No Arabic abstract
This manuscript extends the analysis of a much studied singularly perturbed three-component reaction-diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to stable periodic front motion, including stable traveling breathers, and these results are illustrated by numerical computations.
We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear level, the spectrum is stabilized by using an exponential weight. A-priori estimates for the nonlinear terms of the equation governing the evolution of the perturbations of the front are obtained when perturbations belong to the intersection of the exponentially weighted space with the original space without a weight. These estimates are then used to show that in the original norm, initially small perturbations to the front remain bounded, while in the exponentially weighted norm, they algebraically decay in time.
We revisit the problem of pinning a reaction-diffusion front by a defect, in particular by a reaction-free region. Using collective variables for the front and numerical simulations, we compare the behaviors of a bistable and monostable front. A bistable front can be pinned as confirmed by a pinning criterion, the analysis of the time independant problem and simulations. Conversely, a monostable front can never be pinned, it gives rise to a secondary pulse past the defect and we calculate the time this pulse takes to appear. These radically different behaviors of bistable and monostable fronts raise issues for modelers in particular areas of biology, as for example, the study of tumor growth in the presence of different tissues.
We consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction. We deduce from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the chemical kinetics factor. It follows that the solution converges to the solution of a nonlinear diffusion problem, as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.
In this paper, we study the bifurcate of limit cycles for Bogdanov-Takens system($dot{x}=y$, $dot{y}=-x+x^{2}$) under perturbations of piecewise smooth polynomials of degree $2$ and $n$ respectively. We bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles with switching curve $x=y^{2m}$($m$ is a positive integral) are $(39m+36)n+77m+21(mgeq 2)$ and $50n+52(m=1)$ (taking into account the multiplicity). The upper bounds number of limit cycles with switching lines $x=0$ and $y=0$ are 11 (taking into account the multiplicity) and it can be reached.
This paper aims to explore the temporal-spatial spreading and asymptotic behaviors of West Nile virus by a reaction-advection-diffusion system with free boundaries, especially considering the impact of advection term on the extinction and persistence of West Nile virus. We define the spatial-temporal risk index $R^{F}_{0}(t)$ with the advection rate and the general basic disease reproduction number $R^D_0$ to get the vanishing-spreading dichotomy regimes of West Nile virus. We show that there exists a threshold value $mu^{*}$ of the advection rate, and obtain the threshold results of it. When the spreading occurs, we investigate the asymptotic dynamical behaviors of the solution in the long run and first give a sharper estimate that the asymptotic spreading speed of the leftward front is less than the rightward front for $0<mu<mu^*$. At last, we give some numerical simulations to identify the significant effects of the advection.