Qualitative analysis of a mathematical model for tissue inflammation dynamics

The non-linear differential eguation $$\dot X = \frac{{rX}}{{I + X}} - \frac{{XY}}{{X + Y}},\dot Y = \alpha (I + \theta X - Y)$$ is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.     

Qualitative analysis for a cross-diffusion model of forest

The existence of a bounded global attractor for a cross-diffusion model of forest with homogeneous Dirichlet boundary condition is proved under some condition on the parameters Project supported by the National Natural Science Foundation of China (Grant No. 19671005).     

Rimming flow of a power-law fluid: Qualitative analysis of the mathematical model and analytical solutions

Rimming flow of a non-Newtonian fluid on the inner surface of a horizontal rotating cylinder is investigated. Simple lubrication theory is applied since the Reynolds number is small and liquid film is thin. For the steady-state flow of a power-law fluid the mathematical model reduces to a simple algebraic equation regarding the thickness of the liquid film. The qualitative analysis of this equation is carried out and the existence of two possible solutions is rigorously proved. Based on this qualitative analysis, different regimes of the rimming flow are defined and analyzed analytically. For the particular case, when the flow index in a power-law constitutive equation is equal to 1/2, the problem reduces to the fourth order algebraic equation which is solved analytically by Ferrari method.     

Qualitative analysis for a Wolbachia infection model with diffusion

We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-Schauder degree theory, we investigate the influence of diffusion on the Wolbachia infection dynamics. After identifying the system parameter regions in which diffusion alters the local stability of constant steady-states, we find sufficient conditions under which the system possesses inhomogeneous steady-states. Surprisingly, our mathematical analysis, with the help of numerical simulations, indicates that diffusion is able to lower the threshold value of the infection frequency over which Wolbachia can invade the whole population.     

Qualitative analysis of a ratio-dependent Holling-Tanner model

In this paper, we consider a Holling-Tanner system with ratio-dependence. First, we establish the sufficient conditions for the global stability of positive equilibrium by constructing Lyapunov function. Second, through a simple change of variables, we transform the ratio-dependent Holling-Tanner model into a better studied Liénard equation. As a result, the uniqueness of limit cycle can be solved.     

Qualitative analysis of a delayed non-Fickian model

This article focusses on the mathematical analysis of a delayed integro-differential model in which flux does not obey the classical Fick's law. The well-posedness of the integro-differential model in the Hadamard's sense is established. The dependence on the delay parameter of the total amount of desorpted/sorpted mass is studied. Numerical results that show the effectiveness of the model are included.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

A mathematical analysis of a fish school model

A model proposed in the literature for fish schools of relatively large size is studied for mathematical and qualitative properties. Existence, uniqueness and positivity of solutions are established and bifurcation properties relative to diffusion and alignment parameters are studied.     

A mathematical analysis for a model arising from public goods games

We study the effects of altruistic behaviors in a public goods game model which describes the competition between the farmers and the exploiters. Corresponding to different parametric regions, we analyze in detail the stability of the equilibrium states and obtain attraction regions for stable equilibria. Then using the upper–lower solution method and monotone iterations, we further show that for a family of wave speeds, there exist traveling wave solutions connecting one of the unstable states to the stable state. This answers a conjecture made by Wakano in [J.Y. Wakano, A mathematical analysis on public goods games in the continuous space, Math. Biosci. 201 (2006) 72–89]. The results indicate that when the penalty for the altruistic behavior is small, the growth rate of the population determines its survival or extinction states in the long run. Furthermore, if the two populations have the same total growth rate, altruism in the competition leads to a wide range of co-existent states. Numerical simulations are also presented to illustrate the theoretical results.     

Qualitative analysis for a ratio-dependent predator-prey model with disease and diffusion

This paper is purported to study a reaction diffusion system arising from a ratio-dependent predator-prey model with disease. We study the dynamical behavior of the predator-prey system. The conditions for the permanent and existence of steady states and their stability are established. We can obtain the bounds for positive steady state of the corresponding elliptic system. The non-existence results of non-constant positive solutions are derived.     

An analysis of a mathematical model of trophoblast invasion

We present a mathematical model that describes the initial stages of placental development during which trophoblast cells begin to invade the uterine tissue. We then carry out a mathematical analysis of a simpler submodel that describes the final stages of normal embryo implantation and suggests that as the timescale of interest increases, the dominant migratory mechanism of the trophoblasts switches from chemotaxis to nonlinear random motion.     

A mathematical analysis for a diffusive epidemic model with criss-cross dynamics

Abstract. In this paper, an initial boundary value problem with homogeneous Neumann bound-ary condition is studied for a reaction diffusion system which models the spread of infectious dis-eases within two population groups by means of serf and criss-cross infection mechanism, Exis-tence, uniqueness and houndedness of the nonnegative global solution     

Asymptotic analysis of the mathematical model of a wind-powered vehicle

Mathematical model of a wind-powered vehicle is proposed. The vehicle uses a wing to convert the energy of an upcoming wind flow into the energy of the mechanical motion in such a way that the vehicle goes against the wind. The corresponding dynamical system is constructed. Parametric analysis of the system is carried out using the Poincare-Pontryagin asymptotic approach. The existence of an attracting steady regime is shown. Boundaries of applicability of asymptotic estimations are analyzed using numerical simulation. Results of the parametric analysis were used for construction of a prototype of such a vehicle. Experiments with this prototype qualitatively agree with the mathematical modeling.     

Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuron

This article is devoted to the study of a mathematical model arising in the mathematical modeling of pulse propagation in nerve fibers. A widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations. When considered as part of a particular initial boundary value problem the equation models the electrical activity in a neuron. A small perturbation parameter ε is introduced to the highest order derivative term. The parameter if decreased, speeds up the fast variables of the model equations whereas it does not affect the slow variables. In order to formally reduce the problem to a discussion of the moment of fronts and backs we take the limit ε → 0. This limit is singular and is therefore the solution tends to a slowly moving solution of the limiting equation. This leads to the boundary layers located in the neighborhoods of the boundary of the domain where the solution has very steep gradient. Most of the classical methods are incapable of providing helpful information about this limiting solution. To this effort a parameter robust numerical method is constructed on a piecewise uniform fitted mesh. The method consists of standard upwind finite difference operator. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives. A parameter uniform error estimate for the numerical scheme so constructed is established in the maximum norm. It is then proven that the numerical method is unconditionally stable and provides a solution that converges to the solution of the differential equation. A set of numerical experiment is carried out in support of the predicted theory, which validates computationally the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

On a mathematical model for laser-induced thermotherapy

We study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.     

A simple mathematical model for approximate analysis of tall buildings

The focus of this article is to present a new and simple mathematical model that may be used to determine the optimum location of a belt truss reinforcing system on tall buildings such that the displacements due to lateral loadings would generate the least amounts of stress and strain in building’s structural members. The effect of belt truss and shear core on framed tube is modeled as a concentrated moment applied at belt truss location, this moment acts in a direction opposite to rotation created by lateral loads. The axial deformation functions for flange and web of the frames are considered to be cubic and quadratic functions respectively; developing their stress relations and minimizing the total potential energy of the structure with respect to the lateral deflection, rotation of the plane section, and unknown coefficients of shear lag, the mathematical model is developed. The proposed model shows a good understanding of structural behavior; easy to use, yet reasonably accurate and suitable for quick evaluations during the preliminary design stage which requires less time. Numerical examples are given to demonstrate the ease of application and accuracy of the proposed modeled.