Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Stability of an age-structured epidemiological model for hepatitis C

This article introduces an age-structured epidemiological model for the disease transmission dynamics of hepatitis C. We first show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number ? ₀ is below one, in this case, the disease always dies out, then we prove that at least one endemic steady state exists when the reproductive number ? ₀ is above one, the stability conditions for the endemic steady states are also given.     

Combining empirical likelihood and generalized method of moments estimators: Asymptotics and higher order bias

This paper proposes an estimator combining empirical likelihood (EL) and the generalized method of moments (GMM) by allowing the sample average moment vector to deviate from zero and the sample weights to deviate from n⁻¹. The new estimator may be adjusted through free parameter δ∈(0,1) with GMM behavior attained as δ?0 and EL as δ?1. When the sample size is small and the number of moment conditions is large, the parameter space under which the EL estimator is defined may be restricted at or near the population parameter value. The support of the parameter space for the new estimator may be adjusted through δ. The new estimator performs well in Monte Carlo simulations.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.     

On the number of components in 2-factors of claw-free graphs

In this paper, we prove that if a claw-free graph G with minimum degree δ?4 has no maximal clique of two vertices, then G has a 2-factor with at most (|G|-1)/4 components. This upper bound is best possible. Additionally, we give a family of claw-free graphs with minimum degree δ?4 in which every 2-factor contains more than n/δ components.     

Maps between buildings that preserve a given Weyl distance

Let Δ and Δ′ be two buildings of the same type (W, S), viewed as sets of chambers endowed with“distance” functions δ and δ′, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element ω ε W, we study surjective maps ? : Δ → Δ′ with the property that δ(C, D) = ω if and only if Δ′ (?(C), ?(D)) = ω. The result is that the restrictions of ? to all residues of certain spherical types—determined by ω—are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ? is an isomorphism. In particular, ? is an isomorphism if Δ and Δ′ are 2-spherical and every reduced expression of ω involves all elements of S.     

Symmetric marching technique for the numerical solution of nonseparable elliptic equations

We have developed two methods for the nonseparable elliptic equations of the type ?[middot](a?u)=f, based on the extension of the symmetric marching technique developed in [2]. In method I, symmetric marching has been applied to carry out the iterations in the iterative scheme developed by Concus and Golub [3]. Method II is a noniterative method based on the direct extension of symmetric marching to the above problem. Numerical results are given to illustrate the behavior of the two methods.     

A threshold phenomenon in the Field-Noyes model of the Belousov-Zhabotinskii reaction

The Belousov-Zhabotinskii reaction is one of the most interesting and best understood chemical oscillators. It has been conjectured that certain biological phenomena have important features in common with this reaction. We investigate the Field-Noyes model of this reaction and demonstrate that there is a range of values of the stoichiometric parameter, f, over which the model exhibits “threshold phenomena.” That is, if a perturbation from the steady state exceeds a certain “threshold” value then a solution in the form of a “spike” results followed by its return to the steady state. We show that the underlying mathematical structure of this model resembles very closely the underlying mathematical structure of the Hodgkin-Huxley nerve conduction equations which exhibit the same sort of threshold phenomena.     

Loops and branches of coexistence states in a Lotka-Volterra competition model

A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.     

Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton–Kantorovich theorem. It allows to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we obtain as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.     

Inverse problem of groundwater modeling by iteratively regularized Gauss-Newton method with a nonlinear regularization term

A nonlinear minimization problem ‖F(d)−u‖?min, ‖u−u_δ‖≤δ, is a typical mathematical model of various applied inverse problems. In order to solve this problem numerically in the lack of regularity, we introduce iteratively regularized Gauss-Newton procedure with a nonlinear regularization term (IRGN-NRT). The new algorithm combines two very powerful features: iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process. The convergence analysis is carried out in the presence of noise in the data and in the modified source condition. Numerical simulations for a parameter identification ill-posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method.     

Falkner-Skan equation for flow past a moving wedge with suction or injection

The characteristics of steady two-dimensional laminar boundary layer flow of a viscous and incompressible fluid past a moving wedge with suction or injection are theoretically investigated. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of Falkner-Skan power-law parameter (m), suction/injection parameter (f0) and the ratio of free stream velocity to boundary velocity parameter (λ) are discussed in detail. The numerical results for velocity distribution and skin friction coefficient are given for several values of these parameters. Comparisons with the existing results obtained by other researchers under certain conditions are made. The critical values off ₀,m and λ are obtained numerically and their significance on the skin friction and velocity profiles is discussed. The numerical evidence would seem to indicate the onset of reverse flow as it has been found by Riley and Weidman in 1989 for the Falkner-Skan equation for flow past an impermeable stretching boundary.     

On some questions concerning strong compactness

A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ?< κ, then must GCH fail at some regular cardinal δ?≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We get a negative answer to the first of these questions and positive answers to the second of these questions for a supercompact cardinal κ in the context of the absence of the full Axiom of Choice.     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M₃, solute gradient, S₁, and Taylor number, T_A1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.     

A non-linear population model of diabetes mellitus

A mathematical study of the size of a population ofdiabetes mellitus patients is carried out in this paper. The study also monitors the number of patients with complications. By appropriate definition of a parameter, the mathematical model may be classified as linear or non-linear. The non-linear case is discussed and the critical values of the population are analysed for stability. Numerical methods are developed for solving the model equations and the results of numerical simulations are reported.     

A mathematical model of the photosynthetic carbon metabolism has multiple steady states under the same parameter conditions

This paper analyzes a mathematical model of the photosynthetic carbon metabolism, which incorporates not only the Calvin-Benson cycle, but also another two important metabolic pathways: starch synthesis and photorespiratory pathway. Theoretically, the paper shows the existence of steady states, stability and instability of the steady states, the effects of CO ₂ concentration on steady states. Especially, a critical point is found, the system has only one steady state with CO ₂ concentration in the left neighborhood of the critical point, but has two with CO ₂ concentration in the right neighborhood. In addition, the paper also explores the influence of CO ₂ concentration on the efficiency of photosynthesis. These theoretical results not only provide insight to the kinetic behaviors of the photosynthetic carbon metabolism, but also can be used to help improving the efficiency of photosynthesis in plants.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

K-flows are forward deterministic,backward completely non-deterministic stationary point processes

Stationary processes with discrete time parameter and finitely many states are forward deterministic if and only if they are backward deterministic. In contrast to this we prove in the case of continuous time parameter: Every K-flow in a Lebesgue space is isomorphic to the flow of shifts of a stationary forward deterministic, backward completely non-deterministic process (X_t, t?R¹) with two states and with right-continuous paths having only finitely many jumps in any finite time-interval. The process may be considered as a point process. The result is obtained from a representation theorem for flows, describing increasing sub-σ-algebras of a flow up to “equivalence.”     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.