Pathogen evolution poses a significant challenge to public health, as efforts to control the spread of infectious diseases struggle to keep up with a shifting target. To better understand this adaptive process, we turn to mathematical modelling. Specifically, we use multi-type branching processes to describe a pathogen's stochastic spread among members of a host population or growth within a single host. In each case, there is potential for new pathogen strains with different characteristics to arise through mutation. We first develop a specific model to study the emergence of a newly introduced infectious disease, where the pathogen must adapt to its new host or face extinction in this population. In an extension of previous models, we separate the processes of host-to-host contacts and disease transmission, in order to consider each of their contributions in isolation. We also allow for an arbitrary distribution of host contacts and arbitrary mutational pathways/rates among strains. This framework enables us to assess the impact of these various factors on the chance that the process develops into a large-scale epidemic. We obtain some intriguing results when interpreted in a biological context. Secondly, motivated by a desire to investigate the time course of pathogen evolutionary processes more closely, we derive some novel theoretical results for multi-type branching processes. Specifically, we obtain equations for: (1) the distribution of waiting time for a particular type to arise; and (2) the distribution of population numbers over time, conditioned on a particular type not having yet appeared. A few numerical examples scratch the surface of potential applications for these results, which we hope to develop further. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2010-07-28 11:43:22.984

We investigate the decay mode τ − → π − π + π − ηντ , where the η subsequently decays to π + π − π 0 using 232 fb−1 data acquired by the BABAR detector. The branching fraction of τ − → π − π + π − ηντ is found to be (1.88 ± 0.14 ± 0.11) × 10−4 . The first error on the is measurement is purely statistical and the second error is estimated systematic error. This measurement is consistent with the prior experimental mesaurements at CLEO and BABAR.

In this study, the effects of chain extension and branching on the properties of nanocomposites produced from recycled poly(ethylene terephthalate) and organically modified clay were investigated. As the chain extension/branching agent, maleic anhydride (MA) and pyromellitic dianhydride (PMDA) were used. The nanocomposites were prepared by twin-screw extrusion, followed by injection molding. Recycled poly(ethylene terephthalate), was mixed with 2, 3 and 4 weight % of organically modified montmorillonite. During the second extrusion step, 0.5, 0.75 or 1 weight % of MA or PMDA was added to the products of the first extrusion. As the second extrusion step is reactive extrusion, the anhydrides were added at three different screw speeds of 75, 150, 350 rpm, in order to observe the change of properties with the screw speed. XRD analysis showed that, the interlayer spacing of Cloisite 25A expanded from 19.21 &amp / #506 / to about 28-34 &amp / #506 / after processing with polymer indicating an intercalated structure. PMDA, MA and organoclay content as well as the screw speed did not have a recognizable effect on interlayer distance. In the first extrusion step, nanocomposites containing 3% organoclay content gave significant increase in Young&rsquo / s modulus and decrease in elongation to break values indicating good interfacial adhesion. After the addition of chain extenders, it was observed that both MA and PMDA gave rise to improved mechanical properties of the nanocomposite owing to the branching and chain extending effects that increase the molecular weight. However, PMDA gave better mechanical properties at lower content which makes it a more effective chain extender. DSC analysis showed that MA was more effective in increasing the glass transition temperature and melting temperature in comparison to PMDA.

Diversity in plant architecture is largely generated by the post-embryonic regulation of meristem initiation and activity. In a phenomenon known as apical dominance, the active growth of the shoot apical meristem (SAM) exerts significant inhibitory force on the outgrowth of axillary meristems (AMs) into shoot branches. The degree of branching in plants is a determinant of yield in many crop species and is carefully regulated to ensure that plants only branch at specific stages of development or in response to their environment. Apical dominance has been attributed to the action of the hormone auxin, produced in SAM tissues and transported downwards. A second hormone, cytokinin, acts antagonistically to auxin to promote branching. Nonetheless, the exact mechanism by which these hormones operate is still being elucidated and continued research suggested that novel signals are involved. The recent discovery that strigolactones, previously implicated in parasitic weed germination and mycorrhizal associations, are branching inhibitors supports the existence of additional signals controlling branching in plants. In garden pea (Pisum sativum) strigolactones are synthesised by the coordinated action of the carotenoid cleavage dioxygenase (CCD) family enzymes, RMS1 (RAMOSUS1) and RMS5. These are encoded by MAX4 (MORE AXILLARY GROWTH4) and MAX3 in Arabidopsis thaliana respectively. Mutants for MAX genes have increased amounts of auxin travelling in the polar auxin transport stream (PATS) of inflorescence stems but exhibit increased branching that is insensitive to inhibition by this auxin. Two hypotheses for the action of strigolactones have been presented. The first is that strigolactones modulate the levels of auxin transport proteins, preventing axillary buds from establishing an active auxin transport flow into the primary stem, which inhibits growth. The second is that strigolactones act downstream of auxin signalling to inhibit the action of outgrowth-promoters. Consistent with this latter hypothesis, in pea, rice (Oryza sativa) and petunia (Petunia hybrida), the expression of RMS1/MAX4 orthologues is auxin regulated. These genes are also regulated by feedback signalling in strigolactone pathway mutants and this is proposed to involve an additional novel signal. In Arabidopsis, however, research showed that MAX4 is not regulated by feedback or auxin in the shoot and placed doubt on the importance of this regulation for branching control. The strigolactone biosynthetic pathway offers a novel target for the manipulation of plant architecture and yield while controlling the germination of parasitic weed species that are detrimental to agriculture. Therefore, a greater understanding of the pathway and its regulators is beneficial. The majority of the research in this thesis pre-dates the discovery of strigolactones as the RMS/MAX-derived branching inhibitor, yet aimed to clarify the evolutionary conservation and functional importance of the regulation of strigolactone biosynthetic genes by auxin and feedback signalling in Arabidopsis. Quantitative real-time PCR analysis demonstrated that MAX3 and MAX4 are co-ordinately and systemically regulated by auxin and by feedback throughout development. Both auxin and feedback regulation required the AXR1/TIR1 auxin response pathway, which targets Aux/IAA transcriptional repressors for proteasomal degradation. In particular, correct degradation of the Aux/IAA protein IAA12 appears to be necessary for optimal MAX3 and MAX4 expression. Moreover this regulation affects strigolactone-dependent branching inhibition. Therefore it is proposed that auxin inhibits branching, in part, by positively regulating strigolactone synthesis. As feedback requires AXR1, this also suggests that increased auxin level and/or signalling in the PATS in conditions of reduced strigolactone signalling mediates feedback regulation of the strigolactone pathway. Consistent with this, microarray analysis revealed that in addition to the inflorescence, max mutants have increased global auxin-responsive gene expression associated with the PATS in the vegetative stage. The pea RMS1 gene was the first strigolactone pathway gene demonstrated to be auxin-regulated. Sequencing of the RMS1 promoter and comparative bioinformatic analysis with promoters of other strigolactone synthesis genes revealed a number of conserved, putative regulatory cis-elements that could mediate this regulation and cross-talk with additional branching cues. However a 2.5 kb fragment of the RMS1 promoter was not sufficient to drive transcriptional and translational fusions with GFP and the RMS1 coding region in Arabidopsis. The RMS1 coding region driven by the CAMV 35S promoter complemented the max4 mutant but did not affect branching induced by auxin-depleting treatments. Grafting studies with axr1 and iaa12 mutants, and decapitation and auxin-transport inhibition in max4 mutants, demonstrated that auxin signalling has a function in branching control independent from the regulation of strigolactone synthesis genes. Overall, data obtained herein was incorporated into current models for the interaction of the strigolactone pathway with auxin and cytokinin in the control of shoot branching. It is suggested that both strigolactone and auxin have the capacity to regulate the levels or distribution of each other in interlocking feedback loop that intersects with additional developmental, physiological and environmental cues for the precise control of axillary branching in plants.

Diversity in plant architecture is largely generated by the post-embryonic regulation of meristem initiation and activity. In a phenomenon known as apical dominance, the active growth of the shoot apical meristem (SAM) exerts significant inhibitory force on the outgrowth of axillary meristems (AMs) into shoot branches. The degree of branching in plants is a determinant of yield in many crop species and is carefully regulated to ensure that plants only branch at specific stages of development or in response to their environment. Apical dominance has been attributed to the action of the hormone auxin, produced in SAM tissues and transported downwards. A second hormone, cytokinin, acts antagonistically to auxin to promote branching. Nonetheless, the exact mechanism by which these hormones operate is still being elucidated and continued research suggested that novel signals are involved. The recent discovery that strigolactones, previously implicated in parasitic weed germination and mycorrhizal associations, are branching inhibitors supports the existence of additional signals controlling branching in plants. In garden pea (Pisum sativum) strigolactones are synthesised by the coordinated action of the carotenoid cleavage dioxygenase (CCD) family enzymes, RMS1 (RAMOSUS1) and RMS5. These are encoded by MAX4 (MORE AXILLARY GROWTH4) and MAX3 in Arabidopsis thaliana respectively. Mutants for MAX genes have increased amounts of auxin travelling in the polar auxin transport stream (PATS) of inflorescence stems but exhibit increased branching that is insensitive to inhibition by this auxin. Two hypotheses for the action of strigolactones have been presented. The first is that strigolactones modulate the levels of auxin transport proteins, preventing axillary buds from establishing an active auxin transport flow into the primary stem, which inhibits growth. The second is that strigolactones act downstream of auxin signalling to inhibit the action of outgrowth-promoters. Consistent with this latter hypothesis, in pea, rice (Oryza sativa) and petunia (Petunia hybrida), the expression of RMS1/MAX4 orthologues is auxin regulated. These genes are also regulated by feedback signalling in strigolactone pathway mutants and this is proposed to involve an additional novel signal. In Arabidopsis, however, research showed that MAX4 is not regulated by feedback or auxin in the shoot and placed doubt on the importance of this regulation for branching control. The strigolactone biosynthetic pathway offers a novel target for the manipulation of plant architecture and yield while controlling the germination of parasitic weed species that are detrimental to agriculture. Therefore, a greater understanding of the pathway and its regulators is beneficial. The majority of the research in this thesis pre-dates the discovery of strigolactones as the RMS/MAX-derived branching inhibitor, yet aimed to clarify the evolutionary conservation and functional importance of the regulation of strigolactone biosynthetic genes by auxin and feedback signalling in Arabidopsis. Quantitative real-time PCR analysis demonstrated that MAX3 and MAX4 are co-ordinately and systemically regulated by auxin and by feedback throughout development. Both auxin and feedback regulation required the AXR1/TIR1 auxin response pathway, which targets Aux/IAA transcriptional repressors for proteasomal degradation. In particular, correct degradation of the Aux/IAA protein IAA12 appears to be necessary for optimal MAX3 and MAX4 expression. Moreover this regulation affects strigolactone-dependent branching inhibition. Therefore it is proposed that auxin inhibits branching, in part, by positively regulating strigolactone synthesis. As feedback requires AXR1, this also suggests that increased auxin level and/or signalling in the PATS in conditions of reduced strigolactone signalling mediates feedback regulation of the strigolactone pathway. Consistent with this, microarray analysis revealed that in addition to the inflorescence, max mutants have increased global auxin-responsive gene expression associated with the PATS in the vegetative stage. The pea RMS1 gene was the first strigolactone pathway gene demonstrated to be auxin-regulated. Sequencing of the RMS1 promoter and comparative bioinformatic analysis with promoters of other strigolactone synthesis genes revealed a number of conserved, putative regulatory cis-elements that could mediate this regulation and cross-talk with additional branching cues. However a 2.5 kb fragment of the RMS1 promoter was not sufficient to drive transcriptional and translational fusions with GFP and the RMS1 coding region in Arabidopsis. The RMS1 coding region driven by the CAMV 35S promoter complemented the max4 mutant but did not affect branching induced by auxin-depleting treatments. Grafting studies with axr1 and iaa12 mutants, and decapitation and auxin-transport inhibition in max4 mutants, demonstrated that auxin signalling has a function in branching control independent from the regulation of strigolactone synthesis genes. Overall, data obtained herein was incorporated into current models for the interaction of the strigolactone pathway with auxin and cytokinin in the control of shoot branching. It is suggested that both strigolactone and auxin have the capacity to regulate the levels or distribution of each other in interlocking feedback loop that intersects with additional developmental, physiological and environmental cues for the precise control of axillary branching in plants.

Στην εργασία αυτή παρουσιάζεται η πιθανογεννήτρια του αριθμού των απογόνων της ν-oστης γενιάς μια κλαδωτής ανέλιξης ως το πηλίκο των πιθανογεννήτριων δύο γεωμετρικών κατανομών. Στην βιβλιογραφία, με εξαίρεση δύο συγκεκριμένες περιπτώσεις (πηλίκα πιθανογεννητριών αρνητικής διωνυμικής με γεωμετρική, Kemp, 1979, και γεωμετρικής με Poisson Jayasree and Swamy, 2006), δεν έχει μελετηθεί το γενικότερο πρόβλημα των συνθηκών που επιτρέπουν το πηλίκο δύο πιθανογεννητριών να είναι η πιθανογεννήτρια μιας διακριτής μη αρνητικής τυχαίας μεταβλητής. Εδώ δίνονται οι ικανές και αναγκαίες συνθήκες για τα αντίστοιχα πηλίκα πιθανογεννητριών κατανομών από την οικογένεια Katz ή την οικογένεια Sundt and Jewell με την γεωμετρική κατανομή. Μελετάται επίσης και το πηλίκο απείρως διαιρετών κατανομών με την Poisson και παρουσιάζονται αναλυτικά τέτοια παραδείγματα. Διάφορες ιδιότητες των κατανομών που προκύπτουν εξετάζονται και γίνεται εκτίμηση των παραμέτρων. Στη συνέχεια, παρουσίαζεται μια διδιάστατη κλαδωτή ανέλιξη, δίνεται αναλυτικός τύπος για την πιθανογεννήτρια της από κοινού συνάρτησης κατανομής του πλήθους των δύο ειδών απογόνων της ν-oστης γενιάς, και αποδεικνύεται ότι αυτή μπορεί να γραφεί ως το πηλίκο των πιθανογεννήτριων δύο διδιαστάτων γεωμετρικών κατανομών. Μελετούμε γενικότερα το αντίστοιχο πρόβλημα για διδιάστατες τ.μ. και εξετάζουμε τις ικανές συνθήκες στις περιπτώσεις πηλίκου πιθανογεννητριών της διδιάστατης αρνητικής διωνυμικής με τη διδιάστατη γεωμετρική και της διδιάστατης αρνητικής διωνυμικής με τη διδιάστατη Poisson. Παρουσιάζονται αναγωγικές και αναλυτικές σχέσεις για τις πιθανότητες και τις παραγοντικές ροπές και μελετάται η μορφή των πιθανογεννητριών τόσο των περιθωρίων όσο και των δεσμευμένων κατανομών που προκύπτουν. / In this master thesis we observe, that the probability generating function of the number of the descendants of the n-th generation in a branching process, can be represented as the ratio of the probability generating functions (p.g.f.) of two geometric distributions. In the literature, with the exception of two particular cases (ratio of negative binomial with geometric, Kemp, 1979, and geometric with Poisson, Jayasree and Swamy, 2006), the general problem, for the conditions that allow the ratio of two p.g.f.’s to be the p.g.f. of a discrete non-negative random variable (r.v.), has not been considered. Here, are given the necessary and sufficient conditions for the ratios of the p.g.f. of a distribution from the Katz or the Sundt and Jewell family with the p.g.f. of a Geometric distribution. The ratio of an infinitely divisible r.v. with a Poisson r.v. is also studied and various such examples are presented in detail. Properties of these distributions are given and also parameters estimators are provided. In the sequel, a bivariate branching process is considered and the explicit form for the p.g.f. of the number of two type descendants in the n-th generation is derived. It is proved, that it can be written as the ratio of the p.g.f.’s of two bivariate geometric distributions. The sufficient conditions in the cases of the ratio of the bivariate negative binomial distribution with the bivariate geometric distribution and the bivariate negative binomial distribution with the bivariate Poisson distribution are examined. Recurrence relations and the explicit form of the probabilities and the factorial moments are given and the form of the p.g.f.’s for the marginals and the conditional distributions are studied.

Thesis (MSc)--Stellenbosch University, 2001. / ENGLISH ABSTRACT: A total of 125 trees covering a range of tree diameters were selected from a progeny trial of open pollinated progenies of Pinus patuia for studies on trait assessments and evaluation for branching characteristics. The trees comprised 25 families with each family having five trees. The trial was 36 months old and was located in a relatively high yielding site in Block B at Usutu in Swaziland. The height, diameter, the number of branches and the number of whorls of each tree were recorded. In addition to these objective assessments on individual trees, trees were given scores for branch size, branch angle, stem straightness and crown coarseness. For the detailed branch evaluation study, trees were marked at 20% and 50% of the total tree height and the section between this range was used. On this section the internode length, the internode diameter, the branch diameter and the branch angle were measured. It was observed that the objective assessments correlated strongly with the scoring (subjective) and it was further recommended that a closer look is needed to consider the prospects of employing subjective assessments as a means of reducing costs incurred during objective trait assessments. The branch sampling study revealed that there were prospects of assessing fewer branches on a tree as opposed to assessing all the branches, which could be a major contributor to the high costs in tree breeding. All three methods used to explore this prospect showed some reasonable agreement with each other in this regard. The best sample is to measure all branches in whorl one and two above 20% tree height followed by sampling four branches, two from the first whorl and two from the second whorl. There is a slight difference in these two samples as determined by the coefficents of determination. The study recommends that four branches can be used in the assessment of the branch angle and the branch diameter on the trees when one considered the cost elements. / AFRIKAANSE OPSOMMING: In 'n ondersoek na die evaluering van tak-eienskappe van Pinus patuIa is 125 bome in 'n oop-bestuifde nageslags-proef geselekteer. Hierdie bome is gekies met groottes wat strek oor 'n wye verskeidenheid van borshoogtedeursnee. Die ondersoek het 25 families ingesluit met elke familie wat uit 5 bome bestaan. Die proef is 36 maande oud en is gevestig in'n hoë produksiearea van Blok B, Usutu, in Swaziland. Die hoogte, borshoogtedeurnee, aantal takke en die hoeveelheid takkranse is gemeet en bepaal by elk van die 125 bome. Verder is punte toegeken vir takgrootte, takhoek, graad van stamkromming en kroon grofheid. In die gedetailleerde tak-evaluasie is slegs die stamseksie tussen 200!o en 50% van die totale boomhoogte gebruik. Op hierdie seksie van die boomstam is die internodale afstand, die internodale deursnee, die takdeursnee en die tak-hoek gemeet. Uit die gegewens wat versamel is, het dit geblyk dat daar sterk korrelasie tussen die objektiewe evaluasie en die subjektiewe evaluasie (met gebruik van kwalitatiewe eienskappe op 'n tellingbasis) verkry is. Hieruit word aanbeveel dat meer aandag aan subjektiewe evaluasie (as vereenvoudigde evaluasieprosedure) bestee word om kostes te besnoei. Die gedetailleerde tak-evaluasiestudie het aangedui dat die gebruik van slegs 'n paar takke, in teenstelling met al die takke op die boom, voordelig kan wees om verdere kostes van evaluasie te bespaar. Die beste monster is om al die takke in die eerste twee takkranse bo 20% boomhoogte te meet. Daarnaas is die beste monster die eerste twee grootste takke in die eerste en tweede takkranse. Die gebruik van hierdie kleiner monsters vir evaluering het in sekere opsigte nog beter voorspellings gelewer as in die gevalle waar al die takke in die eerste takkrans gebruik is. Die studie b aan dat In betroubare monster van vier takke (twee uit elke takkrans bo 20% boomhoogte) geneem kan word om die boom se takeienskappe te be-oordeel.

Branching is a highly plastic trait, enabling plants to adapt their growth form in response to environmental stimuli. In flowering plants, shoot branching is regulated through the activity of axillary buds, which grow into branches. Several classes of plant hormones have been shown to play pivotal roles in regulating bud outgrowth. Auxin derived from the primary shoot apex and active branches inhibits bud outgrowth, whereas cytokinin promotes it. Strigolactones also inhibit bud outgrowth, by changing properties of the auxin transport network, increasing the competition between buds. This occurs by modulating access to the polar auxin transport stream (PATS) in the main stem. The PATS provides directional, long distance transport of auxin down the stem, involving basal localisation of the auxin transporter PIN-FORMED1 (PIN1). Buds need to export their auxin across the stem towards the PATS in order to activate, but since PIN1 is mainly expressed in narrow files of cells associated with the stem vasculature, PIN1 itself it is unlikely to facilitate this connectivity. This thesis re-examines the role of auxin transport in the stem, showing that, besides the PIN1-mediated PATS, other auxin transport proteins constitute a more widespread and less polar auxin transport stream, allowing auxin exchange between the PATS and surrounding tissues. Disruption of this transport stream is shown to reduce bud-bud communication and to partially rescue the increased branching observed in strigolactone mutants. Furthermore, it is shown that distinct classes of auxin transport proteins within this stream can differentially affect bud outgrowth mediated by BRANCHED1 (BRC1). BRC1 is a transcription factor proposed to determine bud activation potential. Taken together, the data presented here provide a more comprehensive understanding of the shoot auxin transport network and its role in shoot branching regulation.

O complexo 1,4-bis(2,6-diisopropilfenil)-acenaftenodiimina-dicloroníquel(II) (1), em combinação com metilaluminoxano (MAO) foi utilizado para polimerizar eteno utilizando altas pressões e temperaturas. Foram investigados os efeitos da pressão de eteno, da temperatura, do tempo de reação e da quantidade de catalisador bem como, as propriedades dos polietilenos sintetizados. Os polietilenos obtidos com o sistema (1)/MAO foram altamente ramificados. As ramificações variaram de metil até hexil ou até mais longas, sem adição de comonômero. Os polietilenos não apresentaram metilas isoladas, apresentando uma grande quantidade de metilas 1,4, metilas 1,5 e metilas 1,6 e cadeias longas. A presença de ramificações foi devido ao mecanismo denominado chain-walking. Os valores de ramificações nos PE foram maiores que 105 e menores que 277 ramificações/1000C. O aumento do número de ramificações foi devido ao aumento na temperatura de polimerização e uma diminuição da pressão de eteno. Os PE obtidos com o sistema (1)/MAO apresentaram peso molecular (Mw) elevado entre 44.000 e 105.000 Daltons e valores de polidispersão de 2,0 a 4,0, dependendo das condições reacionais. O peso molecular dos polímeros diminuiu com o aumento da temperatura de polimerização. / The combination of 1,4-bis(2,6-diisopropylphenyl)-acenaphthenediiminedichloronickel( II) (1) and methylaluminoxane (MAO) was highly active in ethylene polymerization under high pressures and temperatures. Herein we investigated the effects of ethylene pressure, reaction temperature, reaction time and amount of catalyst on polymer properties and reaction performance. The polyethylenes obtained with 1/MAO are highly branched. The branches goes from methyl to hexyl or even longer, and this without comonomer addition. These polyethylenes obtained do not shows isolated methyl groups, but shows 1,4-methyl, 1,5 and 1,6 methyl patterns. The branching was due to the so-called chain-walking mechanism. The branch content, which is in the range 105 to 277 branches/1000 C, increased with the temperature rising or the ethylene concentration decrease. The polyethylenes produced with these system have molecular weight between 44.000 and 105.000 Daltons and polydispersions from 2,0 to 4,0 depending on the reactions conditions. The polymer molecular weight tended to decrease with increasing polymerization temperature.

L’objet de cette thèse est d’étudier la probabilité de survie d’un processus de branchement en environnement aléatoire markovien et d’étendre dans ce cadre les résultats connus en milieu aléatoire i.i.d.. le cœur de l’étude repose sur l’utilisation des théorèmes limites locaux pour une marche aléatoire centrée (Sn)n≥0 sur R à pas markoviens et pour (rnn)n≥0, où mn = min (0, S1,... , Sn). Pour traiter le cas d’un environnement aléatoire markovien, nous développons dans un premier temps une étude des théorèmes locaux pour une chaîne semi-markovienne à valeurs réelles en améliorant certains résultats déjà connus et développés initialement par E. L. Presman (voir aussi [21]). Nous utilisons ensuite ces résultats pour l’étude du comportement asymptotique de la probabilité de survie d’un processus de branchement critique en environnement aléatoire markovien. Les résultats principaux de cette thèse (théorème limite local et son application au processus de branchement critique eu milieu aléatoire) ont été acceptés et publiés dans le Comptes Rendus de l‘Académie des Sciences ([20]). Le texte principal de cette mémoire de thèse consisite les détails des preuves. / The purpose of this thesis is to study the survival probability of a branching process in markovian random environment and expand in this framework some known results which have been developed for a branching processus in i.i.d. random environment, the core of the study is based on the use of the local limit theorem for a centered random walk (Sn)n≥o on R with markovian increasements and for (mn)n≥0. where mn = min (O. S1,……. , Sn). In order to treat the case of a markovian random environment, we establish firstly a local limit theorem for a semi-markovian chain on R. which improves certain results developed initially by E. P. Presman (see also [21]). And then we use these results to study the asymptotic behavior of a critical branching process in markovian environment. The main results et this thesis (local limit theorem and its application to the critical branching process in random environment) are accepted and published in Comptes Rendus de l’Académie des Sciences ([20]). The principal text et this thesis contains the details of the proofs.