reaction-diffusion and spatially nonlocal models, thus bridging these approaches.

Figure 1

Schematic representation of the timescales of the signal $\varphi (x,t)$ in the fast signal-dynamics limit (${\tau }_{\varphi }\to 0$). Signal response to two signal release pulses is shown for moderately fast signals. $\varphi$ remains mainly localized in time within the pulse duration $\overline{\delta }$ (dashed rectangle). The rising and decaying parts of the signal are indicated by the two short segments close to the label ${\tau }_{\varphi }$. The vertical dotted line indicates an intermediate time at which signal intensity attains the steady $\mathcal{G}$ profile. In the plot, the lapse between pulses is set to ${\tau }_R$ (the mean interevent time) and the field $\varphi$ spreads according to Eq. (7) [with $\nu =4$ and $\mu =1$ as in Fig. 3].

Figure 2

(a) Profile $\mathcal{G}(x)$ from Eq. (8) and several $\nu$. (b) Corresponding growth rates $\lambda (k)$ of perturbations to the homogeneous solution, as a function of perturbation wave number $k$. Parameters are $D_{\rho }=0.01$, $\mu =1$, $\overline{\epsilon }=r=1$, $D_{\varphi }=\gamma =1$, and $I_0={10}^2$.

Figure 3

Temporal evolution of signal and population fields for the slow (a) and fast (b) signal regimes. The system is a line of length 100 with periodic boundary conditions. Colors indicate field intensity $\varphi (x,t)$ (upper panels), and population density relative to the (fast-signal limit) homogeneous state, $\rho (x,t)/{\rho }_0$ (bottom panels). In the upper panels crosses indicate release instants and positions (not all releases are captured by the finite resolution of the heat map). Data from numerical integration of Eqs. (1)–(3), using a Euler scheme with $\delta t={10}^{-6}$ and $\delta x=1.0$. The dynamics is given by Eq. (7) with $D_{\rho }={10}^{-2}$, $r={\tau }_{\rho }=1$, $\alpha ={10}^2$, $\overline{\delta }={10}^{-2}$, and $\epsilon =10$. Signal dynamics is set by $\mu =1$ and $\nu =4$, scaled by ${\tau }_{\varphi }$, in such a way that $D_{\varphi }/4=\gamma =1/{\tau }_{\varphi }$ and $I_0={10}^2/{\tau }_{\varphi }$. The slow and fast regimes were obtained setting ${\tau }_{\varphi }=1$ and ${\tau }_{\varphi }=1/500$, respectively. Mean interpulse interval is in both cases ${\tau }_R=1/(N(t)\alpha )\simeq {10}^{-1}$ (for a schematic closeup of the signal field in the fast-signal regime see Fig. 1).