Asymptotic behavior of a metapopulation model

We study the behavior of an infinite system of ordinary differential equations modeling the dynamics of a metapopulation, a set of (discrete) populations subject to local catastrophes and connected via migration under a mean field rule; the local population dynamics follow a generalized logistic law. We find a threshold below which all the solutions tend to total extinction of the metapopulation, which is then the only equilibrium; above the threshold, there exists a unique equilibrium with positive population, which, under an additional assumption, is globally attractive. The proofs employ tools from the theories of Markov processes and of dynamical systems.Comment: Published at http://dx.doi.org/10.1214/105051605000000070 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mathematical Modelling of Ecological Systems in Patchy Environments

In this thesis, we incorporate spatial structure into different ecological/epidemiological systems by applying the patch model. Firstly, we consider two specific costs of dispersal: (i) the period of time spent for migration; (ii) deaths during the dispersal process. Together with the delayed logistic growth, we propose a two-patch model in terms of delay differential equation with two constant time delays. The costs of dispersal, by themselves, only affect the population sizes at equilibrium and may even drive the populations to extinction. With oscillations induced by the delay in logistic growth, numerical examples are provided to illustrate the impact of loss by dispersal. Secondly, we study a predator-prey system in a two-patch environment with indirect effect (fear) considered. When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. The benefit of an anti-predation response is also included. We investigate the effect of anti-predation response on population dynamics by analyzing the model with a fixed response level and study the anti-predation strategies from an evolutionary perspective by applying adaptive dynamics. Thirdly, we explore the short-term or transient dynamics of some SIR infectious disease models over a patchy environment. Employing the measurements of reactivity of equilibrium and amplification rates previously used in ecology to study the response of an ecological system to perturbations to an equilibrium, we analyze the impact of the dispersals/travels between patches and other disease-related parameters on short term dynamics of these spatially structured disease models. This contrasts with most existing works on modelling the dynamics of infectious disease which are only interested in long-term disease dynamics in terms of the basic reproduction number

Analysis of a Patch Model for the Dynamical Transmission of Echinococcosis

A patch model for echinococcosis due to dogs migration is proposed to explore the effect of dogs migration among patches on the spread of echinococcosis. We firstly define the basic reproduction number R0. The mathematical results show that the dynamics of the model can be completely determined by R0. If R01, the model is permanence and endemic equilibrium is globally asymptotically stable. According to the simulations, it is shown that the larger diffusion of dogs from the lower epidemic areas to the higher prevalence areas can intensify the spread of echinococcosis. However, the larger diffusion of dogs from the higher prevalence areas to the lower epidemic areas can reduce the spread and is beneficial for disease control

Global stability of a two-patch cholera model with fast and slow transmissions

Please read abstract in the article.The Abdus Salam International Center for Theoretical Physics (ICTP) in Trieste-Italy under the Associateship Scheme, the African Center of Excellence in Information and Communication Technologies (CETIC) in Cameroon and the South African Research Chairs Initiative (SARChI Chair), in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.elsevier.com/locate/matcomhj2021Mathematics and Applied Mathematic

Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of Allee effects

Unlike unconditionally advantageous “Fisherian” variants that tend to spread throughout a species range once introduced anywhere, “bistable” variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or “engineered underdominance” to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications

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

Representing spatial interactions in simple ecological models

The real world is a spatial world, and all living organisms live in a spatial environment. For mathematical biologists striving to understand the dynamical behaviour and evolution of interacting populations, this obvious fact has not been an easy one to accommodate. Space was considered a disposable complication to systems for which basic questions remained unanswered and early studies ignored it. But as understanding of non-spatial systems developed attention turned to methods of incorporating the effects of spatial structure. The essential problem is how to usefully manage the vast amounts of information that are implicit in a fully heterogeneous spatial environment. Various solutions have been proposed but there is no single best approach which covers all circumstances. High dimensional systems range from partial differential equations which model continuous population densities in space to the more recent individual-based systems which are simulated with the aid of computers. This thesis develops a relatively new type of model with which to explore the middle ground between spatially naive models and these fully complex systems. The key observation is to note the existence of correlations in real systems which may naturally arise as a consequence of their dynamical interaction amongst neighbouring individuals in a local spatial environment. Reflecting this fact - but ignoring other large scale spatial structure - the new models are developed as differential equations (pair models) which are based on these correlations. Effort is directed at a first-principles derivation from explicit assumptions with well stated approximations so the origin of the models is properly understood. The first step is consideration of simple direct neighbour correlations. This is then extended to cover larger local correlations and the implications of local spatial geometry. Some success is achieved in establishing the necessary framework and notation for future development. However, complexity quickly multiplies and on occasion conjectures necessarily replace rigorous derivations. Nevertheless, useful models result. Examples are taken from a range of simple and abstract ecological models, based on game theory, predator-prey systems and epidemiology. The motivation is always the illustration of possibilities rather than in depth investigation. Throughout the thesis, a dual interpretation of the models un-folds. Sometimes it can be helpful to view them as approximations to more complex spatial models. On the other hand, they stand as alternative descriptions of space in their own right. This second interpretation is found to be valuable and emphasis is placed upon it in the examples. For the game theory and predator-prey examples, the behaviour of the new models is not radically different from their non-spatial equivalents. Nevertheless, quantitative behavioural consequences of the spatial structure are discerned. Results of interest are obtained in the case of infection systems, where more realistic behaviour an improvement on non-spatial models is observed. Cautiously optimistic conclusions are reached that this, middle road of spatial modelling has an important contribution to make to the field