Similarity solutions for systems arising from an Aedes aegypti model

In a recent paper a new model for the Aedes aegypti mosquito dispersal dynamics was proposed and its Lie point symmetries were investigated. According to the carried group classification, the maximal symmetry Lie algebra of the nonlinear cases is reached whenever the advection term vanishes. In this work we analyze the family of systems obtained when the wind effects on the proposed model are neglected. Wide new classes of solutions to the systems under consideration are obtained.     

A hybrid model for selection of an outsourcing provider

Outsourcing is a good strategy for firms that need to reduce operating costs and improve competitiveness and it is important that firms scientifically select appropriate outsourcing providers. Some efforts have been made to find systematic ways to deal with outsourcing problems, but these efforts incorrectly assumed that the criteria used in the decision process are independent, which is not true in the real world. In this study, we propose a new hybrid multiple criteria decision-making (MCDM) model, which addresses the dependent relationships between the various criteria. The relations-structure among the criteria is built with the aid of the decision-making trial and evaluation laboratory (DEMATEL) method. Decision-makers tend to hold diverse opinions about their preferences due to incomplete information and knowledge, or inherent conflict between various departments. We further used the fuzzy preference programming and the analytic network process (ANP) to form a model for the selection of partners for outsourcing providers. The proposed model can help practitioners improve their decision making process, especially when criteria are numerous and inter-related. The method is demonstrated using data from a Taiwanese airline.     

An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model

Asymptotic expansion of singular integrals is presented for the study of the existence and uniqueness of spiky steady states of Keller–Segel's minimal chemotaxis model. A tool is developed to find asymptotic expansion of eigenpairs of a nonlocal singular eigenvalue problem arising from the study of stability of the spiky steady state. New insights are given to simplify significantly the proofs of the main results in [1].     

Autoimmunity arising from bystander proliferation of T cells in an immune response model

We study a mathematical model of immune response by T cells where the regulatory T cells (Treg) inhibit interleukin 2 secretion. The bystander proliferation to an immune response is modelled. We consider an asymmetry reflecting that the difference between the growth and death rates can be higher for the active T cells and Tregs than for the inactive. This asymmetry leads to a better understanding of the bystander proliferation. An exposure to a pathogen results in an increased proliferation rate of the bystander T cells. If the population of the bystander T cells becomes large enough, autoimmunity can arise, eventually after a long transient period.     

An eigenvalue problem arising in a patch formation model

A simple numerical scheme has been developed for the solution of the eigenvalue problem arising in a patch formation model given by Del Grosso et al. [1]. The scheme is based on finding bounds which separate the eigenvalues. The exact eigenvalues are obtained by solving an algebraic equation given by the corresponding regular Frobenius series solution. At the same time eigenfunctions may also be obtained from this series solution.     

A free boundary problem arising from DCIS mathematical model

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.     

An orienteering model for the search and rescue problem

In this paper, we propose a new model for the search and rescue problem. We focus on the case of a single airborne search asset through a connected space and continuous time with a maximum travel time \(T\) . The intent is to maximize the detection of a cooperative target (search and rescue). The proposed model is based on the assumption of existing a priori information (e.g., result of information fusion process) to establish a spatial distribution of probability of containment in possible geographic locations. The possibility area is defined using a cut threshold on the probability of containment and the search path as well as the allocation of the level of effort to each region in the search space is obtained based on an orienteering model. We illustrate the application of the proposed model on an empirical example.     

A two-cell model of search for an evading target

A form of game-theoretic model is proposed for the problem of search for an evading target. The region of search is divided into two cells, with the evader starting in the first cell and having the objective of moving from there, through the second to some goal area, without being detected. Optimal searcher strategy is derived in analytical form for some special cases of practical interest. The results indicate that if there is any difference in detectability in the two cells, then there is optimally a rather large concentration of search effort in the more favourable cell. The special cases are also compared numerically with some more general situations. Although both the model and the solution are relatively simple, they serve to demonstrate the potential value of this form of approach to the problem, as well as the complexity of the problem for more general situations.     

Mathematical analysis of a model of chemotaxis arising from morphogenesis

We consider non‐negative solution couples (u,v) of with positive parameters χ and λ, where the spatial domain is the interval (0,1). This system appears as a limit case of a model for morphogenesis proposed by Bollenbach et al. (Phys. Rev. E. 75 , 2007). Under suitable boundary conditions, modeling the presence of a morphogen source at x = 0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover, we prove the convergence of the solution to the unique steady state provided that χ is small and λ is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements. Copyright © 2012 John Wiley & Sons, Ltd.     

A free boundary arising from a model for nerve conduction

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions (Trans. Amer. Math. Soc.277 (1983), 1–42) introduced the class of “viscosity” solutions of these equations and proved uniqueness within this class. This paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

Applied mathematical research topics arising from an industrial workshop

Since 1984, a special industrial mathematics workshop the Mathematics—inIndustry Study Group—has operated in Australia on an annualbasis. The study group has considered 63 topics from about 40separate organizations. In many cases, the topics required thedevelopment of appropriate mathematical models in order to givethe insights demanded by their industrial sponsors. In somecases, the topics required completely fresh approaches, therebycontributing to the development of new theories for variousapplications. This paper summarizes the operations of the study group, looksat the beneficial outcomes of the meetings, and present someinnovative work that has arisen from them. Examples addressedinclude: ultrafiltration, heat and mass transfer in a metalprocessing furnace, a vibrocompaction problem, vibrations inautomobile driveshafts, and airline crew scheduling.     

Set of involutions arising from an egglike inversive plane

To every egglike inversive plane there is associated a family of involutions of the point set of such that circles of are the fixed point sets of the involutions in . Korchmaros and Olanda characterized a family of involutions on a set of size n² + 1to be for an egglike inversive plane of order n by four conditions. In this paper, we give an alternative proof where the Galois space PG(3,n) in which is embedded is built up directly by using concepts and results on finite linear spaces.     

Autonomous change of behavior for environmental context: An intermittent search model with misunderstanding search pattern

Although foraging patterns have long been predicted to autonomously adapt to environmental conditions, empirical evidence has been found in recent years. This evidence suggests that the search strategy of animals is open to change so that animals can flexibly respond to their environment. In this study, we began with a simple computational model that possesses the principal features of an intermittent strategy, ie, careful local searches separated by longer steps, as a mechanism for relocation, where an agent in the model follows a rule to switch between two phases, but it could misunderstand this rule, ie, the agent follows an ambiguous switching rule. Thanks to this ambiguity, the agent's foraging strategy can continuously change. First, we demonstrate that our model can exhibit an autonomous change of strategy from Brownian‐type to Lévy type depending on the prey density, and we investigate the distribution of time intervals for switching between the phases. Moreover, we show that the model can display higher search efficiency than a correlated random walk.     

An application of Fourier transforms on finite Abelian groups to an enumeration arising from the Josephus problem

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We analyze an enumeration associated with the Josephus problem by applying a Fourier transform to a multivariate generating function. This yields a formula for the enumeration that reduces to a simple expression under a condition we call local prime abundance. Under this widely held condition, we prove (Corollary 3.4) that the proportion of Josephus permutations in the symmetric group Sn that map t to k (independent of the choice of t and k) is 1/n. Local prime abundance is intimately connected with a well-known result of S.S. Pillai, which we exploit for the purpose of determining when it holds and when it fails to hold. We pursue the first case where it fails, reducing an intractable DFT computation of the enumeration to a tractable one. A resulting computation shows that the enumeration is nontrivial for this case.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=DnZi-Znuk-A.     

Delayed feedback-based bifurcation control in an Internet congestion model

In this paper, the control of Hopf bifurcation in an Internet congestion model with a single link accessed by a single source is presented. By choosing the gain parameter as a bifurcation parameter, it is found that the system without control cannot guarantee a stationary sending rate. Furthermore, Hopf bifurcation occurs when the positive gain parameter of the system exceeds a critical value. For Internet congestion model, a control model based on delayed feedback is proposed and analyzed for delaying the onset of undesirable Hopf bifurcation. Numerical simulations are given to justify the validity of delayed feedback controller in bifurcation control.     

Existence for a model arising from the in situ vitrification process

Existence of a solution is established for a time-dependent problem that can be used to model the in situ vitrification process. Certain properties of the solution are also presented.