Discrete and Continuous Dynamical Systems - B
March 2022 , Volume 27 , Issue 3
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This paper is concerned with the existence of traveling wave solutions for a vaccination model with general incidence. The existence or non-existence of traveling wave solutions for the model with specific incidence were proved recently when the wave speed is greater or smaller than a critical speed respectively. However, the existence of critical traveling wave solutions (with critical wave speed) was still open. In this paper, applying the Schauder's fixed point theorem via a pair of upper- and lower-solutions of the system, we show that the general vaccination model admits positive critical traveling wave solutions which connect the disease-free and endemic equilibria. Our result not only gives an affirmative answer to the open problem given in the previous specific work, but also to the model with general incidence. Furthermore, we extend our result to some nonlocal version of the considered model.
The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.
This paper studies bifurcations in the coupled
In this work, we consider a diffusive tumor-CD4
In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations
This paper deals with a free boundary problem for the cancer invasion model over a one dimensional habitat in the micro-environment, in which the free boundary represents the spreading front and is caused by tumour cells and acid-mediated. In this problem it is assumed that the tumour cells spread from the given initial region, and the spreading front expands at a speed that is proportional to the tumour cell and acids' population gradient at the front. The main objective is to realize the dynamics/variations of the healthy cells, tumour cells, acid-mediated and the free boundary. We prove a spreading-vanishing dichotomy for this model, namely the tumour cells either successfully spreads to infinity as time tends to infinite at the front, or it fails to establish and dies out in long run while the healthy cells stabilizes at a positive steady-state. The long time behavior of solution and criteria for spreading and vanishing are obtained.
In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in
We consider a perturbed linear boundary-value problem for a weakly singular integral equation. Assume that the generating boundary-value problem is unsolvable for arbitrary inhomogeneities. Efficient conditions for the coefficients guaranteeing the appearance of the family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter
Resorting to M.G. Crandall and P.H. Rabinowitz's well-known bifurcation theory we first obtain the local structure of steady states concerning the ratio–dependent predator–prey system with prey-taxis in spatial one dimension, which bifurcate from the homogeneous coexistence steady states when treating the prey–tactic coefficient as a bifurcation parameter. Based on this, then the global structure of positive solution is established. Moreover, through asymptotic analysis and eigenvalue perturbation we find the stability criterion of such bifurcating steady states. Finally, several numerical simulations are performed to show the pattern formation.
The paper deals with the bifurcation diagram and all global phase portraits in the Poincaré disc of a nonsmooth van der Pol-Duffing oscillator with the form
Disease transmission can present significantly different cyclic patterns including small fluctuations, regular oscillations, and singular oscillations with short endemic period and long inter-epidemic period. In this paper we consider the slow-fast dynamics and nonlinear oscillations during the transmission of mosquito-borne diseases. Under the assumption that the host population has a small natural death rate, we prove the existence of relaxation oscillation cycles by geometric singular perturbation techniques and the delay of stability loss. Generation and annihilation of periodic orbits are investigated through local, semi-local bifurcations and bifurcation of slow-fast cycles. It turns out that relaxation oscillation cycles occur only if the basic reproduction number
In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.
In this paper, a one-dimensional Keller-Segel system of parabolic-parabolic type which is defined on the bounded interval with the Dirichlet boundary condition is considered. Under the assumption that initial data is sufficiently small, a unique mild solution to the system is constructed and the continuity of solution for the initial data is shown, by using an argument of successive approximations.
In this paper we study several
We continue to study the nonsmooth van der Pol-Duffing oscillator
We consider infinite horizon optimal control problems with time averaging and time discounting criteria and give estimates for the Cesàro and Abel limits of their optimal values in the case when they depend on the initial conditions. We establish that these limits are bounded from above by the optimal value of a certain infinite dimensional (ID) linear programming (LP) problem and that they are bounded from below by the optimal value of the corresponding dual problem. (These estimates imply, in particular, that the Cesàro and Abel limits exist and are equal to each other if there is no duality gap). In addition, we obtain IDLP-based optimality conditions for the long run average optimal control problem, and we illustrate these conditions by an example.
In this paper, we investigate the global existence, uniqueness and asymptotic stability of time periodic classical solution for a class of extended Fisher-Kolmogorov equations with delays and general nonlinear term. We establish a general framework to investigate the asymptotic behavior of time periodic solutions for nonlinear extended Fisher-Kolmogorov equations with delays and general nonlinear function, which will provide an effective way to deal with such kinds of problems. The discussion is based on the theory of compact and analytic operator semigroups and maximal regularization method.
for some constant
This paper is devoted to a general multipatch cholera epidemic model to investigate disease dynamics in a periodic environment. The basic reproduction number
This study examines dynamic transitions of Brinkman equation coupled with the thermal diffusion equation modeling the surface tension driven convection in porous media. First, we show that the equilibrium of the equation loses its linear stability if the Marangoni number is greater than a threshold, and the corresponding principle of exchange stability (PES) condition is then verified. Second, we establish the nonlinear transition theorems describing the possible transition types associated with the linear instability of the equilibrium by applying the center manifold theory to reduce the infinite dynamical system to a finite dimensional one together with several non-dimensional transition numbers. Finally, careful numerical computations are performed to determine the sign of these transition numbers as well as related transition types. Our result indicates that the system favors all three types of transitions. Unlike the buoyancy forces driven convections, jump and mixed type transition can occur at certain parameter regimes.
This paper is concerned with the pullback random attractors of nonautonomous nonlocal fractional stochastic
Mathematical modeling of human immunodeficiency virus (HIV) and human T-lymphotropic virus type Ⅰ (HTLV-I) mono-infections has received considerable attention during the last decades. These two viruses share the same way of transmission between individuals; through direct contact with certain contaminated body fluids. Therefore, a person can be co-infected with both viruses. In the present paper, we construct and analyze a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD
An adaptive switching feedback control scheme is proposed for classes of discrete-time, positive difference equations, or systems of equations. In overview, the objective is to choose a control strategy which ensures persistence of the state, consequently avoiding zero which corresponds to absence or extinction. A robust feedback control solution is proposed as the effects of different management actions are assumed to be uncertain. Our motivating application is to the conservation of dynamic resources, such as populations, which are naturally positive quantities and where discrete and distinct courses of management actions, or control strategies, are available. The theory is illustrated with examples from population ecology.
Our aim is to consider a distributed optimal control problem for a coupled phase-field system which was introduced by Cahn and Novick-Cohen. First, we establish that the existence of a weak solution, in particular, we also obtain that a strong solution is uniqueness. Then the existence of optimal controls is proved. Finally we derive that the control-to-state operator is Fréchet differentiable and the first-order necessary optimality conditions involving the adjoint system are discussed as well.
We study a threshold phenomenon of rumor outbreak on the SIR rumor spreading model with a variable trust rate depending on the populations of ignorants and spreaders. Rumor outbreak in the SIR rumor spreading model is defined as a persistence of the final rumor size in the large population limit or thermodynamics limit
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