On the geometry of wave solutions of a delayed reaction-diffusion equation
Elena Trofimchuk, Pedro Alvarado, Sergei Trofimchuk
研究了一类非局部时滞单稳态反应扩散方程正有界波解的存在性和几何性。
The aim of this paper is to study the existence and the geometry of positive bounded wave solutions to a non-local delayed reaction-diffusion equation of the monostable type.
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Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays
研究一类时滞反应扩散方程组的行波解。利用Schauder的不动点定理，将行波解的存在性归结为广义上下解的存在性.利用收缩矩形的技巧，得到了时滞扩散方程组行波解的渐近性态。为了说明我们的主要结果，建立了具有分布时滞的扩散Lotka-Volterra竞争系统正行波解的存在性、不存在性和渐近性态.讨论了扩散Lotka-Volterra竞争系统的非单调行波解的存在性。特别地，证明了当存在瞬时自限效应时，种内竞争项中出现的大时滞不影响行波解的存在性和渐近性态.
With the increasing complexity of time-delayed systems, the diversification of boundary types of chemical reaction systems poses a challenge for persistence analysis. This paper focuses on delayed complex balanced mass action systems (DeCBMAS) and derives that some boundaries of a DeCBMAS can not contain an $ω$-limit point of some trajectory with positive initial point by using the method of semilocking set decomposition and the property of the facet, further expanding the range of persistence of delayed complex balanced systems. These findings demonstrate the effectiveness of semilocking set …
Perron Theorem in the Monotone Iteration Method for Traveling Waves in Delayed Reaction-Diffusion Equations
本文利用单调迭代法重新讨论了时滞反应扩散方程行波解的存在性。我们证明了Perron有界解的存在性定理为寻找具有时滞的反应扩散方程组的行波解提供了一个严格的和建设性的框架。该方法尝试了两个经典的例子与延迟：捕食者-食饵和Belousov-Zhabotinskii模型。
This paper is concerned with the existence of traveling wave solutions for diffusive two-species Lotka-Volterra systems with delay in both the reaction and diffusion terms without monotonicity. We extend the partial or cross monotone iteration method to systems that satisfy the partial quasi-monotone condition via construction appropriate upper and lower solutions. This is done by using Schauder's fixed point theorem.
Asymptotic stability of delayed complex balanced reaction networks with non-mass action kinetics
该研究针对具有乘积形式广义动力学的时滞化学反应网络，证明复平衡下每个正化学计量相容类内存在唯一局部渐近稳定的正平衡点，采用对数Lyapunov函数及时延Lyapunov-Krasovskii泛函作为证明工具，并通过示例验证结果。
For the networks that are linear conjugate to complex balanced systems, the delayed version may include two classes of networks: one class is still linear conjugate to the delayed complex balanced network, the other is not. In this paper, we prove the existence of the first class of networks, and emphasize the local asymptotic stability relative to a certain defined invariant set. For the second class of systems, we define a special subclass and derive the local asymptotic stability for the subclass. Two examples are provided to illustrate our results.
In this paper, we investigate a reaction-diffusion-advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our result implies that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction-diffusion-advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation.
Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain
研究无界域上延迟反应扩散方程全局吸引子的存在性及Hausdorff和分形维数上界，通过先验估计和渐近紧性证明存在性，并利用解分解与挤压性质获得仅依赖方程内在特征的维数估计。
In this paper, we investigate a delayed reaction-diffusion-advection equation, which models the population dynamics in the advective heterogeneous environment. The existence of the nonconstant positive steady state and associated Hopf bifurcation are obtained. A weighted inner product associated with the advection rate is introduced to compute the normal forms, which is the main difference between Hopf bifurcation for delayed reaction-diffusion-advection model and that for delayed reaction-diffusion model. Moreover, we find that the spatial scale and advection can affect Hopf bifurcation in th…
In this paper, we derive two sufficient conditions to diagnose the persistence of two classes of delayed complex balanced chemical reaction network systems equipped with mass-action kinetics. One class is identified by $\dim Z_W=\dim\mathscr{S}-1$ ($Z_W$ is defined by \ref{eq:g} while $\mathscr{S}$ is the stoichiometric subspace of the network), the other class is identified through $\dim Z_W=0$ for any semilocking species set $W$ in the network. Then we also derive that delayed complex balanced systems with 2-d stoichiometric subspace are persistent. The results recur those proposed by Anders…
In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretica…
We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spa…
In the context of a spatially extended model for the electrical activity in a pituitary lactotroph cell line, we establish that two delayed bifurcation phenomena from ODEs ---folded node canards and slow passage through Hopf bifurcations--- persist in the presence of diffusion. For canards, the single cell (ODE) model exhibits canard-induced bursting. Numerical simulations of the PDE reveal rich spatio-temporal canard dynamics, and the transitions between different bursts are mediated by spatio-temporal maximal canards. The ODE model also exhibits delayed loss of stability due to slow passage …
Dynamical analysis in a nonlocal delayed reaction-diffusion tumor model with therapy
本文研究了非局部时滞反应扩散肿瘤模型的动力学特性，通过Lyapunov-Schmidt方法分析稳态解的存在性，并探讨了延迟依赖的Hopf分岔现象。
The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic nonlocal delayed reaction-diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition on the finite end. This equation models the spatial-temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first estab…
Double Hopf bifurcation analysis can be used to reveal some complicated dynamical behavior in a dynamical system, such as the existence or coexistence of periodic orbits, quasi-periodic orbits, or even chaos. In this paper, an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation of a reaction-diffusion system with time delay and Neumann boundary condition is rigorously established, by employing the center manifold reduction technique and the normal form method. We find that the dynamical behavior near bifurcation points are proved to be governed by twelve disti…
A Blowup Problem of Reaction Diffusion Equation Related to the Diffusion Induced Blowup Phenomenon
本文研究一类Logistic型反应扩散方程的Cauchy、Neumann和狄利克雷问题的非负解.在对方程系数的各种限制条件下，给出了非负解的有限时间爆破结果。应用这些结果可以构造出具有所谓扩散诱导爆破现象的反应扩散系统，特别是在等扩散速率的情况下。
Topological dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain
研究无界域上时滞反应扩散方程全局吸引子的Hausdorff与分形维数估计，通过辅助Hilbert空间重述方程并利用扰动半群方法证明全局解与吸引子存在性，最终获得维数上界。
We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.
Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism is necessary, using, for instance, impulsive reaction-diffusion equations. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vecto…
Weak mean random attractors for non-local random and stochastic reaction-diffusion equations
本文研究了非局部随机和随机反应扩散方程的弱均随机吸引子存在性，证明了解的存在性与唯一性及吸引子的性质。
A new instability framework in 2-component reaction-diffusion systems
本文针对双组分反应扩散系统提出包含0模Hopf及m和m+1模Turing不稳定的新框架，通过中心流形约化推导余维3分岔正规形，并给出数值验证。
We prove existence and uniqueness of \emph{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in $\real^N$, with $m>1$, $0<p<1$ and the critical value for the weight $$ σ=\frac{2(1-p)}{m-1}. $$ Existence and uniqueness of some specific solution holds true when $m+p\geq2$. On the contrary, no eternal solution exists if $m+p<2$. We also classify exponential self-similar solutions with a different interface behavior when $m+p>2$. Some transformations to reaction-convection-diffusion e…
Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs
本文通过二分Hilbert-Schmidt核将符号场与几何场耦合，建立延迟泛函微分方程，证明其在常数输入下的适定性和紧全局吸引子，并给出主子系统全局稳定的耦合条件。
This paper deals with the existence, monotonicity, uniqueness and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations.
Stability of traveling wavefronts in discrete reaction-diffusion equations with nonlocal delay effects
本文研究时间延迟、空间离散的反应扩散方程的行波解。利用加权能量法和格林函数技术相结合，证明了所有非临界波前都是全局指数稳定的，而当波前周围的初始扰动在负无穷大附近指数衰减到零时，临界波前是全局代数稳定的，与时间延迟的大小无关.
Convergence in Comparable Almost Periodic Reaction-Diffusion Systems with Dirichlet Boundary Condition
研究了具有Dirichlet边界条件的非单调可比概周期反应扩散系统的渐近动力学，该系统与一致稳定的强保序系统具有可比性.利用斜积半流理论，得到了一类可比较的反应扩散方程组一致稳定解的渐近概周期性。
A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propa…
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