Abstract In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.

Pattern formation driven by cross–diffusion in a 2D domain

We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.

Stability Results for Steady, Spatially--Periodic Planforms

Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition

In this paper we derive a reaction-diffusion-chemotaxis model for the dynamics of multiple sclerosis. We focus on the early inflammatory phase of the disease characterized by activated local microglia, with the recruitment of a systemically activated immune response, and by oligodendrocyte apoptosis. The model consists of three equations describing the evolution of macrophages, cytokine and apoptotic oligodendrocytes. The main driving mechanism is the chemotactic motion of macrophages in response to a chemical gradient provided by the cytokines. Our model generalizes the system proposed by Calvez and Khonsari (Math Comput Model 47(7–8):726–742, 2008) and Khonsari and Calvez (PLos ONE 2(1):e150, 2007) to describe Balo’s sclerosis, a rare and aggressive form of multiple sclerosis. We use a combination of analytical and numerical approaches to show the formation of different demyelinating patterns. In particular, a Turing instability analysis demonstrates the existence of a threshold value for the chemotactic coefficient above which stationary structures develop. In the case of subcritical transition to the patterned state, the numerical investigations performed on a 1-dimensional domain show the existence, far from the bifurcation, of complex spatio-temporal dynamics coexisting with the Turing pattern. On a 2-dimensional domain the proposed model supports the emergence of different demyelination patterns: localized areas of apoptotic oligodendrocytes, which closely fit existing MRI findings on the active MS lesion during acute relapses; concentric rings, typical of Balo’s sclerosis; small clusters of activated microglia in absence of oligodendrocytes apoptosis, observed in the pathology of preactive lesions.

Demyelination patterns in a mathematical model of multiple sclerosis

A diffusive predator–prey system with Crowley–Martin functional response is considered. Firstly, the maximal parameter region, where the coexistence equilibrium is stable, is provided, of which the boundary consists of Turing bifurcation curves and Hopf bifurcation curve, and result derived by Shi and Ruan (2015) is improved. Meanwhile, critical conditions for Turing instability are derived, which are necessary and sufficient. Furthermore, considering the degenerated situation where Turing bifurcation and Hopf bifurcation occur simultaneously, conditions for codimension-two Turing–Hopf bifurcation and Turing–Turing bifurcations are given. For Turing–Hopf bifurcation, by analyzing the normal forms truncated to order 3, which are derived by applying normal form method and generic formulas developed by Jiang, An and Shi (2018), it is found that system exhibits spatial, temporal and spatiotemporal patterns, like transient spatially inhomogeneous periodic solutions, as well as tristable phenomena of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. At last, numerical simulations, including transient, bistable and tristable patterns, are illustrated to support our theory results.

Turing–Hopf bifurcation and spatiotemporal patterns in a diffusive predator–prey system with Crowley–Martin functional response

We focus on the morphochemical reaction-diffusion model introduced in Bozzini et al. (2013) and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of Turing instability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis, and derive the normal form equations governing the amplitude of the patterns. These amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations. Hysteretic type phenomena are highlighted also through numerical simulations. We show the occurrence of spatial pattern propagation and derive the Ginzburg-Landau equation describing the envelope of the traveling wavefront.

Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth

In this paper the Turing pattern formation mechanism of a two components reaction-diffusion system modeling the Schnakenberg chemical reaction is considered. In Ref. (Madzavamuse et al., J Math Biol 70(4):709–743, 2015) it was shown how the presence of linear cross-diffusion terms favors the destabilization of the constant steady state. We perform the weakly nonlinear multiple scales analysis to derive the equations for the amplitude of the Turing patterns and to show how the cross-diffusion coefficients influence the occurrence of super-critical or sub-critical bifurcations. We present a numerical exploration of far from equilibrium regimes and prove the existence of multistable stationary solutions.

Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

In this paper the Turing pattern formation mechanism of a two component reactiondiffusion system modeling the Schnakenberg chemical reaction coupled to linear crossdiffusion terms is studied. The linear cross-diffusion terms favors the destabilization of the constant steady state and the mechanism of pattern formation with respect to the standard linear diffusion case, as shown in [1]. Since the subcritical Turing bifurcations of reaction-diffusion systems lead to spontaneous onset of robust, finite-amplitude localized patterns, here a detailed investigation of the Turing pattern forming region is performed to show how the diffusion coefficients for both species (the activator and the inhibitor) influence the occurrence of supercritical or subcritical bifurcations. The weakly nonlinear (WNL) multiple scales analysis is employed to derive the equations for the amplitude of the Turing patterns and to distinguish the supercritical and the subcritical pattern region, both in 1D and 2D domains. Numerical simulations are employed to confirm the WNL theoretical predictions through which a classification of the patterns (squares, rhombi, rectangle and hexagons) is obtained. In particular, due to the hysteretic nature of the subcritical bifurcation, we observe the phenomenon of pattern transition from rolls to hexagons, in agreement with the bifurcation diagram.

S. Lupo

Papers

Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth

Super-critical and sub-critical bifurcations in a reaction-diffusion Schnakenberg model with linear cross-diffusion

Effects of cross-diffusion on Turing patterns in a reaction-diffusion Schnakenberg model.