Abstract. Many reaction-diffusion (RD) systems exhibit spot patterns. When the domain is a periodic rectangle, many such patterns can be constructed explicitly using regular integer lattices, and moreover numerical simulations show that they are prevalent even when starting with random initial conditions. For a wide class of RD systems which includes the Schnakenberg model, these equilibria are local minima of the corresponding Green’s function potential. We use integer lattice enumeration and Floquet theory to classify all stable lattices on a square up to [Formula: see text] spots. We then investigate stability boundaries and bifurcations as the aspect ratio of the rectangle is varied. For a rectangle with a small aspect ratio, we derive explicit thresholds for stability of a pattern consisting of two interlaced stripes. For certain aspect ratios and [Formula: see text], there exists a perfect hexagonal lattice without any defects. We determine stability boundaries of such lattices as the aspect ratio changes. We also explore bifurcations and transitions that result when the lattice is deformed beyond its stable regime. The results are illustrated and confirmed using the Schnakenberg RD system.
Abstract. We study the discrete nonlinear Schrödinger equation with competing powers [Formula: see text] satisfying [Formula: see text]. The physically relevant cases are given by [Formula: see text], [Formula: see text], and [Formula: see text]. In the anticontinuum limit, all intrinsic localized modes are compact and can be classified by their codes, which record one of two nonzero (smaller and larger) states and their sign alternations. By using the spectral stability analysis, we prove that the codes for larger states of the same sign are spectrally and nonlinearly (orbitally) stable, whereas the codes for smaller states of the alternating signs are spectrally stable but have eigenvalues of negative Krein signature. We also identify numerically the spectrally stable codes which consist of stacked combinations of the sign-definite larger states and the sign-alternating smaller states.
We developed a coupled social–climate network model that links opinion dynamics and the climate system, where opinions directly translate into actions and actions reflect underlying opinions, to examine how this feedback shapes collective behavior and global temperature trajectories. In contrast to previous social–climate models that discretized opinions, we assumed opinions on climate change form a continuum, and were thereby able to capture more nuanced interactions. The model shows that resistance to behavior change, elevated mitigation costs, and slow response to climate events can result in a global temperature anomaly in excess of 2[Formula: see text]C. However, this outcome could be avoided by lowering mitigation costs and increasing the rate of interactions between individuals with differing opinions (social learning). Our model is one of the first to demonstrate the emergence of opinion polarization in a human-environment system. We predict that polarization of opinions in a population can be extinguished, and the population will adopt mitigation practices, when the response to temperature change is sensitive, even at higher mitigation costs. It also indicates that even with polarized opinion, an average promitigative opinion in the population can reduce emissions. Finally, our model underscores how frequent and unexpected social or environmental changes, such as policy changes or extreme weather events, can slow climate change mitigation. This analysis helps identify the factors that support achieving international climate goals, such as leveraging peer influence and decreasing stubbornness in individuals, reducing mitigation costs, and encouraging climate-friendly lifestyles. Our model offers a valuable new framework for exploring the integration of social and natural sciences, particularly in the domain of human behavioral change.
Population dynamics in fields such as molecular biology, epidemiology, and ecology exhibit highly stochastic and nonlinear behavior. In gene regulatory systems in particular, oscillations and multistability are especially common. Despite this, none of the currently available stochastic models for population dynamics are both accurate and computationally efficient for long-term predictions. A prominent model in this field, the linear noise approximation (LNA), is computationally efficient for tasks such as simulation, sensitivity analysis, and parameter estimation; however, it is only accurate for linear systems and short-time predictions. Other models may achieve greater accuracy across a broader range of systems, but they sacrifice computational efficiency and analytical tractability. This paper demonstrates that, with specific modifications, the LNA can accurately capture nonlinear dynamics in population processes. We introduce a new framework based on center manifold theory, a classical concept from nonlinear dynamical systems. This approach enables the identification of simple, system-specific modifications to the LNA, tailored to classes of qualitatively similar nonlinear dynamical systems. With these modifications, the LNA can achieve accurate long-term simulations without compromising computational efficiency. We apply our methodology to classes of oscillatory and bistable systems and present multiple examples from molecular population dynamics that demonstrate accurate long-term simulations alongside significant improvements in computational efficiency.
This paper proposes a sparse regression strategy for discovery of ordinary differential equations from incomplete and noisy data. Inference is performed over both equation parameters and state variables using a statistically motivated likelihood function. Sparsity is enforced by a selection algorithm which iteratively removes terms and compares models using statistical information criteria. Large scale optimization is performed using a second-order variant of the Levenb erg-Marquardt method, where the gradient and Hessian are computed via automatic differentiation. The proposed method is illustrated and tested on several systems with varying levels of noisy and incomplete data. Comparisons are made to a state-of-the-art algorithm for system identification, demonstrating competitiveness of the proposed approach.
