Meinhardt Model Research Articles

Stability and bifurcation diagram for a shadow Gierer–Meinhardt system in one spatial dimension

We are concerned with a Neumann problem of a shadow system of the Gierer–Meinhardt model in an interval I=(0,1) . A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each n⩾1 , two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, n⩾2 , are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.

Bogdanov–Takens Bifurcation of Codimension 3 in the Gierer–Meinhardt Model

Bogdanov–Takens Bifurcation of Codimension 3 in the Gierer–Meinhardt Model

3D-Printed Bioreceptive Tiles of Reaction–Diffusion (Gierer–Meinhardt Model) for Multi-Scale Algal Strains’ Passive Immobilization

The current architecture practice is shifting towards Green Solutions designed, produced, and operated domestically in a self-sufficient decentralized fashion, following the UN sustainability goals. The current study proposes 3D-printed bioreceptive tiles for the passive immobilization of multi-scale-length algal strains from a mixed culture of Mougeotia sp., Oedogonium foveolatum, Zygnema sp., Microspora sp., Spirogyra sp., and Pyrocystis fusiformis. This customized passive immobilization of the chosen algal strains is designed to achieve bioremediation-integrated solutions in architectural applications. The two bioreceptive tiles following the reaction-diffusion, activator-inhibitor Grier–Meinhardt model have different patterns: P1: Polar periodic, and P2: Strip labyrinth, with niche sizes of 3000 µm and 500 µm, respectively. The results revealed that P2 has a higher immobilization capacity for the various strains, particularly Microspora sp., achieving a growth rate 1.65% higher than its activated culture density compared to a 1.08% growth rate on P1, followed by P. fusiformis with 1.53% on P2 and 1.3% on P1. These results prove the correspondence between the scale and morphology of the strip labyrinth pattern of P2 and the unbranched filamentous and fusiform large unicellular morphology of the immobilized algal strains cells, with an optimum ratio of 0.05% to 0.75% niche to the cell scale. Furthermore, The Mixed Culture method offered an intertwining net that facilitated the entrapment of the various algal strains into the bioreceptive tile.

Bifurcations and Turing patterns in a diffusive Gierer–Meinhardt model

In this paper, the Hopf bifurcations and Turing bifurcations of the Gierer–Meinhardt activator-inhibitor model are studied. The very interesting and complex spatially periodic solutions and patterns induced by bifurcations are analyzed from both theoretical and numerical aspects respectively. Firstly, the conditions for the existence of Hopf bifurcation and Turing bifurcation are established in turn. Then, the Turing instability region caused by diffusion is obtained. In addition, to uncover the diffusion mechanics of Turing patterns, the dynamic behaviors are studied near the Turing bifurcation by using weakly nonlinear analysis techniques, and the type of spatial pattern was predicted by the amplitude equation. And our results show that the spatial patterns in the Turing instability region change from the spot, spot-stripe to stripe in order. Finally, the results of the analysis are verified by numerical simulations.

Instability mechanisms of repelling peak solutions in a multi-variable activator-inhibitor system.

We study the linear stability properties of spatially localized single- and multi-peak states generated in a subcritical Turing bifurcation in the Meinhardt model of branching. In one spatial dimension, these states are organized in a foliated snaking structure owing to peak-peak repulsion but are shown to be all linearly unstable, with the number of unstable modes increasing with the number of peaks present. Despite this, in two spatial dimensions, direct numerical simulations reveal the presence of stable single- and multi-spot states whose properties depend on the repulsion from nearby spots as well as the shape of the domain and the boundary conditions imposed thereon. Front propagation is shown to trigger the growth of new spots while destabilizing others. The results indicate that multi-variable models may support new types of behavior that are absent from typical two-variable models.

Hopf Bifurcation Analysis in a Gierer–Meinhardt Activator–Inhibitor Model

In this paper, we study the Hopf bifurcation in a Gierer–Meinhardt model of the activator–inhibitor type with different sources. In the absence of diffusion, we determine the dynamics of the corresponding kinetic equations. Then, we investigate the impact of the diffusion rates on the stability of the homogeneous steady state. By choosing a proper bifurcation parameter, we prove that, under some suitable conditions of the parameters, a Hopf bifurcation occurs in the nonhomogeneous system. We compute the normal form of this bifurcation up to the third order. Next, we specify the direction of the Hopf bifurcation by the normal form theory. Moreover, we provide numerical simulations to illustrate our analytical results.

Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gierer–Meinhardt system in one-dimensional domain

In this paper, we study the existence and stability of multiple spikes pattern to the fractional Gierer–Meinhardt model with periodic boundary conditions and the fractional power [Formula: see text]. Specifically, we rigorously establish the existence of symmetric multiple spikes and asymmetric two-spikes solutions by the classical Lyapunov–Schmidt reduction method. We also investigate the stability of the constructed solution by studying its associated large and small eigenvalue problems, where we need to consider two nonlocal eigenvalue problems in their fractional versions. In the study of the large eigenvalue problem, the quantity [Formula: see text] is the critical threshold which determines the stability of [Formula: see text]-peaked solutions. For the symmetric two-spikes pattern we obtain the asymptotic expansion for the critical threshold [Formula: see text] up to the second order. Moreover, we provide some elementary properties of the Green’s function, including the first and second derivatives, they are linked to the location of the spikes and the stability. Among these properties on the Green’s function, we find out that the polygamma function [Formula: see text] plays a crucial role.

Turing–Hopf Bifurcation in Diffusive Gierer–Meinhardt Model

Gierer–Meinhardt system is a molecularly plausible model to describe pattern formation. When gene expression time delay is added, the behavior of the Gierer–Meinhardt model profoundly changes. In this paper, we study the delayed reaction–diffusion Gierer–Meinhardt system with Neumann boundary condition. Necessary and sufficient conditions for the occurrence of Turing instability, Hopf bifurcation and Turing–Hopf bifurcation deduced by diffusion and gene expression time delay are obtained through linear stability analysis and root distribution of the characteristic equation with two transcendental terms. With the aid of the normal form Turing–Hopf bifurcation and numerical simulations, we theoretically and numerically obtain the expected solutions including stable spatially inhomogeneous steady states, stable spatially homogeneous periodic orbit and stable spatially inhomogeneous periodic orbit from Turing–Hopf bifurcation.

Turing instability of periodic solutions for the Gierer–Meinhardt model with cross-diffusion

In this paper, we establish the Gierer–Meinhardt model with cross-diffusion, and study Turing instability of its periodic solutions. Firstly, the stability of periodic solutions for the zero-dimensional system is studied by using the center manifold theory and normal form method. Secondly, according to Hopf bifurcation theorem, the diffusion rate formula for determining Turing instability of periodic solutions is established. Thirdly, by using the implicit function existence theorem and Floquet theory, the conditions of Turing instability of periodic solutions are derived, and it is proved that the periodic solutions of the model will undergo Turing instability. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is actually induced by cross-diffusion.

Stability and Hopf Bifurcation Analysis of a Reduced Gierer–Meinhardt Model

In this paper, we consider a reduction of the Gierer–Meinhardt Activator–Inhibitor model. In the absence of diffusion, we determine the global dynamics of the homogeneous system. Then, we study the effect of the diffusion constants on the stability of a homogeneous steady state. By choosing a proper bifurcation parameter, we prove that, under some suitable conditions on the parameters, a generalized Hopf bifurcation occurs in the inhomogeneos model. We compute the normal form of this bifurcation up to the fifth order. Furthermore, the direction of the Hopf bifurcation is obtained by the normal form theory. Finally, we provide some numerical simulations to justify our theoretical results.

Turing patterns of Gierer–Meinhardt model on complex networks

Turing patterns of Gierer–Meinhardt model on complex networks

Multi-spike solutions of a hybrid reaction–transport model

Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit D a ≪ D h , where D a and D h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate α between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans . We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α → ∞ with a graphical construction for finite α .

Critical traveling wave solution in the Meinhardt type model

The traveling waves problem for the singularly perturbed semilinear parabolic equations is considered in the paper. It is shown that the corresponding problem for a singularly perturbed ODE system can be reduced to a certain problem of lower dimension using a splitting transformation based on the technique of slow and fast integral manifolds. The Meinhardt model is considered as an example.

An Asymptotic Analysis of Localized Three-Dimensional Spot Patterns for the Gierer--Meinhardt Model: Existence, Linear Stability, and Slow Dynamics

Localized spot patterns, where one or more solution components concentrates at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) activator-inhibitor system in the limit of a small activator diffusivity $\varepsilon^2\ll 1$. Our main focus is to classify the different types of multi-spot patterns, and predict their linear stability properties, for different asymptotic ranges of the inhibitor diffusivity $D$. For the range $D={\mathcal O}(\varepsilon^{-1})\gg 1$, although both symmetric and asymmetric quasi-equilibrium spot patterns can be constructed, the asymmetric patterns are shown to be always unstable. On this range of $D$, it is shown that symmetric spot patterns can undergo either competition instabilities or a Hopf bifurcation, leading to spot annihilation or temporal spot amplitude oscillations, respectively. For $D={\mathcal O}(1)$, only symmetric spot quasi-equilibria exist and they are linearly stable on ${\mathcal O}(1)$ time intervals. On this range, it is shown that the spot locations evolve slowly on an ${\mathcal O}(\varepsilon^{-3})$ time scale towards their equilibrium locations according to an ODE gradient flow, which is determined by a discrete energy involving the reduced-wave Green's function. The central role of the far-field behavior of a certain core problem, which characterizes the profile of a localized spot, for the construction of quasi-equilibria in the $D={\mathcal O}(1)$ and $D={\mathcal O}(\varepsilon^{-1})$ regimes, and in establishing some of their linear stability properties, is emphasized.

Concentration phenomena on [formula omitted]-shaped metric graph for the Gierer–Meinhardt model with heterogeneity

In this paper, we consider the Gierer–Meinhardt model with the heterogeneity in both the activator and the inhibitor on the Y-shaped compact metric graph. Using the Lyapunov–Schmidt reduction method, we construct a one-peak stationary solution, which concentrates at a suitable point. In particular, we reveal that the location of concentration point is determined by the interaction of the heterogeneity function for the activator with the geometry of the domain, represented by the associated Green’s function. Moreover, based on our main result, we determine the precise location of concentration point for non-heterogeneity case. Furthermore, we also present the effect of heterogeneity by using a concrete example.

PENYELESAIAN SISTEM PEMBENTUKAN SEL PADA HYDRA MENGGUNAKAN METODE BEDA HINGGA SKEMA EKSPLISIT


 Mathematical models that describes the pattern of cell formation in hydra are expressed in a system of equations known as the Meinhardt model. This model is a continuous model in the form of diffusion equations. Thus, one of the studies which can be applied to Meinhardt equation is discretization. The finite difference model is a numerical method that can describe the discrete form of a continuous differential form. The method used in this study is finite different methods implementing explicit scheme. The advantage of the explicit scheme is easier to use for solving non-linear partial differential equations. This method used finite forward difference for derivatives of 𝑡 and finite centre difference for derivatives of 𝑥 at theactivator 𝑎(𝑥, 𝑡) and inhibitor 𝑏(𝑥, 𝑡). The Steps conducted by analyzing Meinhardt equation andcontinued with discretization such that earn the solution of system cell formation in hydra. According to the research its found that the activator cell population graphic have cell growth disposed ascend by the unit time, be different with the inhibitor cell population disposed descend of cell growth by the unit time.

Pattern Dynamics in a Diffusive Gierer–Meinhardt Model

The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.

Stable asymmetric spike equilibria for the Gierer–Meinhardt model with a precursor field

AbstractPrecursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1D Gierer–Meinhardt (GM) model, we show that a non-constant precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric and two-spike patterns, corresponding to localized peaks in the activator of different heights. These stable, asymmetric patterns co-exist in the same parameter space as symmetric two-spike patterns. This is in contrast to a constant precursor case, for which asymmetric spike patterns are known to be unstable. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach, we analyse the spectrum of the linearization of the GM model around the two-spike steady state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis, a new class of vector-valued non-local eigenvalue problem is derived and analysed.

Some results on the Gierer–Meinhardt model with critical exponent [formula omitted

We consider the following problem with the critical exponent p−1=rut=d1Δu−a1u+upvq+δ1(x,t),x∈Ω,t>0,vt=d2Δv−a2v+urvs+δ2(x,t),x∈Ω,t>0,∂u∂η=∂v∂η=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω.Here q,r,d1,d2,a1 and a2 are positive constants, p>1, s>−1, δ1, δ2, u0 and v0 are nonnegative smooth functions, Ω⊂Rd(d≥1) is a bounded smooth domain. Whether d1≠d2 or d1=d2, we establish the results on the finite-time blowup and global existence of the solution, which improves those of Theorem 1.2 in Li et al. (2017).

Hopf Bifurcation and Turing Instability Analysis for the Gierer–Meinhardt Model of the Depletion Type

The reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diffusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. The result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. Then, we analyze the system equation with the diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system. Finally, we perform the numerical simulations for the obtained results which show that the Turing patterns are either spot or stripe patterns.