A stage-structured delayed reaction-diffusion model for competition between two species.

We formulate a delayed reaction-diffusion model that describes competition between two species in a stream. We divide each species into two compartments, individuals inhabiting on the benthos and individuals drifting in the stream. Time delays are incorporated to measure the time lengths from birth to maturity of the benthic populations. We assume that the growth of population takes place on the benthos and that dispersal occurs in the stream. Our system consists of two linear reaction-diffusion equations and two delayed ordinary differential equations. We study the dynamics of the non-spatial model, determine the existence and global stability of the equilibria, and provide conditions under which solutions converge to the equilibria. We show that the existence of traveling wave solutions can be established through compact integral operators. We define two real numbers and prove that they serve as the lower bounds of the speeds of traveling wave solutions in the system

The spatio-temporal dynamics of neutral genetic diversity

International audienceThe notions of pulled and pushed solutions of reaction-dispersal equations introduced by Garnier et al. (2012) and Roques et al. (2012) are based on a decomposition of the solutions into several components. In the framework of population dynamics, this decomposition is related to the spatio-temporal evolution of the genetic structure of a population. The pulled solutions describe a rapid erosion of neutral genetic diversity, while the pushed solutions are associated with a maintenance of diversity. This paper is a survey of the most recent applications of these notions to several standard models of population dynamics, including reaction-diffusion equations and systems and integro-differential equations. We describe several counterintuitive results, where unfavorable factors for the persistence and spreading of a population tend to promote diversity in this population. In particular, we show that the Allee effect, the existence of a competitor species, as well as the presence of climate constraints are factors which can promote diversity during a colonization. We also show that long distance dispersal events lead to a higher diversity, whereas the existence of a nonreproductive juvenile stage does not affect the neutral diversity in a range-expanding population

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog

Dispersal Evolution in Currents : spatial sorting promotes philopatry in upstream patches

Open Access via the Wiley Jisc Agreeement Funding – This work was funded by MarCRef, a collaboration between the Univ. of Aberdeen [grant number CF10434‐53] and Marine Scotland [grant number RG14645‐10], as part of Rebekka Allgayer's PhD program and Research Councils UK, Natural Environment Research Council, NE/T006935/1. The code for the model is available in a GitHub repository: .Peer reviewedPublisher PD

The Kuramoto model: A simple paradigm for synchronization phenomena

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included

Spatial-temporal patterns in evolutionary ecology and fluid turbulence

This thesis explores the turbulence of ecosystems, and the ecology of turbulence. Ecosystems and turbulent fluids are both highly non-equilibrium and exhibit spatio-temporal complexity during the course of their evolution. It might seem that they are too complicated to extract universal properties, even if there are any. Surprisingly, it turns out that each of them can shed light on the other, enabling them both to be solved. In particular, the techniques used to explore ecosystem dynamics turn out to be exactly what is needed to solve the problem of the laminar-turbulent transition in pipes. Accordingly, this thesis is organized into two parts. Part 1 discusses what governs the rate of evolution and what are the consequences of the interplay between ecology and evolution at different scales. Three different aspects of these underlying questions are included in this part: (1) We first study the phenomenon of anomalous population dynamics known as "rapid evolution", in which a fast evolutionary time scale emerges from intense ecological interactions between species. Specific examples are rotifer-algae and bacteria-phage, where the ecosystem is composed of a predator and its prey. However, a sub-population of mutant prey arises from strong environmental pressure, and the trade-off between selection from reproduction and predation is manifested in the patterns of eco-evolutionary dynamics. We discuss how to solve such system with inherent stochasticity by a generic and systematic analytical approach in the spirit of statistical mechanics, using a stochastic individual-level model. We show that this method can naturally capture the universal behavior of the stochastic dynamics from demographic noise without any additional and more biologically detailed assumptions. (2) Second, we address the question of the role of selection in evolution and its relationship with phenotypic fluctuations. Phenotypic fluctuations have been conjectured to be beneficial characteristics to protect against fluctuating selection from environmental changes. But it is not well-understood how phenotypic fluctuations shape the evolutionary trajectories of organisms. We address these questions in the context of directed evolution experiments on bacterial chemotactic phenotypes. Our stochastic modeling and experiments on the evolution of chemotactic fronts suggest that the strength of selection can determine whether or not phenotypic fluctuations grow or shrink during successive rounds of selection and growth. (3) The third aspect of the first part focuses on the paradox of coexistent stability in microbial ecosystems that display especially intricate evolutionary phenomena. We propose that horizontal gene transfer, an important evolutionary driving force, is also the driving force that can stabilize microbe-virus ecosystems. The particular biological system for our model is that of the marine cyanobacteria Prochlorococcus spp., one of the most abundant organisms on the planet, and its phage predator. Phylogenetic analysis reveals compelling evidence for horizontal gene transfer of photosynthesis genes between the bacteria and phage. We test our hypothesis by building a spatially-extended stochastic individual-level model and show that the presence of viral-mediated horizontal gene transfer can induce collective coevolution and ecosystem stability, leading to a large pan-genome, an accelerated evolutionary timescale, and the emergence of ecotypes that are adapted to the stratified levels of light transmission as a function of ocean depth. The goal of Part 2 is to understand the nature of the transition to turbulence in fluids, which has been a puzzle for more than a century. The novelty of our approach is that we consider transitional turbulence as a non-equilibrium phase transition. Accordingly we attempt to approach this problem by looking for an appropriate long-wavelength effective theory. We report evidence of candidate long-wavelength collective modes in direct numerical simulations of the Navier-Stokes equations in a pipe geometry, where we uncover unexpected spatio-temporal patterns reminiscent of ecological predator-prey dynamics. This finding allows us to construct a minimal Landau theory for transitional turbulence, which resembles a stochastic predator-prey model. This in turn can be mapped into the generic universality class of directed percolation. Stochastic simulations of this spatial-extended individual-level predator-prey model are able to recapitulate the experimentally observed super-exponential dependence of the lifetime of turbulent regions on Reynolds number near the onset of turbulence. We argue that these remarkable scaling phenomena reflect the presence of finite-size effects as the correlation length becomes of order the pipe diameter, leading to a universal finite-size scaling distribution for the velocity fluctuations.Ope

A Rabies Model with Distributed Latent Period and Territorial and Diffusing Rabid Foxes

abstract: Rabies is an infectious viral disease. It is usually fatal if a victim reaches the rabid stage, which starts after the appearance of disease symptoms. The disease virus attacks the central nervous system, and then it migrates from peripheral nerves to the spinal cord and brain. At the time when the rabies virus reaches the brain, the incubation period is over and the symptoms of clinical disease appear on the victim. From the brain, the virus travels via nerves to the salivary glands and saliva. A mathematical model is developed for the spread of rabies in a spatially distributed fox population to model the spread of the rabies epizootic through middle Europe that occurred in the second half of the 20th century. The model considers both territorial and wandering rabid foxes and includes a latent period for the infection. Since the model assumes these two kinds of rabid foxes, it is a system of both partial differential and integral equations (with integration over space and, occasionally, also over time). To study the spreading speeds of the rabies epidemic, the model is reduced to a scalar Volterra-Hammerstein integral equation, and space-time Laplace transform of the integral equation is used to derive implicit formulas for the spreading speed. The spreading speeds are discussed and implicit formulas are given for latent periods of fixed length, exponentially distributed length, Gamma distributed length, and log-normally distributed length. A number of analytic and numerical results are shown pertaining to the spreading speeds. Further, a numerical algorithm is described for the simulation of the spread of rabies in a spatially distributed fox population on a bounded domain with Dirichlet boundary conditions. I propose the following methods for the numerical approximation of solutions. The partial differential and integral equations are discretized in the space variable by central differences of second order and by the composite trapezoidal rule. Next, the ordinary or delay differential equations that are obtained this way are discretized in time by explicit continuous Runge-Kutta methods of fourth order for ordinary and delay differential systems. My particular interest is in how the partition of rabid foxes into territorial and diffusing rabid foxes influences the spreading speed, a question that can be answered by purely analytic means only for small basic reproduction numbers. I will restrict the numerical analysis to latent periods of fixed length and to exponentially distributed latent periods. The results of the numerical calculations are compared for latent periods of fixed and exponentially distributed length and for various proportions of territorial and wandering rabid foxes. The speeds of spread observed in the simulations are compared to spreading speeds obtained by numerically solving the analytic formulas and to observed speeds of epizootic frontlines in the European rabies outbreak 1940 to 1980.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201

Nonlocal Models in Biology and Life Sciences: Sources, Developments, and Applications

Nonlocality is important in realistic mathematical models of physical and biological systems at small-length scales. It characterizes the properties of two individuals located in different locations. This review illustrates different nonlocal mathematical models applied to biology and life sciences. The major focus has been given to sources, developments, and applications of such models. Among other things, a systematic discussion has been provided for the conditions of pattern formations in biological systems of population dynamics. Special attention has also been given to nonlocal interactions on networks, network coupling and integration, including models for brain dynamics that provide us with an important tool to better understand neurodegenerative diseases. In addition, we have discussed nonlocal modelling approaches for cancer stem cells and tumor cells that are widely applied in the cell migration processes, growth, and avascular tumors in any organ. Furthermore, the discussed nonlocal continuum models can go sufficiently smaller scales applied to nanotechnology to build biosensors to sense biomaterial and its concentration. Piezoelectric and other smart materials are among them, and these devices are becoming increasingly important in the digital and physical world that is intrinsically interconnected with biological systems. Additionally, we have reviewed a nonlocal theory of peridynamics, which deals with continuous and discrete media and applies to model the relationship between fracture and healing in cortical bone, tissue growth and shrinkage, and other areas increasingly important in biomedical and bioengineering applications. Finally, we provided a comprehensive summary of emerging trends and highlighted future directions in this rapidly expanding field.Comment: 71 page

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience