Spatiotemporal control over self-assembly of supramolecular hydrogels through reaction-diffusion

Supramolecular self-assembly is ubiquitous in living system and is usually controlled to proceed in time and space through sophisticated reaction-diffusion processes, underpinning various vital cellular functions. In this contribution, we demonstrate how spatiotemporal self-assembly of supramolecular hydrogels can be realized through a simple reaction-diffusion-mediated transient transduction of pH signal. In the reaction-diffusion system, a relatively faster diffusion of acid followed by delayed enzymatic production and diffusion of base from the opposite site enables a transient transduction of pH signal in the substrate. By coupling such reaction-diffusion system with pH-sensitive gelators, dynamic supramolecular hydrogels with tunable lifetimes are formed at defined locations. The hydrogel fibers show interesting dynamic growing behaviors under the regulation of transient pH signal, reminiscent of their biological counterpart. We further demonstrate a proof-of-concept application of the developed methodology for dynamic information encoding in a soft substrate. We envision that this work may provide a potent approach to enable transient transduction of various chemical signals for the construction of new colloidal materials with the capability to evolve their structures and functionalities in time and space.

Modelling count data with partial differential equation models in biology

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth-death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction-diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

Discretised Flux Balance Analysis for Reaction-Diffusion Simulation of Single-Cell Metabolism

Metabolites have to diffuse within the sub-cellular compartments they occupy to specific locations where enzymes are, so reactions could occur. Conventional flux balance analysis (FBA), a method based on linear programming that is commonly used to model metabolism, implicitly assumes that all enzymatic reactions are not diffusion-limited though that may not always be the case. In this work, we have developed a spatial method that implements FBA on a grid-based system, to enable the exploration of diffusion effects on metabolism. Specifically, the method discretises a living cell into a two-dimensional grid, represents the metabolic reactions in each grid element as well as the diffusion of metabolites to and from neighbouring elements, and simulates the system as a single linear programming problem. We varied the number of rows and columns in the grid to simulate different cell shapes, and the method was able to capture diffusion effects at different shapes. We then used the method to simulate heterogeneous enzyme distribution, which suggested a theoretical effect on variability at the population level. We propose the use of this method, and its future extensions, to explore how spatiotemporal organisation of sub-cellular compartments and the molecules within could affect cell behaviour.

Novel Aspects in Pattern Formation Arise from Coupling Turing Reaction-Diffusion and Chemotaxis

Recent experimental studies on primary hair follicle formation and feather bud morphogenesis indicate a coupling between Turing-type diffusion driven instability and chemotactic patterning. Inspired by these findings we develop and analyse a mathematical model that couples chemotaxis to a reaction-diffusion system exhibiting diffusion-driven (Turing) instability. While both systems, reaction-diffusion systems and chemotaxis, can independently generate spatial patterns, we were interested in how the coupling impacts the stability of the system, parameter region for patterning, pattern geometry, as well as the dynamics of pattern formation. We conduct a classical linear stability analysis for different model structures, and confirm our results by numerical analysis of the system. Our results show that the coupling generally increases the robustness of the patterning process by enlarging the pattern region in the parameter space. Concerning time scale and pattern regularity, we find that an increase in the chemosensitivity can speed up the patterning process for parameters inside and outside of the Turing space, but generally reduces spatial regularity of the pattern. Interestingly, our analysis indicates that pattern formation can also occur when neither the Turing nor the chemotaxis system can independently generate pattern. On the other hand, for some parameter settings, the coupling of the two processes can extinguish the pattern formation, rather than reinforce it. These theoretical findings can be used to corroborate the biological findings on morphogenesis and guide future experimental studies. From a mathematical point of view, this work sheds a light on coupling classical pattern formation systems from the parameter space perspective.

A mathematical model for nutrient-limited uniaxial growth of a compressible tissue

We consider the uniaxial growth of a tissue or colony of cells, where a nutrient (or some other chemical) required for cell proliferation is supplied at one end, and is consumed by the cells. An example would be the growth of a cylindrical yeast colony in the experiments described by Vulin et al. (2014). We develop a reaction-diffusion model of this scenario which couples nutrient concentration and cell density on a growing domain. A novel element of our model is that the tissue is assumed to be compressible. We define replicative regions, where cells have sufficient nutrient to proliferate, and quiescent regions, where the nutrient level is insufficient for this to occur. We also define pathlines, which allow us to track individual cell paths within the tissue. We begin our investigation of the model by considering an incompressible tissue where cell density is constant before exploring the solution space of the full compressible model. In a large part of the parameter space, the incompressible and compressible models give qualitatively similar results for both the nutrient concentration and cell pathlines, with the key distinction being the variation in density in the compressible case. In particular, the replicative region is located at the base of the tissue, where nutrient is supplied, and nutrient concentration decreases monotonically with distance from the nutrient source. However, for a highly-compressible tissue with small nutrient consumption rate, we observe a counter-intuitive scenario where the nutrient concentration is not necessarily monotonically decreasing, and there can be two replicative regions. For parameter values given in the paper by Vulin et al. (2014), the incompressible model slightly overestimates the colony length compared to experimental observations; this suggests the colony may be somewhat compressible. Both incompressible and compressible models predict that, for these parameter values, cell proliferation is ultimately confined to a small region close to the colony base.

Fixed-time periodic stabilization of discontinuous reaction-diffusion Cohen-Grossberg neural networks

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction-diffusion Cohen-Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction-diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction-diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Stabilization of reaction-diffusion fractional-order memristive neural networks

This paper investigates the stabilization control of fractional-order memristive neural networks with reaction-diffusion terms. With regard to the reaction-diffusion model, a novel processing method based on Hardy-Poincarè inequality is introduced, as a result, the diffusion terms are estimated associated with the information of the reaction-diffusion coefficients and the regional feature, which may be beneficial to obtain conditions with less conservatism. Then, based on Kakutani's fixed point theorem of set-valued maps, new testable algebraic conclusion for ensuring the existence of the system's equilibrium point is obtained. Subsequently, by means of Lyapunov stability theory, it is concluded that the resulting stabilization error system is global asymptotic/Mittag-Leffler stable with a prescribed controller. Finally, an illustrative example about is provided to show the effectiveness of the established results.

Stability and pinning synchronization of delayed memristive neural networks with fractional-order and reaction-diffusion terms

Global asymptotic stability and synchronization are explored in this paper for fractional delayed memristive neural networks with reaction-diffusion terms (FDRDMNNs) in sense of Riemann-Liouville. First, we introduce diffusion into the existing model of fractional delayed memristive neural networks. Next, in terms of Green's theorem and inequality technique, a less conservative criterion for the asymptotic stability of FDRDMNNs is given by endowing Lyapunov direct method. Then, the appropriate pinning feedback controllers and adaptive controllers are designed to achieve the synchronization of the FDRDMNNs, and two sufficient conditions for global asymptotic synchronization are acquired. In addition, the results based on algebraic inequalities enhance some existing ones. The numerical simulations finally verify the validity of the derived results.

conditions for Turing and wave instabilities in reaction-diffusion systems

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with n components and diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to [Formula: see text], gives a constructive method for choosing diffusion constants. The results are illustrated on a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components.

Heterogeneity in Behaviour and Movement can Influence the Stability of Predator-Prey Periodic Travelling Waves

Cyclic predator-prey systems are often observed in nature. In a spatial setting, these can manifest as periodic traveling waves (PTW). Environmental change and direct human activity are known to, among other effects, increase the heterogeneity of the physical environment, which prey and predator inhabit. Aiming to understand the effects of heterogeneity on predator-prey PTWs, we consider a one-dimensional infinite landscape Rosenzweig-MacArthur reaction-diffusion model, with alternating patch types, and study the PTWs in this system. Applying the method of homogenisation, we show how heterogeneity can affect the stability of PTW solutions. We illustrate how the effects of heterogeneity can be understood and interpreted using Turchin's concept of residence index (encapsuling diffusion rate and patch preference). In particular, our results show that prey heterogeneity acts to modulate the effects of predator heterogeneity, by this we mean that as prey increasingly spend more time in one patch type over another the stability of the PTWs becomes more sensitive to heterogeneity in predator movement and behaviour.

A theory of mechanical stress-induced H(2)O(2) signaling waveforms in Planta

Recent progress in nanotechnology-enabled sensors that can be placed inside of living plants has shown that it is possible to relay and record real-time chemical signaling stimulated by various abiotic and biotic stresses. The mathematical form of the resulting local reactive oxygen species (ROS) wave released upon mechanical perturbation of plant leaves appears to be conserved across a large number of species, and produces a distinct waveform from other stresses including light, heat and pathogen-associated molecular pattern (PAMP)-induced stresses. Herein, we develop a quantitative theory of the local ROS signaling waveform resulting from mechanical stress in planta. We show that nonlinear, autocatalytic production and Fickian diffusion of H₂O₂ followed by first order decay well describes the spatial and temporal properties of the waveform. The reaction-diffusion system is analyzed in terms of a new approximate solution that we introduce for such problems based on a single term logistic function ansatz. The theory is able to describe experimental ROS waveforms and degradation dynamics such that species-dependent dimensionless wave velocities are revealed, corresponding to subtle changes in higher moments of the waveform through an apparently conserved signaling mechanism overall. This theory has utility in potentially decoding other stress signaling waveforms for light, heat and PAMP-induced stresses that are similarly under investigation. The approximate solution may also find use in applied agricultural sensing, facilitating the connection between measured waveform and plant physiology.

Event-triggered H∞/passive synchronization for Markov jumping reaction-diffusion neural networks under deception attacks

The issue of H_∞/passive master-slave synchronization for Markov jumping neural networks with reaction-diffusion terms is investigated in this paper via an event-triggered control scheme under deception attacks. To lighten the burden of limited communication bandwidth as well as ensure the control performance, an event-triggered transmission scheme is developed. Meanwhile, the randomly occurring deception attacks, which received from the event generator are assumed to modify the sign of the control signal, are taken into account. Furthermore, sufficient conditions ensuring the prescribed H_∞/passive performance level of the neural networks, are deduced beyond Lyapunov stability theory, and the controller gains are derived dealing with the matrix convex optimization problem. At last, the availability of the approach proposed is demonstrated via a numerical example.

Optimal control of pattern formations for an SIR reaction-diffusion epidemic model

Patterns arising from the reaction-diffusion epidemic model provide insightful aspects into the transmission of infectious diseases. For a classic SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates. A quantitative indicator, "normal serious prevalent area (NSPA)", is introduced to characterize the relationship between patterns and the extent of the epidemic. The extent of epidemic is positively correlated to NSPA. To effectively reduce NSPA of patterns under the large transmission rates, taken removed (recovery or isolation) rate as a control parameter, we consider the mathematical formulation and numerical solution of an optimal control problem for the SIR reaction-diffusion model. Numerical experiments demonstrate the effectiveness of our method in terms of control effect, control precision and control cost.

Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion

In view of the facts in the infection and propagation of COVID-19, a stochastic reaction-diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.

Modeling Plant Tissue Development Using VirtualLeaf

Cell-based computational modeling and simulation are becoming invaluable tools in analyzing plant development. In a cell-based simulation model, the inputs are behaviors and dynamics of individual cells and the rules describing responses to signals from adjacent cells. The outputs are the growing tissues, shapes, and cell-differentiation patterns that emerge from the local, chemical, and biomechanical cell-cell interactions. In this updated and extended version of our previous chapter on VirtualLeaf (Merks and Guravage, Methods in Molecular Biology 959, 333-352), we present a step-by-step, practical tutorial for building cell-based simulations of plant development and for analyzing the influence of parameters on simulation outcomes by systematically changing the values of the parameters and analyzing each outcome. We show how to build a model of a growing tissue, a reaction-diffusion system on a growing domain, and an auxin transport model. Moreover, in addition to the previous publication, we demonstrate how to run a Turing system on a regular, rectangular lattice, and how to run parameter sweeps. The aim of VirtualLeaf is to make computational modeling more accessible to experimental plant biologists with relatively little computational background.

Control Strategies for a Multi-strain Epidemic Model

This article studies a multi-strain epidemic model with diffusion and environmental heterogeneity. We address the question of a control strategy for multiple strains of the infectious disease by investigating how the local distributions of the transmission and recovery rates affect the dynamics of the disease. Our study covers both full model (in which case the diffusion rates for all subgroups of the population are positive) and the ODE–PDE case (in which case we require a total lock-down of the susceptible subgroup and allow the infected subgroups to have positive diffusion rates). In each case, a basic reproduction number of the epidemic model is defined and it is shown that if this reproduction number is less than one then the disease will be eradicated in the long run. On the other hand, if the reproduction number is greater than one, then the disease will become permanent. Moreover, we show that when the disease is permanent, creating a common safety area against all strains and lowering the diffusion rate of the susceptible subgroup will result in reducing the number of infected populations. Numerical simulations are presented to support our theoretical findings.

Dynamics of a reaction-diffusion SIRS model with general incidence rate in a heterogeneous environment

In this paper, we study a diffusive SIRS-type epidemic model with transfer from the infectious to the susceptible class. Our model includes a general nonlinear incidence rate and spatially heterogeneous diffusion coefficients. We compute the basic reproduction number R 0 of our model and establish the global stability of the disease-free steady state when R 0 < 1 . Furthermore, we study the uniform persistence when R 0 > 1 and perform a bifurcation analysis for a special case of our model. Some numerical simulations are presented to illustrate the dynamics of the solutions as the model parameters are varied.

Hive geometry shapes the recruitment rate of honeybee colonies

Honey bees make decisions regarding foraging and nest-site selection in groups ranging from hundreds to thousands of individuals. To effectively make these decisions, bees need to communicate within a spatially distributed group. However, the spatiotemporal dynamics of honey bee communication have been mostly overlooked in models of collective decisions, focusing primarily on mean field models of opinion dynamics. We analyze how the spatial properties of the nest or hive, and the movement of individuals with different belief states (uncommitted or committed) therein affect the rate of information transmission using spatially-extended models of collective decision-making within a hive. Honeybees waggle-dance to recruit conspecifics with an intensity that is a threshold nonlinear function of the waggler concentration. Our models range from treating the hive as a chain of discrete patches to a continuous line (long narrow hive). The combination of population-thresholded recruitment and compartmentalized populations generates tradeoffs between rapid information propagation with strong population dispersal and recruitment failures resulting from excessive population diffusion and also creates an effective colony-level signal-detection mechanism whereby recruitment to low quality objectives is blocked.

Numerical simulation and stability analysis of a novel reaction-diffusion COVID-19 model

In this study, a novel reaction-diffusion model for the spread of the new coronavirus (COVID-19) is investigated. The model is a spatial extension of the recent COVID-19 SEIR model with nonlinear incidence rates by taking into account the effects of random movements of individuals from different compartments in their environments. The equilibrium points of the new system are found for both diffusive and non-diffusive models, where a detailed stability analysis is conducted for them. Moreover, the stability regions in the space of parameters are attained for each equilibrium point for both cases of the model and the effects of parameters are explored. A numerical verification for the proposed model using a finite difference-based method is illustrated along with their consistency, stability and proving the positivity of the acquired solutions. The obtained results reveal that the random motion of individuals has significant impact on the observed dynamics and steady-state stability of the spread of the virus which helps in presenting some strategies for the better control of it.

Parameter estimation in fluorescence recovery after photobleaching: quantitative analysis of protein binding reactions and diffusion

Fluorescence recovery after photobleaching (FRAP) is a common experimental method for investigating rates of molecular redistribution in biological systems. Many mathematical models of FRAP have been developed, the purpose of which is usually the estimation of certain biological parameters such as the diffusivity and chemical reaction rates of a protein, this being accomplished by fitting the model to experimental data. In this article, we consider a two species reaction–diffusion FRAP model. Using asymptotic analysis, we derive new FRAP recovery curve approximation formulae, and formally re-derive existing ones. On the basis of these formulae, invoking the concept of Fisher information, we predict, in terms of biological and experimental parameters, sufficient conditions to ensure that the values all model parameters can be estimated from data. We verify our predictions with extensive computational simulations. We also use computational methods to investigate cases in which some or all biological parameters are theoretically inestimable. In these cases, we propose methods which can be used to extract the maximum possible amount of information from the FRAP data.

A reaction-diffusion model of spatial propagation of A[Formula: see text] oligomers in early stage Alzheimer's disease

The misconformation and aggregation of the protein Amyloid-Beta (A[Formula: see text]) is a key event in the propagation of Alzheimer's Disease (AD). Different types of assemblies are identified, with long fibrils and plaques deposing during the late stages of AD. In the earlier stages, the disease spread is driven by the formation and the spatial propagation of small amorphous assemblies called oligomers. We propose a model dedicated to studying those early stages, in the vicinity of a few neurons and after a polymer seed has been formed. We build a reaction-diffusion model, with a Becker-Döring-like system that includes fragmentation and size-dependent diffusion. We hereby establish the theoretical framework necessary for the proper use of this model, by proving the existence of solutions using a fixed point method.

Analysis of a two-strain malaria transmission model with spatial heterogeneity and vector-bias

In this paper, we introduce a reaction-diffusion malaria model which incorporates vector-bias, spatial heterogeneity, sensitive and resistant strains. The main question that we study is the threshold dynamics of the model, in particular, whether the existence of spatial structure would allow two strains to coexist. In order to achieve this goal, we define the basic reproduction number [Formula: see text] and introduce the invasion reproduction number [Formula: see text] for strain [Formula: see text]. A quantitative analysis shows that if [Formula: see text], then disease-free steady state is globally asymptotically stable, while competitive exclusion, where strain i persists and strain j dies out, is a possible outcome when [Formula: see text] [Formula: see text], and a unique solution with two strains coexist to the model is globally asymptotically stable if [Formula: see text], [Formula: see text]. Numerical simulations reinforce these analytical results and demonstrate epidemiological interaction between two strains, discuss the influence of resistant strains and study the effects of vector-bias on the transmission of malaria.

Invading and Receding Sharp-Fronted Travelling Waves

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.

Mathematical Modeling of Dynamic Cellular Association Patterns in Seminiferous Tubules

In vertebrates, sperm is generated in testicular tube-like structures called seminiferous tubules. The differentiation stages of spermatogenesis exhibit a dynamic spatiotemporal wavetrain pattern. There are two types of pattern-the vertical type, which is observed in mice, and the helical type, which is observed in humans. The mechanisms of this pattern difference remain little understood. In the present study, we used a three-species reaction-diffusion model to reproduce the wavetrain pattern observed in vivo. We hypothesized that the wavelength of the pattern in mice was larger than that in humans and undertook numerical simulations. We found complex patterns of helical and vertical pattern frequency, which can be understood by pattern selection using boundary conditions. From these theoretical results, we predicted that a small number of vertical patterns should be present in human seminiferous tubules. We then found vertical patterns in histological sections of human tubules, consistent with the theoretical prediction. Finally, we showed that the previously reported irregularity of the human pattern could be reproduced using two factors: a wider unstable wavenumber range and the irregular geometry of human compared with mouse seminiferous tubules. These results show that mathematical modeling is useful for understanding the pattern dynamics of seminiferous tubules in vivo.

Estimating the extent of glioblastoma invasion : Approximate stationalization of anisotropic advection-diffusion-reaction equations in the context of glioblastoma invasion

Glioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection–diffusion–reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalized approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalization is introduced via an analytic solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset.

Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.

Spatial modeling and dynamics of organic matter biodegradation in the absence or presence of bacterivorous grazing

Biodegradation is a pivotal natural process for elemental recycling and preservation of an ecosystem. Mechanistic modeling of biodegradation has to keep track of chemical elements via stoichiometric theory, under which we propose and analyze a spatial movement model in the absence or presence of bacterivorous grazing. Sensitivity analysis shows that the organic matter degradation rate is most sensitive to the grazer's death rate when the grazer is present and most sensitive to the bacterial death rate when the grazer is absent. Therefore, these two death rates are chosen as the primary parameters in the conditions of most mathematical theorems. The existence, stability and persistence of solutions are proven by applying linear stability analysis, local and global bifurcation theory, and the abstract persistence theory. Through numerical simulations, we obtain the transient and asymptotic dynamics and explore the effects of all parameters on the organic matter decomposition. Grazers either facilitate biodegradation or has no impact on biodegradation, which resolves the "decomposition-facilitation paradox" in the spatial context.

Effective nonlocal kernels on reaction-diffusion networks

A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called "effective equation" including the reduced integral kernel (called "effective kernel") in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations.

Coloured Noise from Stochastic Inflows in Reaction-Diffusion Systems

In this paper, we present a framework for investigating coloured noise in reaction-diffusion systems. We start by considering a deterministic reaction-diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady-state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models, we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases, we show that the power spectrum of the coloured noise process and the power spectrum of the reaction-diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction-diffusion system.

Mix and Match: Phenotypic Coexistence as a Key Facilitator of Cancer Invasion

Invasion of healthy tissue is a defining feature of malignant tumours. Traditionally, invasion is thought to be driven by cells that have acquired all the necessary traits to overcome the range of biological and physical defences employed by the body. However, in light of the ever-increasing evidence for geno- and phenotypic intra-tumour heterogeneity, an alternative hypothesis presents itself: could invasion be driven by a collection of cells with distinct traits that together facilitate the invasion process? In this paper, we use a mathematical model to assess the feasibility of this hypothesis in the context of acid-mediated invasion. We assume tumour expansion is obstructed by stroma which inhibits growth and extra-cellular matrix (ECM) which blocks cancer cell movement. Further, we assume that there are two types of cancer cells: (i) a glycolytic phenotype which produces acid that kills stromal cells and (ii) a matrix-degrading phenotype that locally remodels the ECM. We extend the Gatenby-Gawlinski reaction-diffusion model to derive a system of five coupled reaction-diffusion equations to describe the resulting invasion process. We characterise the spatially homogeneous steady states and carry out a simulation study in one spatial dimension to determine how the tumour develops as we vary the strength of competition between the two phenotypes. We find that overall tumour growth is most extensive when both cell types can stably coexist, since this allows the cells to locally mix and benefit most from the combination of traits. In contrast, when inter-species competition exceeds intra-species competition the populations spatially separate and invasion arrests either: (i) rapidly (matrix-degraders dominate) or (ii) slowly (acid-producers dominate). Overall, our work demonstrates that the spatial and ecological relationship between a heterogeneous population of tumour cells is a key factor in determining their ability to cooperate. Specifically, we predict that tumours in which different phenotypes coexist stably are more invasive than tumours in which phenotypes are spatially separated.

Model of pattern formation in marsh ecosystems with nonlocal interactions

Smooth cordgrass Spartina alterniflora is a grass species commonly found in tidal marshes. It is an ecosystem engineer, capable of modifying the structure of its surrounding environment through various feedbacks. The scale-dependent feedback between marsh grass and sediment volume is particularly of interest. Locally, the marsh vegetation attenuates hydrodynamic energy, enhancing sediment accretion and promoting further vegetation growth. In turn, the diverted water flow promotes the formation of erosion troughs over longer distances. This scale-dependent feedback may explain the characteristic spatially varying marsh shoreline, commonly observed in nature. We propose a mathematical framework to model grass-sediment dynamics as a system of reaction-diffusion equations with an additional nonlocal term quantifying the short-range positive and long-range negative grass-sediment interactions. We use a Mexican-hat kernel function to model this scale-dependent feedback. We perform a steady state biharmonic approximation of our system and derive conditions for the emergence of spatial patterns, corresponding to a spatially varying marsh shoreline. We find that the emergence of such patterns depends on the spatial scale and strength of the scale-dependent feedback, specified by the width and amplitude of the Mexican-hat kernel function.

Conserved gene signalling and a derived patterning mechanism underlie the development of avian footpad scales

Background

Vertebrates possess a diverse range of integumentary epithelial appendages, including scales, feathers and hair. These structures share extensive early developmental homology, as they mostly originate from a conserved anatomical placode. In the context of avian epithelial appendages, feathers and scutate scales are known to develop from an anatomical placode. However, our understanding of avian reticulate (footpad) scale development remains unclear.

Results

Here, we demonstrate that reticulate scales develop from restricted circular domains of thickened epithelium, with localised conserved gene expression in both the epithelium and underlying mesenchyme. These domains constitute either anatomical placodes, or circular initiatory fields (comparable to the avian feather tract). Subsequent patterning of reticulate scales is consistent with reaction–diffusion (RD) simulation, whereby this primary domain subdivides into smaller secondary units, which produce individual scales. In contrast, the footpad scales of a squamate model (the bearded dragon, Pogona vitticeps) develop synchronously across the ventral footpad surface.

Conclusions

Widely conserved gene signalling underlies the initial development of avian reticulate scales. However, their subsequent patterning is distinct from the footpad scale patterning of a squamate model, and the feather and scutate scale patterning of birds. Therefore, we suggest reticulate scales are a comparatively derived epithelial appendage, patterned through a modified RD system.

Electronic supplementary material

The online version of this article (10.1186/s13227-019-0130-9) contains supplementary material, which is available to authorized users.

Reaction-diffusion approach to modeling water diffusion in glutinous rice flour particles during dynamic vapor adsorption

Dynamic vapor sorption (DVS) method is now widely adopted to determine water diffusion properties in food materials using a sheet or bulk particles as test samples. The Fickian second law with an instant equilibrium boundary condition, although commonly used, can not accurately model the water adsorption kinetics during DVS tests. Dynamic water adsorption of glutinous rice flour was measured at 30 °C and eleven relative humidity steps, and modeled using the Fickian second law with three kinds of boundary condition. Results indicated that the boundary conditions had great impacts on the predicted values especially for the initial section in the DVS curves, and that modifying boundary conditions could not improve the fitness of the final section which characterized a continuing slight increase of water concentration. A reaction-diffusion model, which assumes two diffusible water populations and describes water transport as a competition between diffusion and reversible adsorption on solid matrix, was developed and found to be able to capture the features of water diffusion in the whole adsorption duration. Implementation of the reaction-diffusion approach to glutinous rice flour indicated that diffusion of the Langmuir water became very slow when its adsorption reached equilibrium, while the diffusion of the non-Langmuir water slowed down when water clustering occurred, at the same time the rates of the surface adsorption and bulk adsorption began to decrease. The model developed in this work would help to deepen our mechanistic understanding of water diffusion during a isothermal adsorption.

Dating and localizing an invasion from post-introduction data and a coupled reaction-diffusion-absorption model

Invasion of new territories by alien organisms is of primary concern for environmental and health agencies and has been a core topic in mathematical modeling, in particular in the intents of reconstructing the past dynamics of the alien organisms and predicting their future spatial extents. Partial differential equations offer a rich and flexible modeling framework that has been applied to a large number of invasions. In this article, we are specifically interested in dating and localizing the introduction that led to an invasion using mathematical modeling, post-introduction data and an adequate statistical inference procedure. We adopt a mechanistic-statistical approach grounded on a coupled reaction-diffusion-absorption model representing the dynamics of an organism in an heterogeneous domain with respect to growth. Initial conditions (including the date and site of the introduction) and model parameters related to diffusion, reproduction and mortality are jointly estimated in the Bayesian framework by using an adaptive importance sampling algorithm. This framework is applied to the invasion of Xylella fastidiosa, a phytopathogenic bacterium detected in South Corsica in 2015, France.

Bayesian Parameter Identification for Turing Systems on Stationary and Evolving Domains

In this study, we apply the Bayesian paradigm for parameter identification to a well-studied semi-linear reaction-diffusion system with activator-depleted reaction kinetics, posed on stationary as well as evolving domains. We provide a mathematically rigorous framework to study the inverse problem of finding the parameters of a reaction-diffusion system given a final spatial pattern. On the stationary domain the parameters are finite-dimensional, but on the evolving domain we consider the problem of identifying the evolution of the domain, i.e. a time-dependent function. Whilst others have considered these inverse problems using optimisation techniques, the Bayesian approach provides a rigorous mathematical framework for incorporating the prior knowledge on uncertainty in the observation and in the parameters themselves, resulting in an approximation of the full probability distribution for the parameters, given the data. Furthermore, using previously established results, we can prove well-posedness results for the inverse problem, using the well-posedness of the forward problem. Although the numerical approximation of the full probability is computationally expensive, parallelised algorithms make the problem solvable using high-performance computing.

Permanence and Extinction of a Diffusive Predator-Prey Model with Robin Boundary Conditions

The main concern of this paper is to study the dynamic of a predator-prey system with diffusion. It incorporates the Holling-type-II and a modified Leslie-Gower functional responses under Robin boundary conditions. More concretely, we study the dissipativeness of the system by using the comparison principle, and we derive a criteria for permanence and for predator extinction.

Dynamics of dengue disease with human and vector mobility

Dengue is a vector borne disease transmitted to humans by Aedes aegypti mosquitoes carrying virus of different serotypes. Dengue exhibits complex spatial and temporal dynamics, influenced by various biological, human and environmental factors. In this work, we study the dengue spread for a single serotype (DENV-1) including statistical models of human mobility with exponential step length distribution, by using reaction-diffusion equations and Stochastic Cellular Automata (SCA) approach. We analyze the spatial and temporal spreading of the disease using parameters from field studies. We choose mosquito density data from Ahmedabad city as a proxy for climate data in our SCA model. We find an interesting result that although human mobility makes the infection spread faster, there is an apparent early suppression of the epidemic compared to immobile humans. The disease extinction time is lesser when human mobility is included.

Mechanisms and Measurements of Scale Invariance of Morphogen Gradients

Morphogen gradients provide positional information to underlying cells that translate the information into differential gene expression and eventually different cell fates. Scale invariance is the property where the gradients of the morphogen adjust proportionately to the size of the domain. Scale invariance of morphogen gradients or patterns of differentiation is a common phenomenon observed between individuals within the same species and between homologous tissues or structures in different species. To determine whether or not a pattern is scale invariant, others and we have developed definitions and measurements of gradient scaling. These include point-wise and global scaling errors as well as global scaling power. Furthermore, there are a number of mathematical conditions for scale invariance of advection-diffusion-reaction models that inform mechanisms of scaling. Herein we provide a deeper perspective on modeling and measurement of scale invariance of morphogen gradients.

Slow Passage Through a Hopf Bifurcation in Excitable Nerve Cables: Spatial Delays and Spatial Memory Effects

It is well established that in problems featuring slow passage through a Hopf bifurcation (dynamic Hopf bifurcation) the transition to large-amplitude oscillations may not occur until the slowly changing parameter considerably exceeds the value predicted from the static Hopf bifurcation analysis (temporal delay effect), with the length of the delay depending upon the initial value of the slowly changing parameter (temporal memory effect). In this paper we introduce new delay and memory effect phenomena using both analytic (WKB method) and numerical methods. We present a reaction-diffusion system for which slowly ramping a stimulus parameter (injected current) through a Hopf bifurcation elicits large-amplitude oscillations confined to a location a significant distance from the injection site (spatial delay effect). Furthermore, if the initial current value changes, this location may change (spatial memory effect). Our reaction-diffusion system is Baer and Rinzel's continuum model of a spiny dendritic cable; this system consists of a passive dendritic cable weakly coupled to excitable dendritic spines. We compare results for this system with those for nerve cable models in which there is stronger coupling between the reactive and diffusive portions of the system. Finally, we show mathematically that Hodgkin and Huxley were correct in their assertion that for a sufficiently slow current ramp and a sufficiently large cable length, no value of injected current would cause their model of an excitable cable to fire; we call this phenomenon "complete accommodation."

A Reliable Spectral Method to Reaction-Diffusion Equations in Entrapped-Cell Photobioreactor Packed with Gel Granules Using Chebyshev Wavelets

A mathematical model of a reaction-diffusion within an entrapped-cell photobioreactor packed with gel granules containing immobilized photosynthetic bacterial cells is discussed. A theoretical model is based on a system of coupled nonlinear reaction-diffusion equations. In this research work, we have developed an efficient wavelet-based spectral approach for solving the proposed model. Analytical expressions for the concentration of substrate and product are established for all values of reaction-diffusion parameters using second kind Chebyshev wavelet method. The analytical results were also compared with Homotopy perturbation method (HPM) and Adomian decomposition method (ADM). Satisfactory agreement with ADM and HPM solutions is observed. Moreover, the use of Chebyshev wavelets is found to be simple, reliable, efficient, and computationally attractive.

Simulating PDGF-Driven Glioma Growth and Invasion in an Anatomically Accurate Brain Domain

Gliomas are the most common of all primary brain tumors. They are characterized by their diffuse infiltration of the brain tissue and are uniformly fatal, with glioblastoma being the most aggressive form of the disease. In recent years, the over-expression of platelet-derived growth factor (PDGF) has been shown to produce tumors in experimental rodent models that closely resemble this human disease, specifically the proneural subtype of glioblastoma. We have previously modeled this system, focusing on the key attribute of these experimental tumors-the "recruitment" of oligodendroglial progenitor cells (OPCs) to participate in tumor formation by PDGF-expressing retrovirally transduced cells-in one dimension, with spherical symmetry. However, it has been observed that these recruitable progenitor cells are not uniformly distributed throughout the brain and that tumor cells migrate at different rates depending on the material properties in different regions of the brain. Here we model the differential diffusion of PDGF-expressing and recruited cell populations via a system of partial differential equations with spatially variable diffusion coefficients and solve the equations in two spatial dimensions on a mouse brain atlas using a flux-differencing numerical approach. Simulations of our in silico model demonstrate qualitative agreement with the observed tumor distribution in the experimental animal system. Additionally, we show that while there are higher concentrations of OPCs in white matter, the level of recruitment of these plays little role in the appearance of "white matter disease," where the tumor shows a preponderance for white matter. Instead, simulations show that this is largely driven by the ratio of the diffusion rate in white matter as compared to gray. However, this ratio has less effect on the speed of tumor growth than does the degree of OPC recruitment in the tumor. It was observed that tumor simulations with greater degrees of recruitment grow faster and develop more nodular tumors than if there is no recruitment at all, similar to our prior results from implementing our model in one dimension. Combined, these results show that recruitment remains an important consideration in understanding and slowing glioma growth.

The possibility of coexistence and co-development in language competition: ecology-society computational model and simulation

Language is characterized by both ecological properties and social properties, and competition is the basic form of language evolution. The rise and decline of one language is a result of competition between languages. Moreover, this rise and decline directly influences the diversity of human culture. Mathematics and computer modeling for language competition has been a popular topic in the fields of linguistics, mathematics, computer science, ecology, and other disciplines. Currently, there are several problems in the research on language competition modeling. First, comprehensive mathematical analysis is absent in most studies of language competition models. Next, most language competition models are based on the assumption that one language in the model is stronger than the other. These studies tend to ignore cases where there is a balance of power in the competition. The competition between two well-matched languages is more practical, because it can facilitate the co-development of two languages. A third issue with current studies is that many studies have an evolution result where the weaker language inevitably goes extinct. From the integrated point of view of ecology and sociology, this paper improves the Lotka–Volterra model and basic reaction–diffusion model to propose an “ecology–society” computational model for describing language competition. Furthermore, a strict and comprehensive mathematical analysis was made for the stability of the equilibria. Two languages in competition may be either well-matched or greatly different in strength, which was reflected in the experimental design. The results revealed that language coexistence, and even co-development, are likely to occur during language competition.

On the importance of modelling the internal spatial dynamics of biological cells

Spatial effects such as cell shape have very often been considered negligible in models of cellular pathways, and many existing simulation infrastructures do not take such effects into consideration. Recent experimental results are reversing this judgement by showing that very small spatial variations can make a big difference in the fate of a cell. This is particularly the case when considering eukaryotic cells, which have a complex physical structure and many subtle control mechanisms, but bacteria are also interesting for the huge variation in shape both between species and in different phases of their lifecycle. In this work we perform simulations that measure the effect of three common bacterial shapes on the behaviour of model cellular pathways. To perform these experiments we develop ReDi-Cell, a highly scalable GPGPU cell simulation infrastructure for the modelling of cellular pathways in spatially detailed environments. ReDi-Cell is validated against known-good simulations, prior to its use in new work. We then use ReDi-Cell to conduct novel experiments that demonstrate the effect that three common bacterial shapes (Cocci, Bacilli and Spirilli) have on the behaviour of model cellular pathways. Pathway wavefront shape, pathway concentration gradients, and chemical species distribution are measured in the three different shapes. We also quantify the impact of internal cellular clutter on the same pathways. Through this work we show that variations in the shape or configuration of these common cell shapes alter model cell behaviour.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

Synchronized bifurcation and stability in a ring of diffusively coupled neurons with time delay

In this study, we consider a ring of diffusively coupled neurons with distributed and discrete delays. We investigate the synchronized stability and synchronized Hopf bifurcation of this system, as well as deriving some criteria by analyzing the associated characteristic transcendental equation and by taking τ and β as the bifurcation parameters, which are parameters that measure the discrete delay and the strength of nearest-neighbor connection, respectively. Our simulations demonstrated that the numerically observed behaviors were in excellent agreement with the theoretically predicted results. In addition, using numerically simulations, we investigated the effects of τ and β, as well as the diffusion on dynamic behavior. Our numerical results showed that the addition diffusion to a stable delay-differential equation (DDE) system may make it unstable and that the diffusion may make the system synchronous, whereas it is asynchronous without the diffusion term.

Gene drive through a landscape: Reaction-diffusion models of population suppression and elimination by a sex ratio distorter

Some genes or gene complexes are transmitted from parents to offspring at a greater-than-Mendelian rate, and can spread and persist in populations even if they cause some harm to the individuals carrying them. Such genes may be useful for controlling populations or species that are harmful. Driving-Y chromosomes may be particularly potent in this regard, as they produce a male-biased sex ratio that, if sufficiently extreme, can lead to population elimination. To better understand the potential of such genes to spread over a landscape, we have developed a series of reaction-diffusion models of a driving-Y chromosome in 1-D and radially-symmetric 2-D unbounded domains. The wild-type system at carrying capacity is found to be unstable to the introduction of driving-Y males for all models investigated. Numerical solutions exhibit travelling wave pulses and fronts, and analytical and semi-analytical solutions for the asymptotic wave speed under bounded initial conditions are derived. The driving-Y male invades the wild-type equilibrium state at the front of the wave and completely replaces the wild-type males, leaving behind, at the tail of the wave, a reduced- or zero-population state of females and driving-Y males only. In our simplest model of a population with one life stage and density-dependent mortality, wave speed depends on the strength of drive and the diffusion rate of Y-drive males, and is independent of the population dynamic consequences (suppression or elimination). Incorporating an immobile juvenile stage of fixed duration into the model reduces wave speed approximately in proportion to the relative time spent as a juvenile. If females mate just once in their life, storing sperm for subsequent reproduction, then wave speed depends on the movement of mated females as well as Y-drive males, and may be faster or slower than in the multiple-mating model, depending on the relative duration of juvenile and adult life stages. Numerical solutions are shown for parameter values that may in part be representative for Anopheles gambiae, the primary vector of malaria in sub-Saharan Africa.

Global sensitivity analysis of a dynamic model for gene expression in Drosophila embryos

It is well known that gene regulation is a tightly controlled process in early organismal development. However, the roles of key processes involved in this regulation, such as transcription and translation, are less well understood, and mathematical modeling approaches in this field are still in their infancy. In recent studies, biologists have taken precise measurements of protein and mRNA abundance to determine the relative contributions of key factors involved in regulating protein levels in mammalian cells. We now approach this question from a mathematical modeling perspective. In this study, we use a simple dynamic mathematical model that incorporates terms representing transcription, translation, mRNA and protein decay, and diffusion in an early Drosophila embryo. We perform global sensitivity analyses on this model using various different initial conditions and spatial and temporal outputs. Our results indicate that transcription and translation are often the key parameters to determine protein abundance. This observation is in close agreement with the experimental results from mammalian cells for various initial conditions at particular time points, suggesting that a simple dynamic model can capture the qualitative behavior of a gene. Additionally, we find that parameter sensitivites are temporally dynamic, illustrating the importance of conducting a thorough global sensitivity analysis across multiple time points when analyzing mathematical models of gene regulation.

Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton

The production of toxins by some species of phytoplankton is known to have several economic, ecological, and human health impacts. However, the role of toxins on the spatial distribution of phytoplankton is not well understood. In the present study, the spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton is investigated. We analyze the linear stability of the system and obtain the condition for Turing instability. In the presence of toxic effect, we find that the distribution of nutrient and phytoplankton becomes inhomogeneous in space and results in different patterns, like stripes, spots, and the mixture of them depending on the toxicity level. We also observe that the distribution of nutrient and phytoplankton shows spatiotemporal oscillation for certain toxicity level.

The stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays and reaction-diffusion terms

The global asymptotic stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays and reaction-diffusion terms is investigated. Under some suitable assumptions and using Lyapunov-Krasovskii functional method, we apply the linear matrix inequality technique to propose some new sufficient conditions for the global asymptotic stability of the addressed model in the stochastic sense. The mixed time delays comprise both the time-varying and continuously distributed delays. The effectiveness of the theoretical result is illustrated by a numerical example.

Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters

This paper deals with a delayed reaction-diffusion three-species Lotka-Volterra model with interval biological parameters and harvesting. Sufficient conditions for the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Then an optimal control problem has been considered. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical evidence shows that the presence of harvesting can impact the existence of species and over harvesting can result in the extinction of the prey or the predator which is in line with reality.