Segregation and pursuit waves in activator-inhibitor systems

We investigate the effects of cross-diffusion on propagating waves in an activator-inhibitor system. The model consists of a piecewise linear approximation of FitzHugh-Nagumo kinetics and a cross-diffusion term for either the activator or the inhibitor. We obtain exact analytic solutions for traveling fronts and solitary pulses and discuss the corresponding speed diagrams. A detailed comparison with the corresponding Rinzel-Keller model for the usually studied case of self-diffusion is performed.

Figure 1

Plot of segregation front profiles for the activator (solid curve) and the inhibitor (dashed curve), computed from Eqs. (14), for $\epsilon =0.1$, $b=3$, and $a=0.2$. Equation (15) yields the front speed $c=0.137$, and fronts propagate to the right.

Figure 2

Plot of the propagation speed $c=c(a,b)$ of segregation fronts as a function of $a$ for various positive values of $b$ and for two ratios of the time scales $\epsilon$, namely, $\epsilon =0.1$ (dashed line) and $\epsilon =0.2$ (solid line). The speed curves are obtained from Eq. (15).

Figure 3

Plot of segregation pulse profiles for the activator (solid curve) and the inhibitor (dashed curve), computed from Eq. (18), for $\epsilon =0.5$, $b=4$, and $a=0.375$. Equation (22) yields the pulse speed $c=0.189$, and pulses propagate to the right.

Figure 4

Plot of the propagation speed $c=c(a,b)$ of a pulse as a function of $a$ for various positive values of $b$ and for a ratio of the time scales of $\epsilon =0.5$. The speed curves are obtained from Eq. (22).