Singular measure traveling waves in an epidemiological model with continuous phenotypes

We consider the reaction-diffusion equation \begin{equation*} u_t=u_{xx}+\mu\left(\int_\Omega M(y,z)u(t,x,z)dz-u\right) + u\left(a(y)-\int_\Omega K(y,z) u(t,x,z)dz\right) , \end{equation*} where $u=u(t,x,y)$ stands for the density of a theoretical population with a spatial ($x\in\mathbb R$) and phenotypic ($y\in\Omega\subset \mathbb R^n$) structure, $M(y,z)$ is a mutation kernel acting on the phenotypic space, $a(y)$ is a fitness function and $K(y,z)$ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation, and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $c^*:=2\sqrt{-\lambda_1}$, where $\lambda_1$ is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation

Modelling dryland vegetation patterns : nonlocal dispersal, temporal variability in precipitation and species coexistence

Spatiotemporal patterns of vegetation are a characteristic feature of dryland ecosystems occurring on all continents except Antarctica. The development of an understanding of their ecosystem dynamics is an issue of considerable socio-economic importance as both the livestock and agricultural sectors in dryland economies heavily depend on ecosystem functioning. Mathematical modelling is a powerful tool to disentangle the complex ecosystem dynamics. In this thesis, I present theoretical models to explore the impact of nonlocal seed dispersal and temporal precipitation variability on dryland vegetation patterns and propose several mechanisms that enable species coexistence within vegetation patterns. To do so, I present extensions of the Klausmeier reaction-advection-diffusion model, a well-established model describing the ecohydrological dynamics of vegetation patterns. Model analyses focus on pattern onset at high precipitation values (i.e. on the transition from uniformly vegetated to spatially patterned states) to assess the impact of nonlocal seed dispersal and precipitation seasonality and intermittency, and on comprehensive bifurcation analyses, including results on pattern existence and stability to investigate coexistence of species in the mathematical framework. Results include the inhibition of pattern onset due to long-range seed dispersal and put emphasis on the functional response of plants to low soil moisture levels to understand effects of rainfall intermittency. Moreover, results suggest that coexistence is facilitated by resource heterogeneities induced by the plant’s spatial self-organisation and highlight the importance of considering out-of-equilibrium solutions.UK Engineering and Physical Sciences Research Council (grant EP/L016508/01) Scottish Funding Council Heriot-Watt Universit

Evolution of Intraspecific Genetic Differences Across Heterogeneous Environments and the Potential Impacts on Environmental Restoration

Genetic differences evolve between seed sources of widely distributed species that can impact restoration success. Using the herbaceous perennial Geum triflorum (Pursh) as a model species, we examine genetic differences that evolve across a species? range in both physiology and fitness. G. triflorum occurs across highly differentiated environments consisting of prairie and alvar habitats. Seeds were collected from 22 populations across three eco-geographic regions. Through a common garden experiment in the prairie environment, I first examined differences in physiological traits between source regions and populations. Significant regional differentiation was observed for a majority of traits assessed, with equal or greater trait variation observed at the population scale. Secondly, differences in fitness between source regions were assessed over a period of three years using the ASTER model. Despite being in the home environment, fitness was considerably lower in prairie-sourced plants than alvar sourced plants, both each year and in total

Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations

We consider a mathematical model for the evolutionary dynamics of tumour cells in vascularised tumours under chemotherapy. The model comprises a system of coupled partial integro-differential equations for the phenotypic distribution of tumour cells, the concentration of oxygen and the concentration of a chemotherapeutic agent. In order to disentangle the impact of different evolutionary parameters on the emergence of intra-tumour phenotypic heterogeneity and the development of resistance to chemotherapy, we construct explicit solutions to the equation for the phenotypic distribution of tumour cells and provide a detailed quantitative characterisation of the long-time asymptotic behaviour of such solutions. Analytical results are integrated with numerical simulations of a calibrated version of the model based on biologically consistent parameter values. The results obtained provide a theoretical explanation for the observation that the phenotypic properties of tumour cells in vascularised tumours vary with the distance from the blood vessels. Moreover, we demonstrate that lower oxygen levels may correlate with higher levels of phenotypic variability, which suggests that the presence of hypoxic regions supports intra-tumour phenotypic heterogeneity. Finally, the results of our analysis put on a rigorous mathematical basis the idea, previously suggested by formal asymptotic results and numerical simulations, that hypoxia favours the selection for chemoresistant phenotypic variants prior to treatment. Consequently, this facilitates the development of resistance following chemotherapy

The evolution of exchange in small-scale societies of the Southern High Plains.

A political economic perspective, which envisions exchange as an activity embedded in broader social, economic, and political institutions, provides the theoretical foundation for understanding the alternative roles that exchange played in small-scale societies that inhabited the region. By necessity, a contextual perspective, which emphasizes both spatial and temporal parameters, is employed to investigate the interrelationships that existed between exchange and broader realms of social life.The Southern High Plains is a vast semiarid environment characterized by erratic climate conditions and incongruent resources distributions. Prior to A.D. 1250 the region was inhabited by small groups of mobile foragers. Except for the transfer of some high quality tool stone, evidence for exchange between these societies is extremely meager suggesting that important resources were obtained through residential mobility. The onset of the Middle Ceramic period around A.D. 1250 was marked by the sudden appearance of Plains Village tradition societies who occupied permanent settlements and practiced subsistence economies based on foraging and horticulture. Settlements of the period vary from single family homesteads to villages containing 250 people. The large numbers of habitation sites documented for the period indicate that the region experienced a dramatic increase in human populations at this time. Coinciding with these significant cultural changes, the Middle Ceramic period also witnessed the emergence of widespread exchange networks. This study examines this development and its meaning in small-scale societies of the region.Durable goods obtained through exchange can be described in a number of ways including the distances which items were traded and their function or meaning in society. In this study, utilitarian items, particularly chipped stone tools produced from high quality materials, are the most abundant exchange items documented. These objects were regularly traded over distances of 100 to 300 km. Nonutilitarian items, including jewelry produced from marine shell and precious stone, smoking pipes, and elaborately decorated ceramics were also obtained from communities located 350 to 550 km away. Given the distances involved, the latter objects are assumed to represent status or prestige items. While nonlocal utilitarian items are widespread throughout the region, status items are notably concentrated at a few communities.This study concludes that initially exchange was regional in scope and provided access to utilitarian items among recently settled populations. Although temporal trends are not well understood, exchange was later elaborated by a few communities and involved the procurement of utilitarian and nonutilitarian objects through long-distance trading expeditions to settlements outside the region. Importantly, this expansion was also accompanied by the appearance of other key developments including land tenure systems, intensified economic production, and regional trade centers. Altogether, these trends are interpreted as evidence for increasing social complexity and the emergence of local leaders who encouraged and organized these activities. Support for this interpretation is derived from the ethnographic record which demonstrates that the subsistence economy and exchange frequently provide important avenues by which emergent leaders distinguish themselves above other members of society. In this study exchange is seen as serving a dual role that simultaneously brought prestige to local leaders and enhanced the status and well-being of the communities they represented