Rotating Spirals in segregated reaction-diffusion systems

TL;DR

The paper addresses rotating spirals in the strong competition limit of planar multi-species reaction-diffusion systems, establishing a half-plane reduction and a codimension-(K−1) condition on segregated boundary traces that guarantees the existence of a rotating-spiral solution with a unique singular point at the origin. It develops a comprehensive half-plane construction via Fourier separation, derives explicit compatibility relations for boundary data, and characterizes the resulting free-boundary geometry as equi-distributed logarithmic spirals (when α ≠ 0) or fixed-angle intersections (when α = 0). The authors also construct explicit eternal solutions of the rotating heat equation and provide a rich single-mode analysis, including Dirichlet, Neumann/Robin, and entire problems expressed in terms of the Θν functions and Bessel-type constructs, yielding detailed nodal and asymptotic structures. Overall, the work advances understanding of spiraling interfaces in segregated reaction-diffusion systems and links disk problems to a tractable half-plane framework with explicit, analyzable solutions.

Abstract

We give a complete characterization of the boundary traces $\varphi_i$ ($i=1,\dots,K$) supporting spiraling waves, rotating with a given angular speed $ω$, which appear as singular limits of competition-diffusion systems of the type \[ \frac{\partial}{\partial t} u_i -Δu_i = μu_i -βu_i \sum_{j \neq i} a_{ij} u_j \text{ in } Ω\times\mathbb{R}^+, \qquad u_i = \varphi_i \text{ on $\partialΩ\times\mathbb{R}^+$}, \qquad u_i(\mathbf{x},0) = u_{i,0}(\mathbf{x}) \text{ for $\mathbf{x} \in Ω$} \] as $β\to +\infty$. Here $Ω$ is a rotationally invariant planar set and $a_{ij}>0$ for every $i$ and $j$. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by $ω\in\mathbb{R}$, which reduce to homogeneous harmonic polynomials for $ω=0$.

Figures (4)

• Figure 1: Counter lines of a numerical simulation (obtained in FreeFem++ MR3043640) in the case of $K=3$ densities, with asymmetric competition such that $\frac{a_{12}}{a_{21}}=\frac{a_{23}}{a_{32}}= \frac{a_{31}}{a_{13}}=10$, and reaction term $\mu=0$. The angular velocity is $\omega=3$ for the picture on the left (counterclockwise spin) and $\omega = -3$ for the picture on the right (clockwise spin). In both cases we obtain a unique singular point at the center of the circle by choosing the same boundary conditions, that verify the necessary and sufficient condition in Theorem \ref{['thm:main_intro']} (see equation \ref{['eq:compX3']}). The rotation affects the shape of the spirals, but not their asymptotic behavior close to the center. This is part of the content of Theorem \ref{['thm:main_intro']}.
• Figure 2: Contour lines of the rotating caloric functions in Remark \ref{['rem_entire_intro']}. Here $\omega = 1$, $k =1$ and $k=2$, respectively. In black the nodal lines: the appearance of arithmetic spirals for $r$ large is rather clear in the picture.
• Figure 3: numerical zeroes of $\mathop{\mathrm{Re}}\nolimits{\Theta}_{1-i}$ (blue) and $\mathop{\mathrm{Im}}\nolimits{\Theta}_{1-i}$ (red). The three zeroes located at $10.36+i23.66$, $20.22+i67.99$, $30.21+i132.04$, satisfy condition \ref{['eqn lambda ring']}.
• Figure 4: nodal sets of the solutions corresponding to the three zeroes in Fig. \ref{['fig:zeroes']}.

Theorems & Definitions (70)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5: Homogeneous boundary conditions
• Theorem 1.6: Entire solutions
• Remark 1.7
• Remark 1.8
• Proposition 2.1
• Remark 2.2