Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities

Emergence of non-trivial solutions from trivial solutions in reaction-diffusion equations for pattern formation

Reaction-diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction-diffusion models: the Schnakenberg model and the Gray-Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models.

Two wrongs do not make a right: the assumption that an inhibitor acts as an inverse activator

Models of biochemical networks are often large intractable sets of differential equations. To make sense of the complexity, relationships between genes/proteins are presented as connected graphs, the edges of which are drawn to indicate activation or inhibition relationships. These diagrams are useful for drawing qualitative conclusions in many cases by the identifying recurring of topological motifs, for example positive and negative feedback loops. These topological features are usually classified under the presumption that activation and inhibition are inverse relationships. For example, inhibition of an inhibitor is often classified the same as activation of an activator within a motif classification, effectively treating them as equivalent. Whilst in many contexts this may not lead to catastrophic errors, drawing conclusions about the behavior of motifs, pathways or networks from these broad classes of topological feature without adequate mathematical descriptions can lead to obverse outcomes. We investigate the extent to which a biochemical pathway/network will behave quantitatively dissimilar to pathway/ networks with similar typologies formed by swapping inhibitors as the inverse of activators. The purpose of the study is to determine under what circumstances rudimentary qualitative assessment of network structure can provide reliable conclusions as to the quantitative behaviour of the network. Whilst there are others, We focus on two main mathematical qualities which may cause a divergence in the behaviour of two pathways/networks which would otherwise be classified as similar; (i) a modelling feature we label 'bias' and (ii) the precise positioning of activators and inhibitors within simple pathways/motifs.

Reactive persistence, spatial management, and conservation of metapopulations: An application to seagrass restoration

Assessing the conditions for persistence of spatially structured populations, especially those that are exploited by humans or threatened by global change, is of critical importance to inform management and conservation efforts. Observations for entire metapopulations are usually incomplete and rarely, if ever, sufficiently long to deduce population persistence from spatial patterns of abundance. Instead, insights based on metapopulation theory are often used for interpreting the demographic trajectories of real populations and for informing management decisions. The classical theoretical tool used to assess conditions for metapopulation persistence is the "invasibility criterion," which characterizes the asymptotic, or long-term, stability of a small colonizing population. Essentially, when the linear operator governing the metapopulation dynamics of an invasion event has a positive eigenvalue, recovery and resistance to extinction (resilience) are implied. The converse, however, is not necessarily the case-an invasion may grow over multiple generations, even when the eigenvalues indicate that extinction will eventually occur, a situation referred to here as "reactive persistence." For the management, restoration, and conservation of real metapopulations subject to continual disturbance, this transient behavior is often more relevant than the asymptotic behavior over long time scales. We develop the theoretical tools for assessing reactive persistence, demonstrating how the conditions for asymptotic and reactive persistence differ in both the patch-occupancy models suited to many terrestrial populations and those where local patch extinctions can be disregarded in the dynamics, often suited to marine species. After presenting the mathematical basis for generalizing the invasibility criterion to include reactive persistence, we illustrate how these concepts and tools can be applied in practice, using as a case study the population ecology and restoration of the seagrass Zostera muelleri (Irmisch ex Ascherson, 1867) in the Port of Gladstone in the Great Barrier Reef World Heritage Area Australia. It is shown how the analysis of the transient dynamics of the Z. muelleri metapopulation can be used to guide restoration efforts. Moreover, it is demonstrated that these reactive persistence concepts provide a more appropriate basis for site prioritization for restoration interventions than traditional stability analysis.

Turing patterns in simplicial complexes

The spontaneous emergence of patterns in nature, such as stripes and spots, can be mathematically explained by reaction-diffusion systems. These patterns are often referred as Turing patterns to honor the seminal work of Alan Turing in the early 1950s. With the coming of age of network science, and with its related departure from diffusive nearest-neighbor interactions to long-range links between nodes, additional layers of complexity behind pattern formation have been discovered, including irregular spatiotemporal patterns. Here we investigate the formation of Turing patterns in simplicial complexes, where links no longer connect just pairs of nodes but can connect three or more nodes. Such higher-order interactions are emerging as a new frontier in network science, in particular describing group interaction in various sociological and biological systems, so understanding pattern formation under these conditions is of the utmost importance. We show that a canonical reaction-diffusion system defined over a simplicial complex yields Turing patterns that fundamentally differ from patterns observed in traditional networks. For example, we observe a stable distribution of Turing patterns where the fraction of nodes with reactant concentrations above the equilibrium point is exponentially related to the average degree of 2-simplexes, and we uncover parameter regions where Turing patterns will emerge only under higher-order interactions, but not under pairwise interactions.

Turing's cascade instability supports the coordination of the mind, brain, and behavior

Turing inspired a computer metaphor of the mind and brain that has been handy and has spawned decades of empirical investigation, but he did much more and offered behavioral and cognitive sciences another metaphor-that of the cascade. The time has come to confront Turing's cascading instability, which suggests a geometrical framework driven by power laws and can be studied using multifractal formalism and multiscale probability density function analysis. Here, we review a rapidly growing body of scientific investigations revealing signatures of cascade instability and their consequences for a perceiving, acting, and thinking organism. We review work related to executive functioning (planning to act), postural control (bodily poise for turning plans into action), and effortful perception (action to gather information in a single modality and action to blend multimodal information). We also review findings on neuronal avalanches in the brain, specifically about neural participation in body-wide cascades. Turing's cascade instability blends the mind, brain, and behavior across space and time scales and provides an alternative to the dominant computer metaphor.

Optimal control of networked reaction-diffusion systems

Patterns in nature are fascinating both aesthetically and scientifically. Alan Turing's celebrated reaction-diffusion model of pattern formation from the 1950s has been extended to an astounding diversity of applications: from cancer medicine, via nanoparticle fabrication, to computer architecture. Recently, several authors have studied pattern formation in underlying networks, but thus far, controlling a reaction-diffusion system in a network to obtain a particular pattern has remained elusive. We present a solution to this problem in the form of an analytical framework and numerical algorithm for optimal control of Turing patterns in networks. We demonstrate our method's effectiveness and discuss factors that affect its performance. We also pave the way for multidisciplinary applications of our framework beyond reaction-diffusion models.

Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality

Synchronization is an important behavior that characterizes many natural and human made systems that are composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure’s non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.

Nontrivial amplification below the threshold for excitable cell signaling

In many asymptotically stable fluid systems, arbitrarily small fluctuations can grow by orders of magnitude before eventually decaying, dramatically enhancing the fluctuation variance beyond the minimum predicted by linear stability theory. Here using influential quantitative models drawn from the mathematical biology literature, we establish that dramatic amplification of arbitrarily small fluctuations is found in excitable cell signaling systems as well. Our analysis highlights how positive and negative feedback, proximity to bifurcations, and strong separation of timescales can generate nontrivial fluctuations without nudging these systems across their excitation thresholds. These insights, in turn, are relevant for a broader range of related oscillatory, bistable, and pattern-forming systems that share these features. The common thread connecting all of these systems with fluid dynamical examples of noise amplification is non-normality.

Endogenous spatial pattern formation from two intersecting ecological mechanisms: the dynamic coexistence of two noxious invasive ant species in Puerto Rico

Endogenous (or autonomous, or emergent) spatial pattern formation is a subject transcending a variety of sciences. In ecology, there is growing interest in how spatial patterns can 'emerge' from internal system processes and simultaneously affect those very processes. A classic situation emerges when a predator's focus on a dominant competitor releases competitive pressure on a subdominant competitor, allowing coexistence of the two. If this idea is formulated spatially, two interesting consequences immediately arise. First, a spatial predator/prey system may take the form of a Turing instability, in which an activator (the dispersing prey population) is contained by a repressor (the more rapidly dispersing predator population) generating a spatial pattern of clusters of prey and predators, and second, an indirect intransitive loop (where A beats B beats C beats A) emerges from the simple fact that the system is spatial. Two common invasive ant species, Wasmannia auropunctata and Solenopsis invicta, and the parasitic phorid flies of S. invicta commonly coexist in Puerto Rico. Emergent spatial patterns generated by the combination of the Turing mechanism and the indirect intransitive loop are likely to be common here. This theoretical framework and the realities of the natural history in the field could explain both the long-term coexistence of these two species, and the highly variable pattern of their occurrence across a large landscape.

Reactivity of communities at equilibrium and periodic orbits

Reactivity measures the transient response of a system following a perturbation from a stable state. For steady states, the theory of reactivity is well developed and frequently applied. However, we find that reactivity depends critically on the scaling used in the equations. We therefore caution that calculations of reactivity from nondimensionalized models may be misleading. The attempt to extend reactivity theory to stable periodic orbits is very recent. We study reactivity of periodically forced and intrinsically generated periodic orbits. For periodically forced systems, we contribute a number of observations and examples that had previously received less attention. In particular, we systematically explore how reactivity depends on the timing of the perturbation. We then suggest ways to extend the theory to intrinsically generated periodic orbits. We investigate several possible global measures of reactivity of a periodic orbit and show that there likely is no single quantity to consistently measure the transient response of a system near a periodic orbit.

Turing Instability and Colony Formation in Spatially Extended Rosenzweig-MacArthur Predator-Prey Models with Allochthonous Resources

While it is somewhat well known that spatial PDE extensions of the Rosenzweig-MacArthur predator-prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing-Hopf mechanisms, in two distinct spatial Rosenzweig-MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator-prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator-prey-quarry-resource-scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing-Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig-MacArthur predator-prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig-MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.

How Spatial Heterogeneity Affects Transient Behavior in Reaction-Diffusion Systems for Ecological Interactions?

Most studies of ecological interactions study asymptotic behavior, such as steady states and limit cycles. The transient behavior, i.e., qualitative aspects of solutions as and before they approach their asymptotic state, may differ significantly from asymptotic behavior. Understanding transient dynamics is crucial to predicting ecosystem responses to perturbations on short timescales. Several quantities have been proposed to measure transient dynamics in systems of ordinary differential equations. Here, we generalize these measures to reaction-diffusion systems in a rigorous way and prove various relations between the non-spatial and spatial effects, as well as an upper bound for transients. This extension of existing theory is crucial for studying how spatially heterogeneous perturbations and the movement of biological species involved affect transient behaviors. We illustrate several such effects with numerical simulations.

Conditions for transient epidemics of waterborne disease in spatially explicit systems

Waterborne diseases are a diverse family of infections transmitted through ingestion of-or contact with-water infested with pathogens. Outbreaks of waterborne infections often show well-defined spatial signatures that are typically linked to local eco-epidemiological conditions, water-mediated pathogen transport and human mobility. In this work, we apply a spatially explicit network model describing the transmission cycle of waterborne pathogens to determine invasion conditions in metacommunities endowed with a realistic spatial structure. Specifically, we aim to define conditions under which pathogens can temporarily colonize a set of human communities, thus triggering a transient epidemic outbreak. To that end, we apply generalized reactivity analysis, a recently developed methodological framework for the study of transient dynamics in ecological systems subject to external perturbations. The study of pathogen invasion is complemented by the detection of the spatial signatures associated with the perturbations to a disease-free system that are expected to be amplified the most over different time scales. Understanding the drivers of waterborne disease dynamics over time scales that are relevant to epidemic and/or endemic transmission is a crucial, cross-disciplinary challenge, as large portions of the developing world still struggle to cope with the burden of these infections.

Construction of quasipotentials for stochastic dynamical systems: An optimization approach

The construction of effective and informative landscapes for stochastic dynamical systems has proven a long-standing and complex problem. In many situations, the dynamics may be described by a Langevin equation while constructing a landscape comes down to obtaining the quasipotential, a scalar function that quantifies the likelihood of reaching each point in the state space. In this work we provide a novel method for constructing such landscapes by extending a tool from control theory: the sum-of-squares method for generating Lyapunov functions. Applicable to any system described by polynomials, this method provides an analytical polynomial expression for the potential landscape, in which the coefficients of the polynomial are obtained via a convex optimization problem. The resulting landscapes are based on a decomposition of the deterministic dynamics of the original system, formed in terms of the gradient of the potential and a remaining "curl" component. By satisfying the condition that the inner product of the gradient of the potential and the remaining dynamics is everywhere negative, our derived landscapes provide both upper and lower bounds on the true quasipotential; these bounds becoming tight if the decomposition is orthogonal. The method is demonstrated to correctly compute the quasipotential for high-dimensional linear systems and also for a number of nonlinear examples.

Topological resilience in non-normal networked systems

The network of interactions in complex systems strongly influences their resilience and the system capability to resist external perturbations or structural damages and to promptly recover thereafter. The phenomenon manifests itself in different domains, e.g., parasitic species invasion in ecosystems or cascade failures in human-made networks. Understanding the topological features of the networks that affect the resilience phenomenon remains a challenging goal for the design of robust complex systems. We hereby introduce the concept of non-normal networks, namely networks whose adjacency matrices are non-normal, propose a generating model, and show that such a feature can drastically change the global dynamics through an amplification of the system response to exogenous disturbances and eventually impact the system resilience. This early stage transient period can induce the formation of inhomogeneous patterns, even in systems involving a single diffusing agent, providing thus a new kind of dynamical instability complementary to the Turing one. We provide, first, an illustrative application of this result to ecology by proposing a mechanism to mute the Allee effect and, second, we propose a model of virus spreading in a population of commuters moving using a non-normal transport network, the London Tube.

Epidemicity thresholds for water-borne and water-related diseases

Determining the conditions that favor pathogen establishment in a host community is key to disease control and eradication. However, focusing on long-term dynamics alone may lead to an underestimation of the threats imposed by outbreaks triggered by short-term transient phenomena. Achieving an effective epidemiological response thus requires to look at different timescales, each of which may be endowed with specific management objectives. In this work we aim to determine epidemicity thresholds for some prototypical examples of water-borne and water-related diseases, a diverse family of infections transmitted either directly through water infested with pathogens or by vectors whose lifecycles are closely associated with water. From a technical perspective, while conditions for endemicity are determined via stability analysis, epidemicity thresholds are defined through generalized reactivity analysis, a recently proposed method that allows the study of the short-term instability properties of ecological systems. Understanding the drivers of water-borne and water-related disease dynamics over timescales that may be relevant to epidemic and/or endemic transmission is a challenge of the utmost importance, as large portions of the developing world are still struggling with the burden imposed by these infections.

Significance of non-normality-induced patterns: Transient growth versus asymptotic stability

Reaction-diffusion models following the original idea of Turing are widely applied to study the propensity of a system to develop a pattern. To this end, an asymptotic analysis is typically performed via the so-called dispersion relation that relates the spectral properties of a spatial operator (diffusion) to the temporal behaviour of the whole initial-boundary value reaction-diffusion problem. Here, we amend this approach by studying the transient growth due to non-normality that can also lead to a pattern development in non-linear systems. We conclude by identification of the significance of this transient growth and by assessing the plausibility of the standard spectral approach. Particularly, the non-normality-induced patterns are possible but require fine parameter tuning.

A necessary condition for dispersal driven growth of populations with discrete patch dynamics

We revisit the question of when can dispersal-induced coupling between discrete sink populations cause overall population growth? Such a phenomenon is called dispersal driven growth and provides a simple explanation of how dispersal can allow populations to persist across discrete, spatially heterogeneous, environments even when individual patches are adverse or unfavourable. For two classes of mathematical models, one linear and one non-linear, we provide necessary conditions for dispersal driven growth in terms of the non-existence of a common linear Lyapunov function, which we describe. Our approach draws heavily upon the underlying positive dynamical systems structure. Our results apply to both discrete- and continuous-time models. The theory is illustrated with examples and both biological and mathematical conclusions are drawn.

Giant Amplification of Noise in Fluctuation-Induced Pattern Formation

The amplitude of fluctuation-induced patterns might be expected to be proportional to the strength of the driving noise, suggesting that such patterns would be difficult to observe in nature. Here, we show that a large class of spatially extended dynamical systems driven by intrinsic noise can exhibit giant amplification, yielding patterns whose amplitude is comparable to that of deterministic Turing instabilities. The giant amplification results from the interplay between noise and nonorthogonal eigenvectors of the linear stability matrix, yielding transients that grow with time, and which, when driven by the ever-present intrinsic noise, lead to persistent large amplitude patterns. This mechanism shows that fluctuation-induced Turing patterns are observable, and are not strongly limited by the amplitude of demographic stochasticity nor by the value of the diffusion coefficients.

WEAK NONLINEAR ANALYSIS OF THE NEOCLASSICAL GROWTH MODEL AT SPATIALLY HOMOGENEOUS CONDITIONS

The neoclassical growth model is being analyzed subject to spatially homogeneous perturbations by using the weak nonlinear method of solution and comparing its results to the numerical solution. The latter expands the analytical tools beyond the investigation of Turing instability. The results identify a Hopf bifurcation at a critical value of a controlling parameter, and their comparison to direct numerical solutions show an excellent match in the neighborhood of this critical value and for amplitudes of oscillations that are not too large.

LINEAR STABILITY ANALYSIS OF THE NEOCLASSICAL GROWTH MODEL TO SPATIALLY HOMOGENEOUS PERTURBATIONS

A linear stability analysis of the stationary solutions for growth of populations with respect to Spatially Homogeneous Perturbations (SHoP) is presented. The Neoclassical growth theory is extended to apply to spatially heterogeneous populations. The latter includes the metabolic mass transfer effects and allows for the recovery of substantial and distinct phenomena observed experimentally, such as the mechanism controlling the LAG phase, a result that holds impressive future potential in diverse applications. The stability conditions are expressed explicitly in terms of the primitive parameters of the original nonlinear system. The results are necessary when undertaking a corresponding linear stability analysis for growth of populations with respect to Spatially Heterogeneous Perturbations (SHeP).

Paradoxical persistence through mixed-system dynamics: towards a unified perspective of reversal behaviours in evolutionary ecology

Counterintuitive dynamics of various biological phenomena occur when composite system dynamics differ qualitatively from that of their component systems. Such composite systems typically arise when modelling situations with time-varying biotic or abiotic conditions, and examples range from metapopulation dynamics to population genetic models. These biological, and related physical, phenomena can often be modelled as simple financial games, wherein capital is gained and lost through gambling. Such games have been developed and used as heuristic devices to elucidate the processes at work in generating seemingly paradoxical outcomes across a spectrum of disciplines, albeit in a field-specific, ad hoc fashion. Here, we propose that studying these simple games can provide a much deeper understanding of the fundamental principles governing paradoxical behaviours in models from a diversity of topics in evolution and ecology in which fluctuating environmental effects, whether deterministic or stochastic, are an essential aspect of the phenomenon of interest. Of particular note, we find that, for a broad class of models, the ecological concept of equilibrium reactivity provides an intuitive necessary condition that must be satisfied in order for environmental variability to promote population persistence. We contend that further investigations along these lines promise to unify aspects of the study of a range of topics, bringing questions from genetics, species persistence and coexistence and the evolution of bet-hedging strategies, under a common theoretical purview.

Competition Between Transients in the Rate of Approach to a Fixed Point

The goal of this paper is to provide and apply tools for analyzing a specific aspect of transient dynamics not covered by previous theory. The question we address is whether one component of a perturbed solution to a system of differential equations can overtake the corresponding component of a reference solution as both converge to a stable node at the origin, given that the perturbed solution was initially farther away and that both solutions are nonnegative for all time. We call this phenomenon tolerance, for its relation to a biological effect. We show using geometric arguments that tolerance will exist in generic linear systems with a complete set of eigenvectors and in excitable nonlinear systems. We also define a notion of inhibition that may constrain the regions in phase space where the possibility of tolerance arises in general systems. However, these general existence theorems do not not yield an assessment of tolerance for specific initial conditions. To address that issue, we develop some analytical tools for determining if particular perturbed and reference solution initial conditions will exhibit tolerance.

Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB

We present two finite-difference algorithms for studying the dynamics of spatially extended predator-prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB: . Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator-prey interactions.

Interactions between pattern formation and domain growth

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.

Periodicity of cell attachment patterns during Escherichia coli biofilm development

The complex architecture of bacterial biofilms inevitably raises the question of their design. Microstructure of developing Escherichia coli biofilms was analyzed under static and laminar flow conditions. Cell attachment during early biofilm formation exhibited periodic density patterns that persisted during development. Several models for the origination of biofilm microstructure are considered, including an activator-inhibitor or Turing model.