The field of stochastic growth encompasses various different processes which are ubiquitously seen across the physical world. In many systems, stochasticity appears quite naturally, where inherent randomness provides the right setting for the tone of motion and interaction, whose symphony leads to the surprising emergence of interesting patterns and structure. Although on the microscopic scale one can be overwhelmed by the randomness arising from the fluctuating interactions between components, on the macroscopic scale, however, one is mesmerized by the emergence of mathematical beauty and symmetry, leading to complex structures with fractal architecture. Competition between components adds an extra degree of complexity and leads to the possibility of critical behaviour and phase transitions. It is an important aspect of many systems, and in order to provide a full explanation of many natural phenomena, we have to understand the role it plays on modifying behaviour. The combination of stochastic growth and competition leads to the emergence of interesting complex patterns. They occur in various systems and in many forms, and thus we treat competition in growth models driven by different laws for the stochastic noise. As a consequence our results are widely applicable and we encourage the reader to find good use for them in their respective field. In this thesis we study stochastic systems containing interacting particles whose motion and interplay lead to directed growth structures on a particular geometry. We show how the effect of the overall geometry in many growth processes can be captured elegantly in terms of a time dependent metric. A natural example we treat is isoradial growth in two dimensions, with domain boundaries of competing microbial species as an example of a system with a homogeneously changing metric. In general, we view domain boundaries as space-time trajectories of particles moving on a dynamic surface and map those into more easily tractable systems with constant metric. This leads to establishing a generic relation between locally interacting, scale invariant stochastic space-time trajectories under constant and time dependent metric. Indeed "the book of nature is written in the language of mathematics" (Galileo Galilei) and we provide a mathematical framework for various systems with various interactions and our results are backed with numerical confirmation.

The book provides an introduction to deterministic (and some stochastic) modeling of spatiotemporal phenomena in ecology, epidemiology, and neural systems. A survey of the classical models in the fields with up to date applications is given. The book begins with detailed description of how spatial dynamics/diffusive processes influence the dynamics of biological populations. These processes play a key role in understanding the outbreak and spread of pandemics which help us in designing the control strategies from the public health perspective. A brief discussion on the functional mechanism of the brain (single neuron models and network level) with classical models of neuronal dynamics in space and time is given. Relevant phenomena and existing modeling approaches in ecology, epidemiology and neuroscience are introduced, which provide examples of pattern formation in these models. The analysis of patterns enables us to study the dynamics of macroscopic and microscopic behaviour of underlying systems and travelling wave type patterns observed in dispersive systems. Moving on to virus dynamics, authors present a detailed analysis of different types models of infectious diseases including two models for influenza, five models for Ebola virus and seven models for Zika virus with diffusion and time delay. A Chapter is devoted for the study of Brain Dynamics (Neural systems in space and time). Significant advances made in modeling the reaction-diffusion systems are presented and spatiotemporal patterning in the systems is reviewed. Development of appropriate mathematical models and detailed analysis (such as linear stability, weakly nonlinear analysis, bifurcation analysis, control theory, numerical simulation) are presented. Key Features Covers the fundamental concepts and mathematical skills required to analyse reaction-diffusion models for biological populations. Concepts are introduced in such a way that readers with a basic knowledge of differential equations and numerical methods can understand the analysis. The results are also illustrated with figures. Focuses on mathematical modeling and numerical simulations using basic conceptual and classic models of population dynamics, Virus and Brain dynamics. Covers wide range of models using spatial and non-spatial approaches. Covers single, two and multispecies reaction-diffusion models from ecology and models from bio-chemistry. Models are analysed for stability of equilibrium points, Turing instability, Hopf bifurcation and pattern formations. Uses Mathematica for problem solving and MATLAB for pattern formations. Contains solved Examples and Problems in Exercises. The Book is suitable for advanced undergraduate, graduate and research students. For those who are working in the above areas, it provides information from most of the recent works. The text presents all the fundamental concepts and mathematical skills needed to build models and perform analyses.

This thesis is concerned with certain properties of stochastic growth models. A stochastic growth model is a model of infection spread, through a population of individuals, that incorporates an element of randomness. The models we consider are variations on the contact process, the simplest stochastic growth model with a recurrent infection. Three main examples are considered. The first example is a version of the contact process on the complete graph that incorporates dynamic monogamous partnerships. To our knowledge, this is the first rigorous study of a stochastic spatial model of infection spread that incorporates some form of social dynamics. The second example is a non-monotonic variation on the contact process, taking place on the one-dimensional lattice, in which there is a random incubation time for the infection. Some techniques exist for studying non-monotonic particle systems, specifically models of competing populations [38] [12]. However, ours is the first rigorous study of a non-monotonic stochastic spatial model of infection spread. The third example is an additive two-stage contact process, together with a general duality theory for multi-type additive growth models.

With descriptions of hundreds of the most important environmental and ecological models, this handbook is a unique and practical reference source. The Handbook of Environmental and Ecological Modeling is ideal for those working in environmental modeling, including regulators and managers who wish to understand the models used to make assessments. Overviews of more than 360 models are easily accessed in this handbook, allowing readers to quickly locate information they need about models available in a given ecosystem. The material in the Handbook of Environmental and Ecological Modeling is logically arranged according to ecosystem. Each of the sixteen chapters of the handbook covers a particular ecosystem, and includes not only the descriptions of the models, but also an overview of the state-of-the-art in modeling for that particular ecosystem. A summary of the spectrum of available models is also provided in each chapter. The extensive table of contents and the easy-to-use index put materials immediately at your fingertips.

The field of theoretical ecology has expanded dramatically in the last few years. This volume gives detailed coverage of the main developing areas in spatial ecological theory, and is written by world experts in the field. Integrating the perspective from field ecology with novel methods for simplifying spatial complexity, it offers a didactical treatment with a gradual increase in mathematical sophistication from beginning to end. In addition, the volume features introductions to those fundamental phenomena in spatial ecology where emerging spatial patterns influence ecological outcomes quantitatively. An appreciation of the consequences of this is required if ecological theory is to move on in the 21st century. Written for reseachers and graduate students in theoretical, evolutionary and spatial ecology, applied mathematics and spatial statistics, it will be seen as a ground breaking treatment of modern spatial ecological theory.