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Reaction-diffusion (RD) systems with time delays have been commonly used in modeling biological systems and can significantly change the dynamics of these systems. For predator-prey model with modified Leslie-Gower and Holling-type III schemes governed by RD equations, instability induced by time delay can generate spiral waves. Considering that populations are usually organized as networks instead of being continuously distributed in space, it is essential to study the predator-prey model on complex networks. In this paper, we investigate instability induced by time delay for the corresponding network organized system and explore pattern formations on several different networks including deterministic networks and random networks. We firstly obtain instability condition via linear stability analysis and then the condition is applied to study pattern formations for the model in question. The simulation results show that wave patterns can be generated on different networks. However, wave patterns on random networks differ significantly from patterns on deterministic networks. Finally, we discuss the influences of network topology on wave patterns from the aspects of amplitude and period, and reveal the ecology significance implied by these results.

In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations: The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time

In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicals immersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

We investigate quantum reaction–diffusion (RD) systems in one-dimension with bosonic particles that coherently hop in a lattice, and when brought in range react dissipatively. Such reactions involve binary annihilation ( A + A → ∅ ) and coagulation ( A + A → A ) of particles at distance d . We consider the reaction-limited regime, where dissipative reactions take place at a rate that is small compared to that of coherent hopping. In classical RD systems, this regime is correctly captured by the mean-field approximation. In quantum RD systems, for noninteracting fermionic systems, the reaction-limited regime recently attracted considerable attention because it has been shown to give universal power law decay beyond mean field for the density of particles as a function of time. Here, we address the question whether such universal behavior is present also in the case of the noninteracting Bose gas. We show that beyond mean-field density decay for bosons is possible only for reactions that allow for destructive interference of different decay channels. Furthermore, we study an absorbing-state phase transition induced by the competition between branching A → A + A , decay A → ∅ and coagulation A + A → A . We find a stationary phase-diagram, where a first and a second-order transition line meet at a bicritical point which is described by tricritical directed percolation. These results show that quantum statistics significantly impact on both the stationary and the dynamical universal behavior of quantum RD systems.

During the past decades, pattern formulation with reaction-diffusion systems has attracted great research interest. Complex networks, from single-layer networks to more complicated multiplex networks, have made great contribution to the development of this area, especially with emergence of Turing patterns. While among vast majority of existing works on multiplex networks, they only take into account the simple case with ordinary diffusion, which is termed as self-diffusion. However, cross-diffusion, as a significant phenomenon, reveals the direction of species' movement, and is widely found in chemical, biological and physical systems. Therefore, we study the pattern formulation on multiplex networks with the presence of both self-diffusion and cross-diffusion. Of particular interest, heterogeneous patterns with abundant characteristics are generated, which cannot arise in other systems. Through linear analysis, we theoretically derive the Turing instabilities region. Besides, our numerical experiments also generate diverse patterns, which verify the theoretical prediction in our work and show the impact of cross-diffusion on pattern formulation on multiplex networks.

Exposing a solution to a temperature gradient can lead to the accumulation of particles on either the cold or warm side. This phenomenon is known as thermophoresis, and its microscopic origin is still debated. Here, we show that thermophoresis can be observed in any system having internal states with different transport properties, and temperature-modulated rates of transitions between the states. These internal degrees of freedom might be configurational, chemical or velocity states. We also derive an expression for the Soret coefficient, which decides whether particles accumulate on the cold or warm side. Our framework can be applied to any chemical reaction system diffusing in a temperature gradient. It also captures the possibility to observe a sign inversion of the Soret coefficient as the competition between chemical and velocity states. We establish thermophoresis as a genuine non-equilibrium effect, originating from internal microscopic currents consistent with the necessity of transporting heat from warm to cold regions.

We investigate the effects of cooperativity between contagion processes that spread and persist in a host population. We propose and analyze a dynamical model in which individuals that are affected by one transmissible agent A exhibit a higher than baseline propensity of being affected by a second agent B and vice versa. The model is a natural extension of the traditional susceptible-infected-susceptible model used for modeling single contagion processes. We show that cooperativity changes the dynamics of the system considerably when cooperativity is strong. The system exhibits discontinuous phase transitions not observed in single agent contagion, multi-stability, a separation of the traditional epidemic threshold into different thresholds for inception and extinction as well as hysteresis. These properties are robust and are corroborated by stochastic simulations on lattices and generic network topologies. Finally, we investigate wave propagation and transients in a spatially extended version of the model and show that especially for intermediate values of baseline reproduction ratios the system is characterized by various types of wave-front speeds. The system can exhibit spatially heterogeneous stationary states for some parameters and negative front speeds (receding wave fronts). The two agent model can be employed as a starting point for more complex contagion processes, involving several interacting agents, a model framework particularly suitable for modeling the spread and dynamics of microbiological ecosystems in host populations.

We consider systems of reaction–diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic λ-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift–diffusion system, provide a survey on the applicability of the theory.