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This paper analyzes the ongoing transition to sustainable energy in Australia, moving from traditional large-scale plants to distributed renewable generation by studying the time series of coal and gas consumption as well as onshore wind and solar. Even though most of energy generation, especially in the form of electricity is currently being generated from coal and gas, a quantitative assessment of their evolution is necessary to understand whether, and to which extent, renewables are competing in the marketplace with conventional production means. A well-accepted innovation diffusion model is used to capture and interpret the underlying dynamics of the competitive transition in generation. The results show that renewables are exerting a competitive pressure on coal and collaborate with gas towards the transition. The view that gas should play a key role in transition is confirmed by our findings, because it is found to have a competitive role towards coal, while aiding the uptake of renewables.

In order to predict sugar beet yield reduction from weeds, obtained data was fitted to some non-linear regression functions. Two independent variables were selected for validating these functions: time and thermal time. The hypotheses of normality, homoscedasticity, independence of residuals, predictive capacity or goodness of fit with respect of the model, and diagnosis of the model, have been verified by means of residuals analysis for both independent variables. In early competition, a hyperbolic function was rejected due to the fact that some parameters seemed insignificant. However, the goodness of fit was similar to other functions. In late competition, some parameters were insignificant, but this hyperbolic function reaches the best goodness of fit. Comparing the validated functions, the logistical equation that includes the inflection point with independent variable time is the one that reaches the best results. The high flexibility of the model may allow the detection of special cases, and thus minimize risk. This study could be done anywhere in the world, as long as the mathematical model is adapted to the climate study area.

A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.

We consider a class of birth-and-death processes describing a population made of $d$ sub-populations of different types which interact with one another. The state space is $\mathbb{Z}^d_+$ (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter $K$ which can be thought as the order of magnitude of the total size of the population at time $0$. For any fixed finite time span, it is well-known that such processes, when renormalized by $K$, are close, in the limit $K\to+\infty$, to the solutions of a certain differential equation in $\mathbb{R}^d_+$ whose vector field is determined by the birth and death rates.We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for $K$ large, the process will stay in the vicinity of the fixed point for a verylong time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd, for short). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time $t$ and the qsd. This bound is exponentially small in $t$, for $t\gg \log K$. As a by-product, we obtainan estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than $\log K$ and much smaller than the mean time to extinction, which is exponentially large as a function of $K$. Let us stress that we are interested in what happens for finite $K$. We obtain results much beyond what large deviation techniques could provide., Nous considérons une classe de processus de naissance-et-mort décrivant une population constituée de d sous-populations de types différents qui interagissent entre elles. L'espace d'état est $\mathbb{Z}^d_+$ (il est donc non borné). Nous supposons que la population s'éteint presque sûrement, de sorte que l'unique distribution de probabilité stationnaire est la masse de Dirac à l'origine. Nous faisons dépendre ces processus d'un paramètre d'échelle K qu'on peut interpréter comme l'ordre de grandeur de la taille totale de la population au temps 0. Etant donné un intervalle de temps, il est bien connu que de tels processus, normalisés par K, sont proches, dans la limite K → +∞, des solutions d'une certaine équation différentielle dans $\mathbb{R}^d_+$ dont le champ de vecteurs est déterminé par les taux de naissance et de mort du processus. Nous considérons le cas où le champ de vecteurs possède un unique point fixe attractif à l'intérieur de l'orthant positif, tandis que l'origine est un point fixe répulsif. On s'attend à ce que, pour K grand, le processus reste dans le voisinage du point fixe attractif pendant très longtemps avant d'être absorbé à l'origine. Afin de décrire précisément ce comportement, nous démontrons l'existence dune distribution quasi-stationnaire (dqs, en abrégé). Nous établissons une borne pour la distance en variation totale entre le processus conditionné à ne pas s'éteindre avant le temps t et la dqs. Cette borne est exponentiellement petite en t pour t log K. En particulier, nous obtenons une estimation du temps moyen d'extinction dans la dqs. Nous quantifions également la distance entre le processus (non conditionné à la non-extinction) et une certaine combinaison convexe de la masse de Dirac à l'origine et de la dqs, ceci pour des temps beaucoup plus grands que log K et beaucoup plus petits que le temps moyen d'extinction, qui est exponentiellement grand en K. Nous attirons l'attention sur le fait que nous sommes intéressés par ce *

Hubs are consolidation and dissemination points in many-to-many flow networks. Hub location problem is to locate hubs among available nodes and allocate non-hub nodes to these hubs. The mainstream hub location studies focus on optimal decisions of one decision-maker with respect to some objective(s) even though the markets that benefit hubbing are oligopolies. Therefore, in this paper, we propose a competitive hub location problem where the market is assumed to be a duopoly. Two decision-makers (or firms) sequentially decide locations of their hubs and then customers choose one firm with respect to provided service levels. Each decision-maker aims to maximize his/her own market share. We propose two problems for the leader (former decision-maker) and follower (latter decision-maker): ( r | X p ) h u b − m e d i a n o i d and ( r | p ) h u b − c e n t r o i d problems, respectively. Both problems are proven to be NP-complete. Linear programming models are presented for these problems as well as exact solution algorithms for the ( r | p ) h u b − c e n t r o i d problem. The performance of models and algorithms are tested by computational analysis conducted on CAB and TR data sets.

Based on the concept of competition for resources, the distribution of trees upon reforestation in windfall areas is studied. As a theoretical model for competition, a Zipf-Pareto model of ranking the distribution of resources is used. Analysis shows that the processes resulting from competitive interactions between the trees of different species proceed slowly in a windfall area where coniferous species get replaced by deciduous ones. In the territory where deciduous species initially dominated, competitive interactions between trees of different species turn out to be formed almost immediately upon natural reforestation after the windfall. By the time the ratio of species stabilizes and becomes a steady state, the ranks of individual species also stabilize. This result is obtained on the basis of a quantitative assessment of the change in leadership between competing species in time using Spearman's rank correlation coefficient. © 2013 Pleiades Publishing, Ltd. This work was supported by the Russian Foundation for Basic Research (grant no. 10 04 00256).

Se estudió la competencia de Commelina erecta originada de semilla y de rizoma con Digitaria sanguinalis y Amaranthus quitensis. Los objetivos del trabajo fueron: 1) estudiar la dinámica del crecimiento de plantas de estas malezas en mezclas binarias y en monoculturas; 2) estimar el rendimiento relativo de cada especie (RR) y el rendimiento relativo total (RRT) de las distintas mezclas y 3) utilizar un modelo dinámico para analizar la importancia de distintos atributos específicos sobre el resultado de la competencia. El crecimiento de las monoculturas de todas las especies ajustó a un modelo sigmoide. Los RR de las distintas malezas variaron en función de la especie acompañante, de la fecha de muestreo y del año. La competencia fue parcial (RRT > 1) únicamente en mezclas que incluyeron plantas de C. erecta originadas de rizomas. A partir de las simulaciones con el modelo, se determinó que el resultado de la competencia depende del tamaño inicial (BI) de rizomas de C. erecta, por otro lado, el análisis de sensibilidad señala la gravitación de la tasa intrínseca de crecimiento (r) en la definición de RR. Bajo condiciones de competencia, los parámetros biológicos que definieron el crecimiento de las especies estudiadas fueron especialmente r y BI. Los resultados destacan la compleja interacción de factores que determinan el balance competitivo, tanto entre ambas formas de propagación de C. erecta, como entre ellas y otras malezas. Competition of C. erecta from seed and rhizome origin with Digitaria sanguinalis and Amaranthus quitensis was studied. The objectives of the study were: 1) determine growth dynamics of plants of these weed species in binary mixtures and in monocultures; 2) estimate the relative yield of each species (RY) and relative yield total (RYT) of the different mixtures, and 3) to use a dynamic model to analyze the importance of different specific attributes on the result of the competition. The growth of the monoculture of all species fitted to a sigmoidal model. The RY of the different species varied relative to the accompanying species, the sampling date and the year. Competition was partial (RYT > 1) only in mixtures including C. erecta plants originated from rhizomes. From the simulations with the model it was determined that the result of the competition depends on the initial size (BI) of C. erecta rhizomes. Furthermore, the sensibility analysis highlights the importance of the intrinsic growth rate (r) in the definition of RY. Under competition conditions the biological parameters defining the growth of the species studied were particularly r and BI. The results emphasize the complex interaction of the factors determining the competitive balance, between both propagation forms of C. erecta, and between those forms and other weed species.

Se estudió la competencia de Commelina erecta originada de semilla y de rizoma con Digitaria sanguinalis y Amaranthus quitensis. Los objetivos del trabajo fueron: 1) estudiar la dinámica del crecimiento de plantas de estas malezas en mezclas binarias y en monoculturas; 2) estimar el rendimiento relativo de cada especie (RR) y el rendimiento relativo total (RRT) de las distintas mezclas y 3) utilizar un modelo dinámico para analizar la importancia de distintos atributos específicos sobre el resultado de la competencia. El crecimiento de las monoculturas de todas las especies ajustó a un modelo sigmoide. Los RR de las distintas malezas variaron en función de la especie acompañante, de la fecha de muestreo y del año. La competencia fue parcial (RRT > 1) únicamente en mezclas que incluyeron plantas de C. erecta originadas de rizomas. A partir de las simulaciones con el modelo, se determinó que el resultado de la competencia depende del tamaño inicial (BI) de rizomas de C. erecta, por otro lado, el análisis de sensibilidad señala la gravitación de la tasa intrínseca de crecimiento (r) en la definición de RR. Bajo condiciones de competencia, los parámetros biológicos que definieron el crecimiento de las especies estudiadas fueron especialmente r y BI. Los resultados destacan la compleja interacción de factores que determinan el balance competitivo, tanto entre ambas formas de propagación de C. erecta, como entre ellas y otras malezas.Competition of C. erecta from seed and rhizome origin with Digitaria sanguinalis and Amaranthus quitensis was studied. The objectives of the study were: 1) determine growth dynamics of plants of these weed species in binary mixtures and in monocultures; 2) estimate the relative yield of each species (RY) and relative yield total (RYT) of the different mixtures, and 3) to use a dynamic model to analyze the importance of different specific attributes on the result of the competition. The growth of the monoculture of all species fitted to a sigmoidal model. The RY of the different species varied relative to the accompanying species, the sampling date and the year. Competition was partial (RYT > 1) only in mixtures including C. erecta plants originated from rhizomes. From the simulations with the model it was determined that the result of the competition depends on the initial size (BI) of C. erecta rhizomes. Furthermore, the sensibility analysis highlights the importance of the intrinsic growth rate (r) in the definition of RY. Under competition conditions the biological parameters defining the growth of the species studied were particularly r and BI. The results emphasize the complex interaction of the factors determining the competitive balance, between both propagation forms of C. erecta, and between those forms and other weed species.

The population and community level consequences of positive interactions between plants remain poorly explored. In this study we incorporate positive resource-mediated interactions in classic resource competition theory and investigate the main consequences for plant population dynamics and species coexistence. We focus on plant communities for which water infiltration rates exhibit positive dependency on plant biomass and where plant responses can be improved by shading, particularly under water limiting conditions. We show that the effects of these two resource-mediated positive interactions are similar and additive. We predict that positive interactions shift the transition points between different species compositions along environmental gradients and that realized niche widths will expand or shrink. Furthermore, continuous transitions between different community compositions can become discontinuous and bistability or tristability can occur. Moreover, increased infiltration rates may give rise to a new potential coexistence mechanism that we call controlled facilitation. © 2009 Elsevier Inc. All rights reserved.

In this paper, we develop a new innovation diffusion model for two competing products, which allows us to evaluate the effect of competition both on the dynamics of within-product and cross-product word-of-mouth and on the definition of the residual market potential of each product. This model, which we call Lotka–Volterra model with churn, LVch, generalizes another model for competition, the unbalanced competition and regime change diachronic model (UCRCD), which assumes a common residual market and a delayed entrance for the second product. We compare the performance of these models in describing the competition between two blockbuster formats in the music industry, the compact cassette and the compact disc. In particular, we analyze the evolution of these technologies in the U.S. market for pre-recorded music, for which annual sales data are available from 1973 to 2012, and find that the LVch model outperforms the UCRCD. An interesting aspect of this application relies on the fact that there is a single product, the music album, which is commercialized in two different formats, so that competition arises between formats and not between two products in the same commercial category.

Texto completo: acesso restrito. p. 773-803 Submitted by Suelen Reis (suelen_suzane@hotmail.com) on 2012-12-12T11:47:29Z No. of bitstreams: 1 1-s2.0-S0895717702002273-main.pdf: 3087586 bytes, checksum: ab940c0ba1c57b163cb27c16f0b8a5ee (MD5) Made available in DSpace on 2012-12-12T11:47:29Z (GMT). No. of bitstreams: 1 1-s2.0-S0895717702002273-main.pdf: 3087586 bytes, checksum: ab940c0ba1c57b163cb27c16f0b8a5ee (MD5) Previous issue date: 2002-11 In this paper, we propose and analyse a mathematical model of cancer treatment by chemotherapy where there is metastasis from a primary to a secondary site by the cancer cells. We view the interactions between normal and cancer cells as being competitive for available resources, and we think of the chemotherapy agent as a predator on both normal and cancer cells. The metastasis may be time delayed. The analysis is carried out both analytically (where possible) and numerically. In particular, we look at the existence and stability of equilibria of our model, which consists of six differential equations. As well, we investigate (numerically) the behavior of solutions for various values of the parameters.

Ana dağıtım üsleri (ADÜ) her noktadan diğer her noktaya akışın olduğu ağlarda toplama ve dağıtma noktalarıdır. ADÜ yer seçimi problemi, ADÜ?lerin yerlerinin belirlenmesi ve ADÜ olmayan noktaların bu ADÜ?lere atanması olarak tanımlanmaktadır.ADÜ?lerin kullanıldığı sektörlerde çok sayıda firma rekabet halinde olsa da ana akım ADÜ yer seçimi çalışmaları tek karar vericinin amaç fonksiyonları üzerinde yoğunlamıştır. Bu tezde iki karar vericinin olduğu bir ADÜ yer seçimi problemi incelenmiştir. Karar vericiler sırayla ADÜ yerlerini seçmekte ve müşteriler sağlanan hizmet seviyelerine göre bunlardan birini tercih etmektedir. Karar vericiler kendi pazar paylarını enbüyüklemeye çalışmaktadır. Ekonomi, perakende yer seçimi ve yöneylem araştırması alanlarındaki çalışmalarının incelenmesinin ardından lider (ilk karar verici) ve takipçi (sonraki karar verici) için iki farklı problem tanımlanmıştır. Problemler kombinatoriyal eniyileme problemleri olarak tanımlanmış ve karmaşıklık sınıflarının NP-zor olduğu ispatlanmıştır. Bu problemler için doğrusal modeller sunulmuştur. Ayrıca takipçinin problemi için doğrusal modelden daha az bilgisayar hafızası ve çalışma süresine ihtiyaç duyan kesin çözüm algoritmalar geliştirilmiştir. Modeller ve algoritmaların performansı ADÜ çalışmalarında sıkça kullanılan iki veri kümesi üzerindeki sayısal çalışmayla incelenmiştir. Hubs are consolidation and dissemination points in many-to-many flow networks. The hub location problem is to locate hubs among available nodes and allocate non-hub nodes to these hubs. The mainstream hub location studies focus on optimal decisions of one decision-maker with respect to some objective(s) even though the markets that benefit hubbing are oligopolies. Therefore, in this thesis, we propose a competitive hub location problem where the market is assumed to be a duopoly. Two decision-makers (or firms) sequentially decide the locations of their hubs and then customers choose the firm according to provided service levels. Each decision-maker aims to maximize his/her market share. Having investigated the existing studies in the field of economy, retail location and operation research, we propose two problems for the leader (former decision-maker) and follower (latter decision-maker): (rXp) hub-medianoid and (rp) hub-centroid problems. After defining them as combinatorial optimization problems, the problems are proved to be NP-hard. Linear programming models are presented for these problems as well as exact solution algorithms for the (rp) hub-centroid problem that outperform the linear model in terms of memory requirement and CPU time. The performance of models and algorithms are tested by the computational analysis conducted on two well-known data sets from the hub location literature. 90

In this paper we study the scaling behavior of the fluctuations in the steady state $W_S$ with the system size $N$ for a surface growth process given by the competition between the surface relaxation (SRM) and the Ballistic Deposition (BD) models on degree uncorrelated Scale Free networks (SF), characterized by a degree distribution $P(k)\sim k^{-\lambda}$, where $k$ is the degree of a node. It is known that the fluctuations of the SRM model above the critical dimension ($d_c=2$) scales logarithmically with $N$ on euclidean lattices. However, Pastore y Piontti {\it et. al.} [A. L. Pastore y Piontti {\it et. al.}, Phys. Rev. E {\bf 76}, 046117 (2007)] found that the fluctuations of the SRM model in SF networks scale logarithmically with $N$ for $\lambda, Comment: 8 pages, 6 figures

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model. Henrique Oliveira and Saber Elaydi

Bu çalısmada rekabette üstünlük saglamaya dönük modelleri incelemek ve tasarlanan yeni bir modelle gıda sektöründe faaliyet gösteren bir firmada uygulama yapmak amacıyla hazırlanmıstır. Bu amaçla önce rekabet konusunda önemli üç model incelenmis, bu modellerden esinlenerek firmanın kendi stratejilerini çizebilecegi yeni bir model tasarlanmıstır. Tasarlanan yeni modele göre incelenen firmanın önce yaptıgı isler analiz edilmis, analiz sonucunda firmanın kar getirmeyen birçok ise kar elde ettigi islerden daha fazla zaman ayırdıgı ortaya çıkmıstır. Karlıgı düsük olan isler elimine edilmis ve karlılıgı yüksek olan islerle ilgili detaylı bir analiz sonucu firmanın yeni rekabet modeli ortaya konmustur., This study has been prepared to examine the models aimed at providing advantage in competition, and to practise with a new model in a firm active in food sector. Thus, the studies of the management groups? that have important studies on the competition have been examined, and a new model which the firm can create its own strategies inspired by these models has been planned. In accordance with the new model planned, the works that the firm carried out have been analysed, and it has been found that as a result of the analysis, the firm spent more time on most works having little profit than the works having much profit. The works having little profit have been eliminated, and the new competition model of the firm has been put forward as a result of a detailed analysis on the works having much profit.

Most mathematical models in the biological literature that describe inherently spatial phenomena of interacting populations consist of systems of ordinary differential equations obtained under global dispersion assumptions, thus leaving out any spatial structure. Interacting particle systems are Markov processes with state space $F^S$ where $F$ is a finite set of colors and where $S$ is a spatial structure, typically $\Z^d$. They are ideally suited to study the consequences of the inclusion of a spatial structure in the form of local interactions. We investigate the mathematical properties (stationary distribution, geometry of the configurations, phase transitions) of various multicolor particle systems defined on $\Z^d$. Each of these systems is intended to model local interactions within a spatially structured community of populations. More precisely, the biological processes we investigate are ecological succession, allelopathy or competition between an inhibitory species and a susceptible species, multi-species host-symbiont interactions, and persistent gene flow from transgenic crops to wild relatives through pollination in a heterogeneous environment. The mathematical techniques are probabilistic, including coupling, duality, multiscale arguments, oriented percolation, asymptotic properties of random walks, and large deviations estimates.; La plupart des modèles mathématiques introduits dans la littérature biologique décrivant des phénomènes spatiaux de populations en interaction consistent en des systèmes d'équations différentielles ordinaires obtenues sous des hypothèses de dispersion globale, excluant par conséquent toute structure spatiale. Les systèmes de particules, au contraire, sont des processus de Markov d'espace d'états $F^S$ où $F$ est un ensemble fini de couleurs et $S$ est une structure spatiale, typiquement $\Z^d$. Ils sont en ce sens parfaitement adaptés à l'étude des conséquences de l'inclusion d'une structure spatiale sous forme d'interactions locales. Nous étudions les propriétés mathématiques (mesures stationnaires, géométrie des configurations, transitions de phases) de différents systèmes de particules multicolores définis sur $\Z^d$. Chacun de ces systèmes est déstiné à modéliser les interactions locales au sein d'une communauté de populations structurée spatialement. Plus précisément, les processus biologiques étudiés sont la succession écologique, l'allélopathie ou compétition entre une espèce inhibitrice et une espèce sensible, les interactions multispécifiques hôtes-symbiontes, et les migrations continues de gènes des cultures transgéniques par pollinisation en milieu hétérogène. Les techniques mathématiques sont purement probabilistes, incluant le couplage, la dualité, les arguments multi-échelle, la percolation orientée, les propriétés asymptôtiques des marches aléatoires, ou encore les estimations de grandes déviations.

Most mathematical models in the biological literature that describe inherently spatial phenomena of interacting populations consist of systems of ordinary differential equations obtained under global dispersion assumptions, thus leaving out any spatial structure. Interacting particle systems are Markov processes with state space $F^S$ where $F$ is a finite set of colors and where $S$ is a spatial structure, typically $\Z^d$. They are ideally suited to study the consequences of the inclusion of a spatial structure in the form of local interactions. We investigate the mathematical properties (stationary distribution, geometry of the configurations, phase transitions) of various multicolor particle systems defined on $\Z^d$. Each of these systems is intended to model local interactions within a spatially structured community of populations. More precisely, the biological processes we investigate are ecological succession, allelopathy or competition between an inhibitory species and a susceptible species, multi-species host-symbiont interactions, and persistent gene flow from transgenic crops to wild relatives through pollination in a heterogeneous environment. The mathematical techniques are probabilistic, including coupling, duality, multiscale arguments, oriented percolation, asymptotic properties of random walks, and large deviations estimates., La plupart des modèles mathématiques introduits dans la littérature biologique décrivant des phénomènes spatiaux de populations en interaction consistent en des systèmes d'équations différentielles ordinaires obtenues sous des hypothèses de dispersion globale, excluant par conséquent toute structure spatiale. Les systèmes de particules, au contraire, sont des processus de Markov d'espace d'états $F^S$ où $F$ est un ensemble fini de couleurs et $S$ est une structure spatiale, typiquement $\Z^d$. Ils sont en ce sens parfaitement adaptés à l'étude des conséquences de l'inclusion d'une structure spatiale sous forme d'interactions locales. Nous étudions les propriétés mathématiques (mesures stationnaires, géométrie des configurations, transitions de phases) de différents systèmes de particules multicolores définis sur $\Z^d$. Chacun de ces systèmes est déstiné à modéliser les interactions locales au sein d'une communauté de populations structurée spatialement. Plus précisément, les processus biologiques étudiés sont la succession écologique, l'allélopathie ou compétition entre une espèce inhibitrice et une espèce sensible, les interactions multispécifiques hôtes-symbiontes, et les migrations continues de gènes des cultures transgéniques par pollinisation en milieu hétérogène. Les techniques mathématiques sont purement probabilistes, incluant le couplage, la dualité, les arguments multi-échelle, la percolation orientée, les propriétés asymptôtiques des marches aléatoires, ou encore les estimations de grandes déviations.

Texto completo: acesso restrito. p. 773-803 Submitted by Suelen Reis (suelen_suzane@hotmail.com) on 2012-12-12T11:47:29Z No. of bitstreams: 1 1-s2.0-S0895717702002273-main.pdf: 3087586 bytes, checksum: ab940c0ba1c57b163cb27c16f0b8a5ee (MD5) Made available in DSpace on 2012-12-12T11:47:29Z (GMT). No. of bitstreams: 1 1-s2.0-S0895717702002273-main.pdf: 3087586 bytes, checksum: ab940c0ba1c57b163cb27c16f0b8a5ee (MD5) Previous issue date: 2002-11 In this paper, we propose and analyse a mathematical model of cancer treatment by chemotherapy where there is metastasis from a primary to a secondary site by the cancer cells. We view the interactions between normal and cancer cells as being competitive for available resources, and we think of the chemotherapy agent as a predator on both normal and cancer cells. The metastasis may be time delayed. The analysis is carried out both analytically (where possible) and numerically. In particular, we look at the existence and stability of equilibria of our model, which consists of six differential equations. As well, we investigate (numerically) the behavior of solutions for various values of the parameters.

This work is a study of three interacting particle systems that are modified versions of the contact process. The contact process is a spin system defined on a graph and is commonly taken as a model for the spread of an infection in a population; transmission of the infection happens by proximity (contact). The first two models we consider – the grass-bushes-trees model and the multitype contact process – are models for competition between species in Ecology. The third model – annealed approximation to boolean networks – approximately describes the transmission of information among genes in a cell. We consider the grass-bushes-trees model on the set of integers, ℤ. Each point of ℤ is a region of space. In the continuous-time dynamics, at each instant each region can be either empty (state 0) or occupied by an individual of one of two existing species (states 1 and 2). Occupants of both species die at rate 1, leaving their regions empty, and send descendents to neighboring regions at rate λ. An individuals of type 1 may be born on a region previously occupied by an individual of type 2, but the converse is forbidden. We take the "heaviside" initial configuration in which all sites to the left of the origin are occupied by type 1 individuals and all sites to the right of the origin are occupied by type 2 individuals. If the birth of new individuals is allowed to occur at sites that are not adjacent to the parent, and if the rate λ is supercritical for the usual contact process on ℤ, we see the formation of an interface region in which both types coexist. Addressing a conjecture of Cox and Durrett (1995), we prove that the size of this region is stochastically tight. The multitype contact process on ℤ is a process identical to the grass-bushes-trees model in every respect except that no births can occur at previously occupied sites; in particular, the model is symmetric for both species. We again start the process from the heaviside configuration and prove that the size of the interface region is tight. In addition, we prove that the position of the interface, when properly rescaled, converges to Brownian motion. Finally, we give necessary and sufficient conditions on the initial configuration so that one of the two species becomes extinct with probability one and also so that both species are present at all times with positive probability. Lastly, we consider a model proposed by Derrida and Pomeau (1986) and recently studied by Chatterjee and Durrett (2009); it is defined as an approximation to S. Kauffman's boolean networks (1969). The model starts with the choice of a random directed graph on n vertices; each node has r input nodes pointing at it. A discrete time threshold contact process is then considered on this graph: at each instant, each site has probability q of choosing to receive input; if it does, and if at least one of its inputs were occupied by a 1 at the previous instant, then it is labeled with a 1; in all other cases, it is labeled with a 0. r and q are kept fixed and n is taken to infinity. Improving a result of Chatterjee and Durrett, we show that if qr > 1, then the time of persistence of the dynamics is exponential in n.