[Paper Review] Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

Konstantinos Spiliotis, Lucia Russo+2 more|arXiv (Cornell University)|Mar 23, 2023

Ecosystem dynamics and resilienceEnvironmental Science3 citations

TL;DR

This study presents a numerical bifurcation analysis of a reaction-diffusion PDE-ODE model for vegetation dynamics in water-limited environments, focusing on Turing instabilities and symmetry-breaking patterns. It reveals secondary bifurcations leading to multistable asymmetric solutions, demonstrating that nonlinearities beyond linear stability analysis give rise to complex, far-from-equilibrium patterns such as skewed and inverted bell-shaped profiles, with key transitions occurring at precipitation rates around 1.14 and 1.35.

ABSTRACT

We study the mechanisms of pattern formation for vegetation dynamics in water-limited regions. Our analysis is based on a set of two partial differential equations (PDEs) of reaction-diffusion type for the biomass and water and one ordinary differential equation (ODE) describing the dependence of the toxicity on the biomass. We perform a linear stability analysis in the one-dimensional finite space, we derive analytically the conditions for the appearance of Turing instability that gives rise to spatio-temporal patterns emanating from the homogeneous solution, and provide its dependence with respect to the size of the domain. Furthermore, we perform a numerical bifurcation analysis in order to study the pattern formation of the inhomogeneous solution, with respect to the precipitation rate, thus analyzing the stability and symmetry properties of the emanating patterns. Based on the numerical bifurcation analysis, we have found new patterns, which form due to the onset of secondary bifurcations from the primary Turing instability, thus giving rise to a multistability of asymmetric solutions.

Motivation & Objective

• To investigate the mechanisms of vegetation pattern formation in semi-arid ecosystems using a reaction-diffusion model with biomass, water, and toxicity dynamics.
• To analytically derive conditions for Turing instability and its dependence on domain size in a one-dimensional finite domain.
• To perform numerical bifurcation analysis with respect to precipitation rate to track stable and unstable inhomogeneous solutions.
• To identify and characterize symmetry-breaking bifurcations and multistability in the system’s solution branches.
• To reveal novel asymmetric patterns arising from secondary bifurcations beyond the primary Turing instability.

Proposed method

• Formulate a coupled system of two PDEs (biomass and water) and one ODE (toxicity) based on ecological feedback mechanisms.
• Perform linear stability analysis around the homogeneous equilibrium to analytically determine the onset of Turing instability and its dependence on domain size.
• Use numerical continuation techniques to compute full bifurcation diagrams of steady-state solutions with respect to the precipitation rate parameter.
• Track both stable and unstable solution branches, including symmetric and asymmetric patterns, using path-following algorithms.
• Analyze symmetry properties of solutions by comparing profiles near and far from the Turing bifurcation point.
• Identify secondary bifurcations such as pitchforks that break reflection symmetry and lead to distinct asymmetric equilibria.

Experimental results

Research questions

• RQ1 What are the analytical conditions for the onset of Turing instability in the vegetation PDE-ODE model, and how do they depend on domain size?
• RQ2 How does the precipitation rate govern the emergence and stability of inhomogeneous vegetation patterns?
• RQ3 What role do secondary bifurcations play in generating asymmetric, non-symmetric solutions beyond the primary Turing instability?
• RQ4 How is reflection symmetry preserved or broken along solution branches, and what does this imply for pattern diversity?
• RQ5 What is the nature of multistability in the system, and how do multiple stable states coexist across different precipitation regimes?

Key findings

• Turing bifurcation occurs at a critical precipitation rate of $ p_{c_1} = 1.14 $, marking the onset of symmetry-breaking pattern formation.
• Near $ p_{c_1} $, two symmetric solutions emerge: a bell-shaped and an inverted bell-shaped profile, symmetric with respect to the homogeneous solution.
• Far from $ p_{c_1} $, the symmetry between the bell-shaped and inverted bell-shaped profiles is broken due to nonlinear effects.
• Secondary bifurcations at $ p ightarrow 1.35 $ (PF1) lead to the emergence of two new asymmetric, unstable steady-state branches: skewed-left and skewed-right patterns.
• These asymmetric solutions are conjugate to each other and preserve symmetry along their respective branches, indicating a reflection-conjugate structure.
• Multistability is observed across multiple parameter regimes: from $ LP4 $ to $ PF2 $, three stable solutions coexist (bare soil, symmetric bell-shaped/inverted bell-shaped), and from $ PF2 $ to $ PF1 $, four stable states coexist, including the inverted bell-shaped and symmetric patterns.