Differential Equations arising from Organising Principles in Biology

This workshop brought together experts in modeling and analysis of organising principles of multiscale biological systems such as cell assemblies, tissues and populations. We focused on questions arising in systems biology and medicine which are related to emergence, function and control of spatial and inter-individual heterogeneity in population dynamics. There were three main areas represented of differential equation models in mathematical biology. The first area involved the mathematical description of structured populations. The second area concerned invasion, pattern formation and collective dynamics. The third area treated the evolution and adaptation of populations, following the Darwinian paradigm. These problems led to differential equations, which frequently are non-trivial extensions of classical problems. The examples included but were not limited to transport-type equations with nonlocal boundary conditions, mixed ODE-reaction-diffusion models, nonlocal diffusion and cross-diffusion problems or kinetic equations

Effects of impulsive harvesting and an evolving domain in a diffusive logistic model

In order to understand how the combination of domain evolution and impulsive harvesting affect the dynamics of a population, we propose a diffusive logistic population model with impulsive harvesting on a periodically evolving domain. Initially the ecological reproduction index of the impulsive problem is introduced and given by an explicit formula, which depends on the domain evolution rate and the impulsive function. Then the threshold dynamics of the population under monotone or nonmonotone impulsive harvesting is established based on this index. Finally numerical simulations are carried out to illustrate our theoretical results, and reveal that a large domain evolution rate can improve the population survival, no matter which impulsive harvesting takes place. Contrary, impulsive harvesting has a negative effect on the population survival, and can even lead to the extinction of the population.Comment: 26 pages, 8 figure

Evolution dynamics of some population models in heterogeneous environments

Spatial and/or temporal evolutions are very important topics in epidemiology and ecology. This thesis is devoted to the study of the global dynamics of some population models incorporating with environmental heterogeneities. Vector-borne diseases such as West Nile virus and malaria, pose a threat to public health worldwide. Both vector life cycle and parasite development are highly sensitive to climate factors. To understand the role of seasonality on disease spread, we start with a periodic West Nile virus transmission model with time-varying incubation periods. Apart from seasonal variations, another important feature of our environment is the spatial heterogeneity. Hence, we incorporate the movement of both vectors and hosts, temperature-dependent incubation periods, seasonal fluctuations and spatial heterogeneity into a general reaction-diffusion vector-borne disease model. By using the theory of basic reproduction number, R₀, and the theory of infinite dimensional dynamical systems, we derive R₀ and establish a threshold-type result for the global dynamics in terms of R₀ for each model. As biological invasions have significant impacts on ecology and human society, how the growth and spatial spread of invasive species interact with environment becomes an important and challenging problem. We first propose an impulsive integro-differential model to describe a single invading species with a birth pulse in the reproductive stage and a nonlocal dispersal stage. Next, we study the propagation dynamics for a class of integro-difference two-species competition models in a spatially periodic habitat

Partial Differential Equations in Ecology

Partial differential equations (PDEs) have been used in theoretical ecology research for more than eighty years. Nowadays, along with a variety of different mathematical techniques, they remain as an efficient, widely used modelling framework; as a matter of fact, the range of PDE applications has even become broader. This volume presents a collection of case studies where applications range from bacterial systems to population dynamics of human riots

Three Population Models Applied to Competition, Disease and Invasion

In this work, we present three diffrent types of population models. The first two models are examined in the context of optimal control problems. The third involves the construction of an invasion model using a significant amount of data. The first model describes the interaction of three populations, motivated by a combat scenario. One of the three populations can switch the mode of alliance with the other two populations between cooperation and competition. The other two populations always compete with each other. In this system of parabolic partial differential equations, the control is the function which measures the strength of alliance. The second model is a metapopulation SIR model for the spread of rabies among raccoons. This system of ordinary differential equations considers subpopulations connected via movement of individuals between subpopulations. The strength of the connectivity between two subpopulations is inversely proportional to the geographical distance between them. We apply control theory to find the best strategy (timing and location) for vaccination to control the disease. The third problem involves construction of a model of the spread of Eurasian collared doves in the U.S. using an integrodifference equation. We investigate the effect of spatial variation of the length of the growing season on the growth rate of the collared dove. Since the growing season length affects the breeding season length, we take into account the difference in the number of clutches in estimating the number of offspring produced each breeding season

Applications of stability theory to ecological problems

The goal of ecology is to investigate the interactions among organisms and their environment. However, ecological systems often exhibit complex dynamics. The application of mathematics to ecological problems has made tremendous progress over the years and many mathematical methods and tools have been developed for the exploration, whether analytical or numerical, of these dynamics. Mathematicians often study ecological systems by modelling them with partial differential equations (PDEs). Calculating the stability of solutions to these PDE systems is a classical question. This thesis first explores the concept of stability in the context of predator-prey invasions. Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised. Vegetation pattern formation in water-limited environments is another topic where stability theory plays a key role; indeed in mathematical models, these patterns are often the results of the dynamics that arise from perturbations to an unstable homogeneous steady state. Vegetation patterns are widespread in semi-deserts and aerial photographs of arid and semi-arid ecosystems have shown several kilometers square of these patterns. On hillsides in particular, vegetation is organised into banded spatial patterns. We first choose a domain in parameter space and calculate the boundary of the region in parameter space where pattern solutions exist. Finally we conclude with investigating how changes in the mean annual rainfall affect the properties of pattern solutions. Our work also highlights the importance of research on the calculation of the absolute spectrum for non-constant solutions