Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus

TL;DR

The paper analyzes long-time behavior and local bifurcations in a one-dimensional torus nonlocal aggregation–diffusion model with two interacting species. It extends scalar results to a two-species system under BV kernels, using a fixed-point formulation and Crandall–Rabinowitz theory to classify all bifurcations from homogeneous states with respect to self- and cross-interactions, including stability exchanges. Global asymptotic stability of the uniform state is established for small couplings via relative-entropy methods, while a detailed Fréchet-derivative framework up to third order yields precise branch directions and phase relations for emergent patterns. The work demonstrates that stable segregation can arise even in purely attractive regimes, connecting to the Differential Adhesion Hypothesis and cell sorting, and provides a rigorous mathematical foundation for predicting pattern formation in multi-species adhesion-diffusion models.

Abstract

We investigate stationary states, including their existence and stability, in a class of nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions. For the scalar case, we extend previous results by showing that key model features, such as existence, regularity, bifurcation structure, and stability exchange, continue to hold under a mere bounded variation hypothesis. For the corresponding two-species system, we carry out a fully rigorous bifurcation analysis using the bifurcation theory of Crandall & Rabinowitz. This framework allows us to classify all solution branches from homogeneous states, with particular attention given to those arising from the self-interaction strength and the cross-interaction strength, as well as the stability of the branch at a point of critical stability. The analysis relies on an equivalent classification of solutions through fixed points of a nonlinear map, followed by a careful derivation of Fréchet derivatives up to third order. An interesting application to cell-cell adhesion arises from our analysis, yielding stable segregation patterns that appear at the onset of cell sorting in a modelling regime where all interactions are purely attractive.

Figures (6)

• Figure 1: A depiction of the results of Theorem \ref{['thm:local_bifurcations_scalar']} and \ref{['thm:local_bifurcations_scalar_2']}. The left panel displays the Fourier coefficients of $W$ (blue squares) and the associated possible bifurcation points $\alpha_k$ (red dots). The right panel displays a typical bifurcation diagram. Green lines denote a stable branch, dashed red lines denote an unstable branch, and the stability of the yellow branches is unknown. See Remark \ref{['remark:scalar_bifurcation_remark']} for further discussion.
• Figure 2: A visualisation of the stability region from Proposition \ref{['prop:local_stability']} in the $(\chi_1 \alpha_1, \chi_2 \alpha_2)$-plane for $\gamma=0$ (lightest shade of green), $\gamma = 1.5$ (darker shade of green), and $\gamma = 3.0$ (darkest shade of green). The line $S^*=0$ (the cyan-colored line) is defined as in \ref{['eq:S_star']}; it divides the stability region according to where the minimum of \ref{['eq:gamma_minimum_1']} is achieved. The speckled area corresponds to the region $S^*<0$, while the untextured area corresponds to the region $S^* > 0$. For a fixed value of $\gamma>0$, those points $(\chi_1 \alpha_1, \chi_2 \alpha_2)$ falling within these shaded regions correspond with local asymptotic stability of the homogeneous solution. Whenever there holds $\alpha^*(\pm W) < \infty$, the stability region vanishes for $\gamma$ sufficiently large. Bifurcation points then occur at the boundary of these regions. The points $P_1$, $P_2$ are reference points for subsequent bifurcation diagrams. See Remark \ref{['remark:local_stability_remark']} for further discussion.
• Figure 3: A visualisation of the results of Theorems \ref{['thm:bifurcations_alpha1_1']} and \ref{['thm:main_results_local_stability_alpha_system']} for the two-species system. The top row corresponds with the point $P_1$, while the bottom row corresponds with the point $P_2$, where $P_i$ are marked as blue stars in Figure \ref{['fig:stability_region_1']}. Green lines denote a stable branch, dashed red lines denote an unstable branch, and the stability of the yellow branches is unknown. Further details are provided in Remark \ref{['remark:alpha1_bifurcations_remark']}.
• Figure 4: A visualisation of the interplay between the negative and positive Fourier modes as found in Theorem \ref{['thm:main_results_local_stability_gamma_system']} for the two-species system when bifurcating with respect to $\gamma$. The blue line depicts the continuous version of the bifurcation points $\gamma_k$ defined in \ref{['eq:critical_gamma']}; the dark purple regions on the $x$-axis denote the possible range of the inputs, namely, $\mathop{\mathrm{Ran}}\nolimits (h_k) = (-\infty, -\alpha^*(W)] \cup [\alpha^*(-W), +\infty)$. The left panel uses the parameter values $P_1$, while the right panel uses the parameter values $P_2$, both of which are depicted in Figure \ref{['fig:stability_region_1']}. See Remark \ref{['remark:gamma_bifurcation_remark']} for further discussion.
• Figure 5: A visualisation of the results of Theorems \ref{['thm:bifurcations_gamma_1']} and \ref{['thm:main_results_local_stability_gamma_system']} for the two-species system. The top and bottom row again correspond with points $P_1$ and $P_2$, respectively. Green lines denote a stable branch, dashed red lines denote an unstable branch, and the stability of the yellow branches is unknown. See Remark \ref{['remark:gamma_bifurcation_remark']} for further discussion.

Theorems & Definitions (35)

• Theorem 1.1: Global asymptotic stability of homogeneous state, scalar case
• Theorem 1.2: Description of local bifurcations, scalar case (Carrillo2020)
• Theorem 1.3: Point of critical stability & stability exchange, scalar case
• Remark 1.4
• Theorem 2.1: Global asymptotic stability of homogeneous state, system case
• Remark 2.2
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• Theorem 2.6: Description of local bifurcations w.r.t. $\alpha_1 \geq 0$