Positive steady states for a prey-predator model with some nonlinear diffusion terms

This paper discusses a prey-predator system with strongly coupled nonlinear diffusion terms. We give a sufficient condition for the existence of positive steady state solutions. Our proof is based on the bifurcation theory. Some a priori estimates for steady state solutions will play an important role in the proof.     

Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion

A diffusive one-prey and two-competing-predators system under homogeneous Dirichlet boundary conditions is studied. First, we obtain sufficient conditions for the extinction and existence of global attractor of the time-dependent system by means of the comparison principle. Second, we discuss the existence and nonexistence of coexistence states, and give sufficient conditions for the existence of coexistence states by using the fixed point index theory. In addition, we investigate the bifurcation from a double eigenvalue by virtue of space decomposition and implicit function theorem. Finally, some numerical simulations are made to verify and complement the theoretical analysis.     

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.     

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.     

Global solutions for a nonlinear wave equation

In this work the existence of a global solution for the mixed problem associated to the nonlinear equation

is proved in a Hilbert space framework by using Galerkin method.     

Existence and orbital stability of standing waves for some nonlinear Schrödinger equations, perturbation of a model case

The following nonlinear Schrödinger equation is studied

     

Numerical modelling of EHD effects on heat transfer and bubble shapes of nucleate boiling

In this article, mathematical and numerical models are developed to study pure electrohydrodynamic (EHD) effects on heat transfer and bubble shapes when an initial bubble attached to a superheated horizontal wall in nucleate boiling. In the modelling of EHD effects on heat transfer, an undeformed bubble is considered; the electric body force and Joule heat are added to the momentum and energy equations; governing equations for heat, fluid flow and electric fields are coupled numerically and solved using a non-orthogonal body-fitted mesh system with necessary interfacial treatments at the gas–liquid boundary. While, to study the pure effect of EHD on the deformation of the bubble, the evaluation of a deformable bubble without heat transfer is simulated by volume of fluid (VOF) method based on an axial symmetric Cartesian coordinate system. The simulations indicate that EHD can effectively enhance heat transfer rate of nucleate boiling by influencing the motion of the ring vortex around the bubble and that bubble can be elongated due to the pull in axial direction and push in the negative radial direction by the electric field force.     

Haar wavelet solutions of nonlinear oscillator equations

In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing–van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.     

Self-similar solutions of a nonlinear friction equation in higher dimensions

We study the class of self-similar solutions of certain multi-dimensional kinetic models of granular flows, which have been recently introduced in connection with the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. The importance of these solutions in connection with the cooling of the dissipative gas is subsequently discussed.

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Sunto Si studia la classe delle soluzioni di similarità di alcune equazioni cinetiche per flussi granulari in più dimensioni. Queste equazioni sono state introdotte di recente in connessione con il limite quasi elastico di un’ equazione di Boltzmann per collisioni dissipative con coefficiente di restituzione variabile. Nella seconda parte del lavoro si discute l’importanza di tali soluzioni nello studio del raffreddamento del gas dissipativo.

     

A nonlinear HIV/AIDS model with contact tracing

A nonlinear mathematical model is proposed and analyzed to study the effect of contact tracing on reducing the spread of HIV/AIDS in a homogeneous population with constant immigration of susceptibles. In modeling the dynamics, the population is divided into four subclasses of HIV negatives but susceptibles, HIV positives or infectives that do not know they are infected, HIV positives that know they are infected and that of AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives and all infectives move with constant rates to develop AIDS. The model is analyzed using the stability theory of differential equations and numerical simulation. The model analysis shows that contact tracing may be of immense help in reducing the spread of AIDS epidemic in a population. It is also found that the endemicity of infection is reduced when infectives after becoming aware of their infection do not take part in sexual interaction.     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.     

Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model

We consider a relatively simple model for pool-boiling processes. This model involves only the temperature distribution within the heater and describes the heat exchange with the boiling medium via a nonlinear boundary condition imposed on the fluid-heater interface. This results in a standard heat-transfer problem with a nonlinear Neumann boundary condition on part of the boundary. In a recent paper [Speetjens M, Reusken A, Marquardt W. Steady-state solutions in a nonlinear pool-boiling model. IGPM report 256, RWTH Aachen. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2006.11.002] we analysed this nonlinear heat-transfer problem for the case of two space dimensions and in particular studied the qualitative structure of steady-state solutions. The study revealed that, depending on system parameters, the model allows both multiple homogeneous and multiple heterogeneous temperature distributions on the fluid-heater interface. In the present paper we show that the analysis from Speetjens et al. (doi:10.1016/j.cnsns.2006.11.002) can be generalised to the physically more realistic case of three space dimensions. A fundamental shift-invariance property is derived that implies multiplicity of heterogeneous solutions. We present a numerical bifurcation analysis that demonstrates the multiple solution structure in this mathematical model by way of a representative case study.     

Dynamical behavior of an epidemic model with a nonlinear incidence rate

In this paper, we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action. By carrying out global qualitative and bifurcation analyses, it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region. When such a region exists, it is shown that the model undergoes a Bogdanov-Takens bifurcation, i.e., it exhibits a saddle-node bifurcation, Hopf bifurcations, and a homoclinic bifurcation. Existence of none, one or two limit cycles is also discussed.     

Exact solutions for a nonlinear model

In this paper we show new exact solutions for a type of generalized sine-Gordon equation which is obtained by constructing a Lagrange function for a dynamical coupled system of oscillators. We convert it into a nonlinear system by perturbing the potential energy from a point of view of an approach proposed by Fermi [1].     

Nonlinear dynamical analysis of hydro-turbine governing system with a surge tank

This paper brings attention to a new nonlinear mathematical model of a hydro-turbine governing system with a surge tank. The nonlinear mathematical model, which is described by state-space equations, is composed of Francis turbine system, electrical generator system, conduit system and governor system. Furthermore, the nonlinear dynamical behaviors of the system with different parameters are studied exhaustively including bifurcation diagrams, time waveforms, phase orbits, Poincare maps, spectrograms and power spectrums. Fortunately, some interesting phenomenons are found from numerical simulation results. More important, all of the above analyses supply some theory bases for designing and running of a hydro-turbine governing system.     

New enclosure algorithms for the verified solutions of nonlinear Volterra integral equations

This paper is concerned with new algorithms which provide the sharp bounds that are guaranteed to contain the exact solutions of nonlinear Volterra integral equations. We develop new enclosure algorithms based on the interval methods which was first introduced by Moore in [24] together with the Taylor polynomials to improve the accuracy of the scheme by reducing the width of interval solutions. The modified methods calculate a priori bound automatically in parallel with the computation of solutions of integral equations. We will show that the accuracy of the proposed algorithms is dependent on the number of interval subdivisions. Some numerical experiments are also included to demonstrate the validity and applicability of the scheme and showing a marked improvement in comparison with the recent existing numerical results.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,