Stability of Spiky Solutions in a Competition Model with Cross-Diffusion

Source abstract

We consider the Shigesada–Kawasaki–Teramoto model of species segregation in the limit of high cross-diffusion rate of one species, and small diffusion rate of another. Recently, steady states in the shape of an inverted spike were constructed in this limit in one dimension [Y. Lou, W.-M. Ni, and S. Yotsutani, Discrete Contin. Dyn. Syst., 10 (2004), pp. 435–458; Y. Wu and Q. Xu, Discrete Contin. Dyn. Syst., 29 (2011), pp. 367–385]. In this paper we consider the stability properties of such spiky states. We show that K symmetric spikes are stable if the domain length is sufficiently large. More precisely, we derive a sequence of thresholds $0=L_1<L_2<L_3<L_4<L_5\ldots$ such that K spikes on the domain of size $2KL$ are stable if and only if $L>L_K$. When $K=2$, the instability of a small eigenvalue is triggered first, resulting in a very slow drift of the two spikes, with eventual absorption of one by the other. When $K\geq3$, the primary instability is due to a large eigenvalue, resulting in a quick death of one or more spikes. We also extend the construction of one-dimensional steady states to a radially symmetric two-dimensional spike at the center of a disk. In one dimension, hypergeometric functions are utilized to study the large eigenvalues; thresholds for small eigenvalues are derived indirectly by classifying the bifurcations of asymmetric patterns. Full numerical simulations in one and two dimensions are performed to confirm the asymptotic results and to explore some of the dynamical scenarios away from the equilibrium state.