Diffusivity Estimation for Activator–Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton

Pasemann, Gregor; Flemming, Sven; Alonso, Sergio; Beta, Carsten; Stannat, Wilhelm

doi:10.1007/s00332-021-09714-4

“…(i) Applying again formula (26) we obtain the same estimates as in the proof of Theorem 5.1 thus the conclusion follows.…”

Section: Proofsupporting

confidence: 63%

“…For recent theoretical works, the reader may check [27], [7], [2] or [3] to name just a few. The theory for models with Brownian motion has reached certain level of maturity so that it enables several interesting practical applications, such as [1] or [26].…”

Section: Introductionmentioning

confidence: 99%

Pathwise least-squares estimator for linear SPDEs with additive fractional noise

Kříž,

Šnupárková

2022

Preprint

0

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This paper deals with the drift estimation in linear stochastic evolution equations (with emphasis on linear SPDEs) with additive fractional noise (with Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-squares estimator contains stochastic integrals of divergence type, we address the problem of its pathwise (and robust to observation errors) evaluation by comparison with the pathwise integral of Stratonovich type and using its chain-rule property. The resulting pathwise LSE is then defined implicitly as a solution to a non-linear equation. We study its numerical properties (existence and uniqueness of the solution) as well as statistical properties (strong consistency and the speed of its convergence). The asymptotic properties are obtained assuming fixed time horizon and increasing number of the observed Fourier modes (space asymptotics). We also conjecture the asymptotic normality of the pathwise LSE.

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“…Proof. (i) Applying again formula (26) we obtain the same estimates as in the proof of Theorem 5.1 thus the conclusion follows. (ii) For conciseness, we consider the canonical case (all the involved series have power growth) only.…”

Section: P Kříž and J šNupárkovásupporting

confidence: 60%

“…For recent theoretical works, the reader may check [27], [7], [2] or [3] to name just a few. The theory for models with Brownian motion has reached certain level of maturity so that it enables several interesting practical applications, such as [1] or [26].…”

Section: Introductionmentioning

confidence: 99%

Pathwise least-squares estimator for linear SPDEs with additive fractional noise

Kříž¹,

Šnupárková²

2022

Electron. J. Statist.

0

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This paper deals with the drift estimation in linear stochastic evolution equations (with emphasis on linear SPDEs) with additive fractional noise (with Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-squares estimator contains stochastic integrals of divergence type, we address the problem of its pathwise (and robust to observation errors) evaluation by comparison with the pathwise integral of Stratonovich type and using its chain-rule property. The resulting pathwise LSE is then defined implicitly as a solution to a non-linear equation. We study its numerical properties (existence and uniqueness of the solution) as well as statistical properties (strong consistency and the speed of its convergence). The asymptotic properties are obtained assuming fixed time horizon and increasing number of the observed Fourier modes (space asymptotics). We also conjecture the asymptotic normality of the pathwise LSE.

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“…For instance, Khan et al [4] explored the stability of a unique equilibrium point, bifurcations, and chaos control for the discrete activator-inhibitor system. Pasemann et al [5] developed a theory for diffusivity estimation for the activator-inhibitor model. Guo et al [6] discussed the Turing patterns of the activator-inhibitor system on regular lattice networks.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

Berkal

¹

,

Almatrafi

²

2023

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In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains.

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Diffusivity Estimation for Activator–Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton

Cited by 13 publications

References 40 publications

Pathwise least-squares estimator for linear SPDEs with additive fractional noise

Pathwise least-squares estimator for linear SPDEs with additive fractional noise

Pathwise least-squares estimator for linear SPDEs with additive fractional noise

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

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