Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat

International audienceThis paper is concerned with the existence and qualitative properties of pulsating fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. We focus especially on the influence of the spatial period and, under various assumptions on the reaction terms and by using different types of arguments, we show several existence results when the spatial period is small or large. We also establish some properties of the set of periods for which there exist non-stationary fronts. Furthermore, we prove the existence of stationary fronts or non-stationary partial fronts at any period which is on the boundary of this set. Lastly, we characterize the sign of the front speeds and we show the global exponential stability of the non-stationary fronts for various classes of initial conditions

Transition fronts for periodic bistable reaction-diffusion equations

International audienceThis paper is concerned with the existence and qualitative properties of transition fronts for spatially periodic reaction-diffusion equations with bistable nonlinearities. The notion of transition fronts connecting two stable steady states generalizes the standard notion of pulsating fronts. In this paper, we prove that the time-global solutions in the class of transition fronts share some common features. In particular, we establish a uniform estimate for the mean speed of transition fronts, independently of the spatial scale. Under the a priori existence of a pulsating front with nonzero speed or under a more general condition guaranteeing the existence of such a pulsating front, we show that transition fronts are reduced to pulsating fronts, and thus are unique up to shift in time. On the other hand, when the spatial period is large, we also obtain the existence of a new type of transition fronts which are not pulsating fronts. This example, which is the first one in periodic media, shows that even in periodic media, the notion of generalized transition fronts is needed to describe the set of solutions connecting two stable steady states

Average speeds of time almost periodic traveling waves for rapidly/slowly oscillating reaction-diffusion equations

This paper is concerned with the propagation dynamics of time almost periodic reaction-diffusion equations. Assuming the existence of a time almost periodic traveling wave connecting two stable steady states, we focus especially on the asymptotic behavior of average wave speeds in both rapidly oscillating and slowly oscillating environments. We prove that, in the rapidly oscillating case, the average speed converges to the constant wave speed of the homogenized equation; while in the slowly oscillating case, it approximates the arithmetic mean of the constant wave speeds for a family of equations with frozen coefficients. In both cases, we provide estimates on the convergence rates showing that, in comparison to the limiting speeds, the deviations of average speeds for almost periodic traveling waves are at most linear in certain sense. Furthermore, our explicit formulas for the limiting speeds indicate that temporal variations have significant influences on wave propagation. Even in periodic environments, it can alter the propagation direction of bistable equations

Genome-wide gene phylogeny of CIPK family in cassava and expression analysis of partial drought-induced genes

Cassava is an important food and potential biofuel crop that is tolerant to multiple abiotic stressors. The mechanisms underlying these tolerances are currently less known. CBL-interacting protein kinases (CIPKs) have been shown to play crucial roles in plant developmental processes, hormone signaling transduction, and in the response to abiotic stress. However, no data is currently available about the CPK family in cassava. In this study, a total of 25 CIPK genes were identified from cassava genome based on our previous genome sequencing data. Phylogenetic analysis suggested that 25 MeCIPKs could be classified into four subfamilies, which was supported by exon-intron organizations and the architectures of conserved protein motifs. Transcriptomic analysis of a wild subspecies and two cultivated varieties showed that most MeCIPKs had different expression patterns between wild subspecies and cultivatars in different tissues or in response to drought stress. Some orthologous genes involved in CIPK interaction networks were identified between Arabidopsis and cassava. The interaction networks and co-expression patterns of these orthologous genes revealed that the crucial pathways controlled by CIPK networks may be involved in the differential response to drought stress in different accessions of cassava. Nine MeCIPK genes were selected to investigate their transcriptional response to various stimuli and the results showed the comprehensive response of the tested MeCIPK genes to osmotic, salt, cold, oxidative stressors, and ABA signaling. The identification and expression analysis of CIPK family suggested that CIPK genes are important components of development and multiple signal transduction pathways in cassava. The findings of this study will help lay a foundation for the functional characterization of the CIPK gene family and provide an improved understanding of abiotic stress responses and signaling transduction in cassava