A note on a periodic Parabolic-ODE chemotaxis system

Negreanu, Mihaela; Tello, J. Ignacio; Vargas, A.M.

doi:10.1016/j.aml.2020.106351

Cited by 11 publications

(16 citation statements)

References 11 publications

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“…We have obtained, under suitable assumptions, that the solution of a chemotaxis system is globally bounded in time by using an Alikakos-Moser iteration, and it fulfills a periodic asymptotic behavior. A possible future work is the consideration of the nonconstant chemosensitivity, χ(u, v), as in [23][24][25][26][27][28][29][30][31][32][33][34][35][36], and to also consider biological systems with two species, two chemotactic terms and one chemical substance verifying a similar equation as in (1). Furthemore, a parabolic-parabolic-ordinary system with periodic terms serves as a model for some chemotaxis phenomena and appears naturally in the interaction of two biological species and a chemical.…”

Section: Discussionmentioning

confidence: 99%

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Dynamics in a Chemotaxis Model with Periodic Source

Negreanu¹,

Vargas²

2022

Mathematics

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We consider a system of two differential equations modeling chemotaxis. The system consists of a parabolic equation describing the behavior of a biological species “u” coupled to an ODE patterning the concentration of a chemical substance “v”. The growth of the biological species is limited by a logistic-like term where the carrying capacity presents a time-periodic asymptotic behavior. The production of the chemical species is described in terms of a regular function h, which increases as “u” increases. Under suitable assumptions we prove that the solution is globally bounded in time by using an Alikakos-Moser iteration, and it fulfills a certain periodic asymptotic behavior. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system showing that the model considered here can provide very interesting and complex dynamics.

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Section: Discussionmentioning

confidence: 99%

“…Finally, in Section 5 we perform a brief numerical study of the system under consideration. Some of the results presented in this paper were announced in [33].…”

Section: Theoremmentioning

confidence: 93%

Dynamics in a Chemotaxis Model with Periodic Source

Negreanu¹,

Vargas²

2022

Mathematics

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“…25) and we know from Lemma 2.3 that ∇ 𝑢 is bounded in 𝐿 2 (︀ 0, 𝑇 𝑓 ; 𝐿 2 (Ω) )︀ . Taking into account that 𝑇 and Φ are bounded in 𝐿 ∞ (0, 𝑇 𝑓 ; 𝐿 ∞ (Ω)), it holds that |∇ 𝑇 | ≤ 𝐶 (|∇ Φ| + |∇ 𝑢|) .…”

mentioning

confidence: 92%

“…Recently, a PDE-ODE model with chemotaxis is studied in [18] obtaining asymptotic stability results using a proper transformation and energy estimates. Another PDE-ODE with chemotaxis problem is considered in [24], see also [25], modelling the evolution of biological species and they obtain analytical results concerning the bifurcation of constant steady states and global existence of solutions for a range of initial data. In [14] a parabolic-ODE problem is analysed, and it is shown that, under several conditions, any stationary solution is locally stable.…”

Section: Introductionmentioning

confidence: 99%

A Glioblastoma PDE-ODE model including chemotaxis and vasculature

2022

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In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the main parameters determining different width of the ring formed by proliferative and necrotic cells and different regular/irregular behaviour of the tumor surface.

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“…The regular function h increases as "U " increases and states the production of the chemical species. In the recently published article [16], the authors prove that for all sufficiently smooth initial data U (x, 0) = U 0 (x), V (x, 0) = V 0 (x), x ∈ Ω, the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞), with Ω ⊂ R n , for n ≥ 1. We prove that the convergence in space and in time of the classical solution is maintained for the discrete model.…”

Section: Introductionmentioning

confidence: 99%

On the convergence of the generalized finite difference method for solving a chemotaxis system with no chemical diffusion

Benito

García

Gavete³

et al. 2020

Comp. Part. Mech.

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This paper focuses on the numerical analysis of a discrete version of a nonlinear reaction-diffusion system consisting of an ordinary equation coupled to a quasilinear parabolic PDE with a chemotactic term. The parabolic equation of the system describes the behavior of a biological species, while the ordinary equation defines the concentration of a chemical substance. The system also includes a logistic-like source, which limits the growth of the biological species and presents a time-periodic asymptotic behavior. We study the convergence of the explicit discrete scheme obtained by means of the Generalized Finite Difference method and prove that the non-negative numerical solutions in two dimensional space preserve the asymptotic behavior of the continuous ones. Using different functions and long-time simulations, we illustrate the efficiency of the developed numerical algorithms in the sense of the convergence in space and in time.

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A note on a periodic Parabolic-ODE chemotaxis system

Cited by 11 publications

References 11 publications

Dynamics in a Chemotaxis Model with Periodic Source

Dynamics in a Chemotaxis Model with Periodic Source

A Glioblastoma PDE-ODE model including chemotaxis and vasculature

On the convergence of the generalized finite difference method for solving a chemotaxis system with no chemical diffusion

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