# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete and Continuous Dynamical Systems - B

March 2022 , Volume 27 , Issue 3

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2022, 27(3): 1209-1225 doi: 10.3934/dcdsb.2021087 +[Abstract](1035) +[HTML](407) +[PDF](396.3KB)
Abstract:

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.

2022, 27(3): 1227-1262 doi: 10.3934/dcdsb.2021088 +[Abstract](1302) +[HTML](478) +[PDF](6960.4KB)
Abstract:

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.

2022, 27(3): 1263-1284 doi: 10.3934/dcdsb.2021089 +[Abstract](1206) +[HTML](479) +[PDF](531.17KB)
Abstract:

This paper studies bifurcations in the coupled \begin{document}$2$\end{document} dimensional almost periodic Ricker map. We establish criteria for stability of an almost periodic solution in terms of the Lyapunov exponents of a corresponding dynamical system and use them to find a bifurcation function. We find that if the almost periodic coefficients of all the maps are identical, then the bifurcation function is the same as the one obtained in the one dimensional case treated earlier, and that this result holds in \begin{document}$N$\end{document} dimension under modest coupling constraints. In the general two-dimensional case, we compute the Lyapunov exponents numerically and use them to examine the stability and bifurcations of the almost periodic solutions.

2022, 27(3): 1285-1300 doi: 10.3934/dcdsb.2021090 +[Abstract](1349) +[HTML](439) +[PDF](357.55KB)
Abstract:

In this work, we consider a diffusive tumor-CD4\begin{document}$^+$\end{document}-cytokine interactions model with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria, including the system persistence and the stability conditions. We also give the existence of nonconstant positive steady states (i.e., a stationary pattern), which indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems, linearization method, the method of energy integral and the Leray-Schauder degree.

2022, 27(3): 1301-1322 doi: 10.3934/dcdsb.2021091 +[Abstract](1243) +[HTML](428) +[PDF](397.23KB)
Abstract:

In this paper, we investigate initial boundary value problems for Kirchhoff-type diffusion equations \begin{document}$\partial_{t}^{\beta}u+M(\|u\|_{H_0^{s}(\Omega)}^2)(-\Delta)^{s}u = \gamma|u|^{\rho}u+g(t,x)$\end{document} with the Caputo time fractional derivatives and fractional Laplacian operators. We establish a new compactness theorem concerning time fractional derivatives. By Galerkin method, let \begin{document}$0<\rho<\frac{4s}{N-2s}$\end{document} when \begin{document}$\gamma<0$\end{document}, and \begin{document}$0<\rho<\min\{\frac{4s}{N},\frac{2s}{N-2s}\}$\end{document} when \begin{document}$\gamma>0$\end{document}, then we obtain the global existence and uniqueness of weak solutions for Kirchhoff problems. Furthermore, we get the decay properties of weak solutions in \begin{document}$L^2(\Omega)$\end{document} and \begin{document}$L^{\rho+2}(\Omega)$\end{document}. Remarkably, the decay rate differs from that in the case \begin{document}$\beta = 1$\end{document}.

2022, 27(3): 1323-1343 doi: 10.3934/dcdsb.2021092 +[Abstract](1080) +[HTML](391) +[PDF](386.46KB)
Abstract:

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.

2022, 27(3): 1345-1377 doi: 10.3934/dcdsb.2021093 +[Abstract](1014) +[HTML](399) +[PDF](479.07KB)
Abstract:

In this paper we investigate the regularity of global attractors and of exponential attractors for two dimensional quasi-geostrophic equations with fractional dissipation in \begin{document}$H^{2\alpha+s}(\mathbb{T}^2)$\end{document} with \begin{document}$\alpha>\frac{1}{2}$\end{document} and \begin{document}$s>1.$\end{document} We prove the existence of \begin{document}$(H^{2\alpha^-+s}(\mathbb{T}^2),H^{2\alpha+s}(\mathbb{T}^2))$\end{document}-global attractor \begin{document}$\mathcal{A},$\end{document} that is, \begin{document}$\mathcal{A}$\end{document} is compact in \begin{document}$H^{2\alpha+s}(\mathbb{T}^2)$\end{document} and attracts all bounded subsets of \begin{document}$H^{2\alpha^-+s}(\mathbb{T}^2)$\end{document} with respect to the norm of \begin{document}$H^{2\alpha+s}(\mathbb{T}^2).$\end{document} The asymptotic compactness of solutions in \begin{document}$H^{2\alpha+s}(\mathbb{T}^2)$\end{document} is established by using commutator estimates for nonlinear terms, the spectral decomposition of solutions and new estimates of higher order derivatives. Furthermore, we show the existence of the exponential attractor in \begin{document}$H^{2\alpha+s}(\mathbb{T}^2),$\end{document} whose compactness, boundedness of the fractional dimension and exponential attractiveness for the bounded subset of \begin{document}$H^{2\alpha^-+s}(\mathbb{T}^2)$\end{document} are all in the topology of \begin{document}$H^{2\alpha+s}(\mathbb{T}^2).$\end{document}

2022, 27(3): 1379-1395 doi: 10.3934/dcdsb.2021094 +[Abstract](1420) +[HTML](457) +[PDF](333.4KB)
Abstract:

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 \begin{document}$\varepsilon$\end{document} with singularity at the point \begin{document}$\varepsilon = 0$\end{document} are established.

2022, 27(3): 1397-1420 doi: 10.3934/dcdsb.2021095 +[Abstract](1080) +[HTML](404) +[PDF](1485.6KB)
Abstract:

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.

2022, 27(3): 1421-1446 doi: 10.3934/dcdsb.2021096 +[Abstract](908) +[HTML](400) +[PDF](1074.8KB)
Abstract:

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 \begin{document}$\dot{x} = y$\end{document}, \begin{document}$\dot{y} = a_1x+a_2x^3+b_1y+b_2|x|y$\end{document}, where \begin{document}$a_i, b_i$\end{document} are real and \begin{document}$a_2b_2\neq0$\end{document}, \begin{document}$i = 1, 2$\end{document}. The system is an equivariant system. When the sum of indices of equilibria is \begin{document}$-1$\end{document}, i.e., \begin{document}$a_2>0$\end{document}, it is proven that the bifurcation diagram includes one Hopf bifurcation surface, one pitchfork bifurcation surface and one heteroclinic bifurcation surface. Although the vector field is only \begin{document}$C^1$\end{document}, we still obtain that the heteroclinic bifurcation surface is \begin{document}$C^{\infty}$\end{document} and a generalized Hopf bifurcation occurs. Moreover, we also find that the heteroclinic bifurcation surface and the focus-node surface have exactly one intersection curve.

2022, 27(3): 1447-1469 doi: 10.3934/dcdsb.2021097 +[Abstract](1273) +[HTML](434) +[PDF](3607.82KB)
Abstract:

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 \begin{document}$\mathcal{R}_0$\end{document} is greater than 1, while small fluctuations and regular oscillations exist under much less restrictive conditions. Our results here provide a sound explanation for different cyclic patterns exhibited in the transmission of mosquito-borne diseases.

2022, 27(3): 1471-1496 doi: 10.3934/dcdsb.2021098 +[Abstract](1631) +[HTML](430) +[PDF](547.86KB)
Abstract:

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.

2022, 27(3): 1497-1510 doi: 10.3934/dcdsb.2021099 +[Abstract](1178) +[HTML](432) +[PDF](359.43KB)
Abstract:

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.

2022, 27(3): 1511-1547 doi: 10.3934/dcdsb.2021100 +[Abstract](1139) +[HTML](415) +[PDF](430.08KB)
Abstract:

In this paper we study several \begin{document}$L^2$\end{document}-constrained variational problems associated with a three component system of nonlinear Schrödinger equations with three wave interaction. We consider the existence and the orbital stability of minimizers for these variational problems. We also investigate an asymptotic expansion of the minimal energy and the asymptotic behavior of a minimizer for the variational problem when the attractive effect of three wave interaction is sufficiently large.

2022, 27(3): 1549-1589 doi: 10.3934/dcdsb.2021101 +[Abstract](912) +[HTML](383) +[PDF](2216.23KB)
Abstract:

We continue to study the nonsmooth van der Pol-Duffing oscillator \begin{document}$\dot{x} = y$\end{document}, \begin{document}$\dot{y} = a_1x+a_2x^3+b_1y+b_2|x|y$\end{document}, where \begin{document}$a_i, b_i$\end{document} are real and \begin{document}$a_2b_2\neq0$\end{document}, \begin{document}$i = 1, 2$\end{document}. Notice that the sum of indices of equilibria is \begin{document}$-1$\end{document} for \begin{document}$a_2>0$\end{document} and \begin{document}$1$\end{document} for \begin{document}$a_2<0$\end{document}. When \begin{document}$a_2>0$\end{document}, the nonsmooth van der Pol-Duffing oscillator has been studied completely in the companion paper. Attention goes to the bifurcation diagram and all global phase portraits in the Poincaré disc of the nonsmooth van der Pol-Duffing oscillator for \begin{document}$a_2<0$\end{document} in this paper. The bifurcation diagram is more complex, which includes two Hopf bifurcation surfaces, one pitchfork bifurcation surface, one homoclinic bifurcation surface, one double limit cycle bifurcation surface and one bifurcation surface for equilibria at infinity. When \begin{document}$b_2>0$\end{document} is fixed, this nonsmooth van der Pol-Duffing oscillator cannot be changed into a near-Hamiltonian system for small \begin{document}$a_1, b_1$\end{document}. Moreover, the global dynamics of the nonsmooth van der Pol-Duffing oscillator and the van der Pol-Duffing oscillator are different.

2022, 27(3): 1591-1610 doi: 10.3934/dcdsb.2021102 +[Abstract](943) +[HTML](394) +[PDF](349.06KB)
Abstract:

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.

2022, 27(3): 1611-1627 doi: 10.3934/dcdsb.2021103 +[Abstract](891) +[HTML](373) +[PDF](347.95KB)
Abstract:

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.

2022, 27(3): 1629-1645 doi: 10.3934/dcdsb.2021104 +[Abstract](984) +[HTML](384) +[PDF](380.98KB)
Abstract:

Let \begin{document}${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$\end{document} denote a BMO space on \begin{document}$\mathbb{R}^{n}$\end{document} associated to a Neumann operator \begin{document}$\mathcal{L}: = -\Delta_{N}$\end{document}. In this article we will show that a function \begin{document}$f\in {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$\end{document} is the trace of the solution of \begin{document}${\mathbb L}u = u_{t}+ \mathcal{L} u = 0, u(x, 0) = f(x),$\end{document} where \begin{document}$u$\end{document} satisfies a Carleson-type condition

for some constant \begin{document}$C>0$\end{document}. Conversely, this Carleson condition characterizes all the \begin{document}${\mathbb L}$\end{document}-carolic functions whose traces belong to the space \begin{document}${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$\end{document}. Furthermore, based on the characterization of \begin{document}${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$\end{document} mentioned above, we prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the intial data \begin{document}$u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})}$\end{document}.

2022, 27(3): 1647-1670 doi: 10.3934/dcdsb.2021105 +[Abstract](1069) +[HTML](438) +[PDF](715.58KB)
Abstract:

This paper is devoted to a general multipatch cholera epidemic model to investigate disease dynamics in a periodic environment. The basic reproduction number \begin{document}$\mathcal{R}_0$\end{document} is introduced and a threshold type of result is established in terms of \begin{document}$\mathcal{R}_0$\end{document}. Specifically, we show that when \begin{document}$\mathcal{R}_0<1$\end{document}, the disease-free steady state is globally attractive if either immigration of hosts is homogeneous or immunity loss of human hosts can be neglected; when \begin{document}$\mathcal{R}_0>1$\end{document}, the disease is uniformly persistent and our system admits at least one positive periodic solution. Numerical simulations are carried out to illustrate the impact of asymptotic infections and population dispersal on the spread of cholera. Our result indicates that (a) neglecting asymptotic infections may underestimate the risk of infection; (b) travel can help the disease to become persistent (resp. eradicated) in the network, even though the disease dies out (resp. persists) in each isolated patch.

2022, 27(3): 1671-1694 doi: 10.3934/dcdsb.2021106 +[Abstract](985) +[HTML](426) +[PDF](1234.17KB)
Abstract:

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.

2022, 27(3): 1695-1724 doi: 10.3934/dcdsb.2021107 +[Abstract](981) +[HTML](378) +[PDF](418.49KB)
Abstract:

This paper is concerned with the pullback random attractors of nonautonomous nonlocal fractional stochastic \begin{document}$p$\end{document}-Laplacian equation with delay driven by multiplicative white noise defined on bounded domain. We first prove the existence of a continuous nonautonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise. We then show pullback asymptotical compactness of solutions and the existence of tempered random attractors by utilizing the Arzela-Ascoli theorem and appropriate uniform estimates of the solutions. Finally, we establish the upper semicontinuity of the random attractors when time delay approaches zero.

2022, 27(3): 1725-1764 doi: 10.3934/dcdsb.2021108 +[Abstract](1001) +[HTML](341) +[PDF](3432.97KB)
Abstract:

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\begin{document}$4^{+}$\end{document}T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing (active) HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by two routes of transmission, virus-to-cell (VTC) and cell-to-cell (CTC). Both active and silent HIV-infected cells can infect the susceptible CD\begin{document}$4^{+}$\end{document}T cells by CTC mechanism. On the other side, HTLV-I has only one mode of transmission via direct cell-to-cell contact. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We calculate all possible equilibria and define the key threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle's invariance principle. We have discussed the influence of CTL immune response on the co-infection dynamics. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

2022, 27(3): 1765-1787 doi: 10.3934/dcdsb.2021109 +[Abstract](923) +[HTML](360) +[PDF](862.72KB)
Abstract:

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.

2022, 27(3): 1789-1825 doi: 10.3934/dcdsb.2021110 +[Abstract](1462) +[HTML](458) +[PDF](437.68KB)
Abstract:

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.

2022, 27(3): 1827-1851 doi: 10.3934/dcdsb.2021111 +[Abstract](1020) +[HTML](346) +[PDF](669.64KB)
Abstract:

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 \begin{document}$(n\to \infty)$\end{document}, where \begin{document}$1/n$\end{document} is the initial population of spreaders. We present a rigorous proof for the existence of threshold on the final size of the rumor with respect to the basic reproduction number \begin{document}$\mathcal{R}_0$\end{document}. Moreover, we prove that a phase transition phenomenon occurs for the final size of the rumor (as an order parameter) with respect to the basic reproduction number and provide a criterion to determine whether the phase transition is of first or second order. Precisely, we prove that there is a critical number \begin{document}$\mathcal{R}_1$\end{document} such that if \begin{document}$\mathcal{R}_1>1$\end{document}, then the phase transition is of the first order, i.e., the limit of the final size is not a continuous function with respect to \begin{document}$\mathcal{R}_0$\end{document}. The discontinuity is a jump-type discontinuity and it occurs only at \begin{document}$\mathcal{R}_0 = 1$\end{document}. If \begin{document}$\mathcal{R}_1<1$\end{document}, then the phase transition is second order, i.e., the limit of the final size is continuous with respect to \begin{document}$\mathcal{R}_0$\end{document} and its derivative exists, except at \begin{document}$\mathcal{R}_0 = 1$\end{document}, and the derivative is not continuous at \begin{document}$\mathcal{R}_0 = 1$\end{document}. We also present numerical simulations to demonstrate our analytical results for the threshold phenomena and phase transition order criterion.

2021 Impact Factor: 1.497
5 Year Impact Factor: 1.527
2021 CiteScore: 2.3