58 research outputs found
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
Fractional derivative models for the spread of diseases
This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution.This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution
Forced waves and their asymptotic behaviors in a Lotka-Volterra competition model with spatio-temporal nonlocal effect under climate change
In this paper, we propose a modified Lotka-Volterra competition model under climate change, which incorporates both spatial and temporal nonlocal effect. First, the theoretical analyses for forced waves of the model are performed, and the existence of the forced waves is proved by using the cross-iteration scheme combining with appropriate upper and lower solutions. Second, the asymptotic behaviors of the forced waves are derived by using the linearization and limiting method, and we find that the asymptotic behaviors of forced waves are mainly determined by the leading equations. In addition, some typical numerical examples are provided to illustrate the analytical results. By choosing three kinds of different kernel functions, it is found that the forced waves can be both monotonic and non-monotonic
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, Rā, and the theory of infinite dimensional
dynamical systems, we derive Rā and establish a threshold-type result for the global
dynamics in terms of Rā 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
Front propagation in the non-local Fisher-KPP equation
Tkachov P. Front propagation in the non-local Fisher-KPP equation. Bielefeld: UniversitƤt Bielefeld; 2017
Study of Malaria Transmission Dynamics by Mathematical Models
This Ph.D thesis focuses on modeling transmission and dispersal of one of the most common infectious disease, Malaria. Firstly, an integro-differential equation system is derived, based on the classical Ross-Macdonald model, toemphasize the impacts of latencies on disease dynamics. The novelty lies in the fact that different distributionfunctions are used to describe the variance of individual latencies. The theoretical results of this projectindicate that latencies reduce the basic reproduction number. Secondly, a patch model is derived to examine how travels of human beings affects the transmission and spread of Malaria. Due to coexistence of latency and dispersal, the model turns out to be a system of delay differential equations on patches with non-local infections. The results from this work indicate that although malaria has been eradicated in many countries since the 1980s, re-emergence of the disease is possible, and henceprecautionary measures should be taken accordingly. Thirdly, since there are more than five species of Malaria Plasmodium causing human malaria, and they are currently distributed in different geographic regions, co-invasion by multiple species of malaria may arise. We propose multi-species models to explore co-infection at within-host level and co-existence at the between-host level. The analysis shows that competition exclusion dominates at the within-host level, meaning that longterm co-infection of a single host by multiple species can be generically excluded. However, at thebetween-host level, long term co-existence of multiple species in a region is possible
Agent-based and continuous models of hopper bands for the Australian plague locust: How resource consumption mediates pulse formation and geometry
Locusts are significant agricultural pests. Under favorable environmental
conditions flightless juveniles may aggregate into coherent, aligned swarms
referred to as hopper bands. These bands are often observed as a propagating
wave having a dense front with rapidly decreasing density in the wake. A
tantalizing and common observation is that these fronts slow and steepen in the
presence of green vegetation. This suggests the collective motion of the band
is mediated by resource consumption. Our goal is to model and quantify this
effect. We focus on the Australian plague locust, for which excellent field and
experimental data is available. Exploiting the alignment of locusts in hopper
bands, we concentrate solely on the density variation perpendicular to the
front. We develop two models in tandem; an agent-based model that tracks the
position of individuals and a partial differential equation model that
describes locust density. In both these models, locust are either stationary
(and feeding) or moving. Resources decrease with feeding. The rate at which
locusts transition between moving and stationary (and vice versa) is enhanced
(diminished) by resource abundance. This effect proves essential to the
formation, shape, and speed of locust hopper bands in our models. From the
biological literature we estimate ranges for the ten input parameters of our
models. Sobol sensitivity analysis yields insight into how the band's
collective characteristics vary with changes in the input parameters. By
examining 4.4 million parameter combinations, we identify biologically
consistent parameters that reproduce field observations. We thus demonstrate
that resource-dependent behavior can explain the density distribution observed
in locust hopper bands. This work suggests that feeding behaviors should be an
intrinsic part of future modeling efforts.Comment: 26 pages, 11 figures, 3 tables, 3 appendices with 1 figure; revised
Introduction, Sec 1.1, and Discussion; cosmetic changes to figures; fixed
typos and made clarifications throughout; results unchange
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