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This chapter discusses partial differential equation models. Partial differential equations can describe the dynamics of phenotype distributions of polymorphic populations, and they allow for a mathematically concise formulation from which some analytical insights can be obtained. It has been argued that because partial differential equations can describe polymorphic populations, results from such models are fundamentally different from those obtained using adaptive dynamics. In partial differential equation models, diversification manifests itself as pattern formation in phenotype distribution. More precisely, diversification occurs when phenotype distributions become multimodal, with the different modes corresponding to phenotypic clusters, or to species in sexual models. Such pattern formation occurs in partial differential equation models for competitive as well as for predator–prey interactions.

When a dynamical system has significant spatial extent, its nonlinear dynamics can lead to the spontaneous formation of spatial patterns. Such systems provide models for how nature might have developed ordered, spatial structures from disordered states. Examples are given from fluid flow, transport models, coupled-oscillator modes, cellular automata, transport models, and reaction-diffusion systems. Diffusion-limited aggregation, viscous fingering, and dielectric breakdown provide further examples of pattern formation. Fractal structures make another appearance in this new context. This chapter also explores the somewhat controversial topic of self-organized criticality which has been put forward as an explanation for the occurrence of fractal structures in nature.

This introductory chapter presents some ideas and key concepts in materials science. These include developments in the field of materials science, examples demonstrating the critical role played by material instabilities and pattern formation, the control of microstructure formation, and new materials design.

Defects play a critical role in many material phenomena, and their relationship to instabilities and pattern formation is well-established. This chapter presents a brief description of the main types of crystal defects, describing their geometry and main characteristics. It then focuses on determining their elastic field and interaction forces.

This chapter examines the complex nature of pattern formation in excitable systems. Many physical, chemical, and biological systems are considered excitable, in which two “components” interact in such a way as to alter each other's state through (nonlinear) processes of inhibition or amplification. Starting from a homogeneous rest state, many systems of this type can spontaneously generate persistent spatiotemporal patterns when subjected to some perturbation. After providing an overview of excitable systems, the chapter discusses the hodgepodge machine and its numerical implementation in the Python code. It then describes four spatial patterns generated by the hodgepodge machine, namely: waves, spirals, spaghettis, and cells. It also explains the spontaneous formation of spatiotemporal patterns. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

Developmental biology is devoted to explaining how a variety of interacting processes generate the heterogeneous shapes, size, and structural features of an organism as it develops from embryo to adult. Although it is commonplace to associate sciences with theories, many presentations of developmental biology make no reference to a theory or theories of development. Instead of seeing this absence as an indicator of immaturity or an invitation to reconstruct a theory in different guise, the lack of reference to theories marks a significant aspect of scientific epistemology: the erotetic (or question-based) organization of developmental biology. This organization consists in stable, broad domains of problems (problem agendas) that correspond to ontogenetic phenomena (pattern formation, differentiation, growth, and morphogenesis). These are on display in both textbooks and journal articles, and shoulder the burden for guiding and directing investigative projects. Problem agendas are configured by arrays of component questions at different levels of abstraction, pertaining to a variety of developmental phenomena that are empirically and conceptually linked by considerations of temporal occurrence and spatial composition. This erotetic structure tracks the goals of researchers and shapes expectations about explanatory adequacy. The resulting epistemic picture not only makes sense of the operative discourse of reasoning practices accessible to developmental biologists when evaluating inductive inferences or explanations but also is applicable to other sciences, underwrites exploratory experimentation in biology, explains the stability and transformation of problems through history, and yields novel perspectives on progress and incommensurability.

Traditionally, quantitative social scientists have, for mathematical convenience, assumed that human collective behavior gravitates toward equilibrium. As part of this tradition, mainstream economics (and human behavioral ecology) has as its core the rational agent, who gathers all available information on a problem and then chooses the optimal decision. Bounded rationality (e.g., in behavioral economics) assumes more local optimization under fixed agent preferences. This chapter considers relaxing the assumptions of rationality much further, by assuming the “zero intelligence” agent, who cannot act with purpose or intent, and cannot learn. By yielding complex collective patterns without requiring complexity of individual behavior, the zero-intelligence approach provides a better basis for understanding societies as open, non-equilibrium systems under constant flux. It deliberately assumes as little as possible, in order to identify the most general characteristics first, which can then reveal the effects of making agent behavior incrementally more complicated. In essence, this chapter argues that building upon a model of ignorance is more effective than relaxing a model of omniscience.

In the spirit of explanatory pluralism, this chapter argues that causal and non-causal explanations of a phenomenon are compatible, each being useful for bringing out different sorts of insights. First the chapter reviews the author’s model-based account of scientific explanation, which can accommodate causal and non-causal explanations alike. Then it distills from the literature an important core conception of non-causal explanation. This non-causal form of model-based explanation is illustrated using the example of how Earth scientists in a subfield known as aeolian geomorphology are explaining the formation of regularly-spaced sand ripples. The chapter concludes that even when it comes to everyday “medium-sized dry goods” such as sand ripples, where there is a complete causal story to be told, one can find examples of non-causal scientific explanations.

This chapter is entirely devoted to characterizing non-linear dynamics on multilayer networks. Special attention is given to recent results on the stability of synchronization that extend the Master Stability Function approach to the multilayer networks scenario. Discontinous synchronization transitions on multiplex networks recently reported in the literature are also discussed, and their application discussed in the context of brain networks. This chapter also presents an overview of the major results regarding pattern formation in multilayer networks, and the proposed characterization of multivariate time series using multiplex visibility graphs. Finally, the chapter discusses several approaches for multiplex network control where the dynamical state of a multiplex network needs to be controlled by eternal signals placed on replica nodes satisfying some structural constraints.

Ecological systems are prone to dramatic shifts between alternative stable states. In reality, these shifts are often caused by slow forces external to the system that eventually push it over a tipping point. Theory predicts that when ecological systems are brought close to a tipping point, the dynamical feedback intrinsic to the system interact with intrinsic noise and extrinsic perturbations in characteristic ways. The resulting phenomena thus serve as “early warning signals” for shifts such as population collapse. In this chapter, we review the basic (qualitative) theory of such systems. We then illustrate the main ideas with a series of models that both represent fundamental ecological ideas (e.g. density-dependence) and are amenable to mathematical analysis. These analyses provide theoretical predictions about the nature of measurable fluctuations in the vicinity of a tipping point. We conclude with a review of empirical evidence from laboratory microcosms, field manipulations, and observational studies.

The dynamics of reactive flows lies at the heart of several important applications, such as combustion, heterogeneous catalysis, pollutant conversion, pattern formation in biology and many others. In general, LB is well suited to describe reaction-diffusion applications with flowing species. This chapter provides the basic guidelines to include reactive phenomena within the LBE formalism. Reactive flows obey the usual fluid equations, augmented with a reactive source term, accounting for species transformations due to chemical reactions. Such term comes typically in the form of a polynomial product of the mass densities of the reacting species.