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This paper delves into a systematically reduced plant system proposed by Ja\"ibi et al. [Phys. D, 2020] in arid area. They used the method of geometric singular perturbation to study the existence of abundant orbits. Instead, we deliberate the stability and distributed patterns of this system. For a non-diffusive scenario for the model, we scrutinize the local and global stability of equilibria and derive conditions for the existence or non-existence of the limit cycle. The bifurcation behaviors are also explored. For the spatial model, we investigate Hopf, Turing, Hopf-Turing, Turing-Turing bifurcations. Specially, the evolution process from periodic solutions to spatially nonconstant steady states is observed near the Hopf-Turing bifurcation point. And mixed nonconstant steady states near the Turing-Turing bifurcation point are observed. Furthermore, it's found that there exist gap, spot, stripe and mixed patterns. The seed-dispersal rate enables the transformation of pattern structures. Reasonable control of system parameters may prevent desertification from occurring.

We propose a lattice dynamical system that arises in a discrete diffusive two-group epidemic model with latency in a patchy environment. The model considers the SIS form and latency of the disease in group 1, while the SIR form without latency of the disease in group 2. The system includes double nonlocal effects, one effect is the nonlocal diffusion of individuals in isolated patches or niches, while the other effect is the distributed transmission delay representing the incubation of the disease. We demonstrate that there is a threshold value $c^*$ that can determine the persistence or disappearance of the disease. If $c\geq c^*$, then there is an epidemic wave connecting the disease-free equilibrium and endemic equilibrium. In this case, the disease will evolve to endemic. If $0

In this paper, we consider a kind of shallow water wave model called the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. We firstly consider the unperturbed KP-BBM equation. Then by using the geometric singular perturbation (GSP) theory, especially the invariant manifold theory, method of dynamical system and Melnikov function, the existence of solitary wave solutions of perturbed KP-BBM equation is proved. In other words, we dissuss the equation under different nonlinear terms. Finally, we validate our results with numerical simulations., Comment: We declare that the authors are ranked in alphabetic order of their names and all of them have the same contributions to this paper

In this manuscript, we investigate the behavior of stellar structure through embedding approach in $f(R, \phi, X)$ modified theory of gravity, where $R$ denotes the Ricci scalar, $\phi$ represents the scalar potential and $X$ indicates the kinetic potential. For this purpose, we consider the spherically symmetric space-time with anisotropic fluid. We further choose three different stars i.e. LMC X-4, Cen X-3, and EXO 1785-248 to demonstrate the behavior of stellar structures. We further compare the Schwarzschild space-time as exterior geometry with spherically symmetric space-time to calculate the values of unknown parameters. In this regard, we investigate the graphical features of stellar spheres such es energy density, pressure components, anisotropic component, equation of state parameters, stability analysis and energy conditions. Furthermore, we investigate some extra conditions such as mass function, compactness factor and surface redshift respectively. Conclusively, all the compact stars under observations are realistic, stable, and are free from any physical or geometrical singularities. We find that the embedding class one solution for anisotropic compact stars is viable and stable., Comment: 20 pages, 21 figures

In this paper, the variational formulation for steady periodic stratified water waves in two-layer flows is given. The critical points of a natural energy functional is proved to be the solutions of the governing equations. And the second variation of the functional is also presented.

In this paper, we study steady 2D periodic blood flow propagating along blood vessels with free boundary conditions. In particular, we focus on the dynamic behavior of flows with harmonic vorticity and time dependent body force. An equivalent formulation with fix boundary is developed by utilizing flow force functions. The local bifurcation result is obtained by Crandall-Rabinowitz theorem. The existence of a local $C^1$-curve of small-amplitude solution is strictly proved.

The constant vorticity {\bf two-layer water wave} in the $\beta$-plane approximation with centripetal forces is investigated in this paper. Different from the works (Chu and Yang\cite[JDE, 2020]{chu} and Chu and Yang \cite[JDE, 2021]{chu2}) on the singe-layer wave flows, we consider the two-layer water wave model containing a free surface and an interface. The interface separates two layers with different features such as velocity field, pressure and vorticity. We prove that if the change in pressure in the $y$-axis direction is bounded, then the pressure is a function only related to depth and the surfaces of the water flows. And the inner wave will not affect the pressure function, if the water flow densities in each layer are equal. Furthermore, the explicit expressions of the velocity, pressure are given for the two-layer water flows. It is interesting that our method and results are also valid for the multi-layer water waves. Let the number of layers of water waves $n$ tend to infinity, we prove that the squence of pressure in the lowest layer $\{P_1^n(x,y,z,t)\}_{n\geq1}$ is uniformly convergent, if the density of each layer is bounded and the each surface of wave flows is uniformly convergent.

Hartman-Grobman theorem was initially extended to the non-autonomous cases by Palmer. Usually, dichotomy is an essential condition of Palmer's linearization theorem. Is Palmer's linearization theorem valid for the systems with trichotomy? In this paper, we obtain new versions of the linearization theorem if linear system admits exponential trichotomy on $\mathbb{R}$. { Furthermore, the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems are H\"{o}lder continuous}. In addition, if a system is periodic, we find the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems do not have periodicity or asymptotical periodicity. To the best of our knowledge, this is the first paper studying the linearization with exponential trichotomy.

Diffusion-driven instability and bifurcation analysis are studied in a predator-prey model with herd behavior and quadratic mortality by incorporating multiple Allee effect into prey species. The existence and stability of the equilibria of the system are studied. And bifurcation behaviors of the system without diffusion are shown. The sufficient and necessary conditions for Turing instability occurring are obtained. And the stability and the direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that too large diffusion rate of prey prevents Turing instability from emerging. Finally, we summarize our findings in the conclusion.

In this paper, we study the topological conjugacy between the linear generalized ODEs (for short, GODEs) \[ \frac{dx}{d\tau}=D[A(t)x] \] and their nonlinear perturbation \[ \frac{dx}{d\tau}=D[A(t)x+F(x,t)] \] on Banach space $\mathscr{X}$, where $A:\mathbb{R}\to\mathscr{B}(\mathscr{X})$ is a bounded linear operator on $\mathscr{X}$ and $F:\mathscr{X}\times \mathbb{R}\to \mathscr{X}$ is Kurzweil integrable. GODEs are completely different from the classical ODEs. Note that the GODEs in Banach space are defined via its solution. $\frac{dx}{d\tau}$ is only a notation and %$\frac{dx}{d\tau}$ it does not indicate that the solution has a derivative. The solution of the GODEs can be discontinuous and even the number of discontinuous points is countable, so that many classical theorems and tools are no longer applicable to the GODEs. For instances, the chain rule and the multiplication rule of derivatives, differential mean value theorem, and integral mean value theorem are not valid for the GODEs. In this paper, we study the linearization and its H\"{o}lder continuity of the GODEs. Firstly, we construct the formula for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense. Afterwards, we establish a Hartman-Grobman type linearization theorem which is a bridge connecting the linear GODEs with their nonlinear perturbations. Further, we show that the conjugacies are both H\"{o}lder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense) and other nontrivial estimate techniques. %The GODEs include measure differential equations, impulsive differential equations, functional differential equations and the classical ordinary differential equations as special cases. Finally, applications to the measure differential equations and impulsive differential equations, our results are very effective.

This paper concerns a second-order differential equation with indefinite and repulsive singularities. It is the first time to study differential equation containing both indefinite and repulsive singularities simultaneously. A set of sufficient conditions are obtained for the existence of positive periodic solutions. The theoretical underpinnings of this paper are the positivity of Green's function and fixed point theorem in cones. Our results improve and extend the results in previous literatures. Finally, three examples and their numerical simulations (phase diagrams and time diagrams of periodic solutions) are given to show the effectiveness of our conclusions.

Choosing ${\kappa}$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$-dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa}\in (0,1-\frac{\sqrt3}{3})\cup (1+\frac{\sqrt3}{3},2)$, while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt{15}/3$ for $\kappa=0$ or $\kappa=2$. The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

In this paper, we analyze the dynamics of a generalized Rotation-Camassa-Holm equation, which is the $\theta$-equation augmented with the Coriolis effect, induced by the earth rotation. The generalized Rotation-Camassa-Holm equation (named as Rotation-$\theta$ equation) is a generalization of a family of models (including the Rotation-Camassa-Holm equation for $\theta=\frac{1}{3}$, asymptotic Rotation-Camassa-Holm equation for $\theta=1$ and the Rotation-Degasperis-Procesi (DP) equation for $\theta=\frac{1}{4}$). Our study is conducted via the bifurcation method and qualitative theory of dynamical systems. The existence of not only smooth solitary wave solutions, periodic wave solutions, but also peakons and periodic peakon solutions is shown. The chosen values $\theta$, allow us to assess the difference in behavior between the classical $\theta$-equation and the Rotation-$\theta$ equation. We conclude that the Coriolis effect does affect the traveling wave solutions. We summarize the bifurcations and explicit expressions of waves solutions in three theorems in Section 4. A conclusion ends the paper.

Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a convenient and powerful tool to succinctly determine eigenvectors from eigenvalues. The identity relates the eigenvector component to the eigenvalues of $A$ and the minor $M_j$, which is formulated in an elegant form as \[ \lvert v_{i,j} \rvert^2\prod_{k=1;k\ne i}^{n-1}({\lambda_i}(A)-{\lambda_k}(A))=\prod_{k=1}^{n-1}({\lambda_i}(A)-{\lambda_k}(M_j)). \,\,\,%\mbox{(\cite{tao-eig,D-P-T-Z})} \] In fact, it has been widely applied in various fields such as numerical linear algebra, random matrix theory, inverse eigenvalue problem, graph theory, neutrino physics and so on. In this paper, we extend the eigenvector-eigenvalue identity to the quaternion division ring, which is non-commutative. A version of eigenvector-eigenvalue identity for the quaternion matrix is established. Furthermore, we give a new method and algorithm to compute the eigenvectors from the right eigenvalues for the quaternion Hermitian matrix. A program is designed to realize the algorithm to compute the eigenvectors. An open problem ends the paper. Some examples show a good performance of the algorithm and the program.

Microbial pesticides can avoid many of negative effects of traditional chemical pesticides. To modelling the inhibiting effect, in this paper we propose a model of entomopathogenic nematodes killing the target insects, and inhibiting the birth rate of the target insects simultaneously. In the model, we achieve the purpose of restricting or eliminating pests by continuously releasing nematodes. By analyzing the stability of the equilibrium of the model and the stability of the Hopf bifurcation periodic solution, the best solution to control pests is obtained, and our conclusions are verified by examples and numerical simulations., Comment: Many studies have shown that some species of microorganism A (or plant extracts) can inhibit other species of microorganism B (bacteria or fungi), and this inhibiting effect can reduce the birth rate of microorganism B. Consequently, the density of B is inversely proportional to the density of A. Thus, using an inverse proportional function to modelling the inhibiting effect is very reasonable

Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of A and B (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of A (see Theorem 4.1) by the method of variation and approximation theory and an explicit expression of B by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

The phase structure of ideal Bose gas system within different boundary conditions, i.e., the periodic boundary condition and Dirichlet boundary condition in this work, in an infinite volume, is investigated. It is found that the ground states of ideal Bose gas within those two boundary conditions are both topologically nontrivial, which can not be classified by the traditional symmetry breaking theory. The ground states are different topological phases corresponding to those two boundary conditions, which can be distinguished by the off--diagonal particle number susceptibility. Moreover, this result is universal. The boundary condition may play an important role in pining the critical endpoint of QCD diagram on the approach of the lattice simulations and the computation of some solvable statistical models ., Comment: I found some mistakes in this version manuscript

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete transform, we form a quaternion function space whose elements have sampling expansions. Moreover, through formulating boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and polynomials with quaternion coefficients. We show that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are eigenvalues of the matrix. Several examples are given to illustrate the results.

This paper presents new results on the limit cycles of a Li\'enard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [33,34]. The results in [34] are only valid for the smooth system. We emphasize that our main results are valid for discontinuous systems. Moreover, we show the presence and an explicit upper bound for the amplitude of the two limit cycles, and we estimate the position of the double-limit-cycle bifurcation surface in the parameter space. Until now, there is no result to determine the amplitude of the two limit cycles. The existing results on the amplitude of limit cycles guarantee that the Li\'enard system has a unique limit cycle. Finally, some applications and examples are provided to show the effectiveness of our results. We revisit a co-dimension-3 Li\'enard oscillator (see [21,32]) in Application 1. Li and Rousseau [21] studied the limit cycles of such a system when the parameters are small. However, for the general case of the parameters (in particular, the parameters are large), the upper bound of the limit cycles remains open. We completely provide the bifurcation diagram for the one-equilibrium case. Moreover, we determine the amplitude of the two limit cycles and estimate the position of the double-limit-cycle bifurcation surface for the one-equilibrium case. Application 2 is presented to study the limit cycles of a class of the Filippov system.

In this paper, we use the two-flavor Nambu-Jona-Lasinio (NJL) model with the proper time regularization to study the finite-volume effects of QCD chiral phase transition. Within a cubic volume of finite size $L$, we choose the stationary wave condition (SWC) as the real physical spatial boundary conditions of quark fields and compare our results with that by means of commonly used (anti-)period boundary condition (APBC or PBC). It is found that the results by means of SWC are obviously different to the results from the APBC or PBC. Although the three boundary conditions give the same chiral crossover transition curve in the infinite volume limit, the limit size $L_0$ (when $L\geq L_{0}$, the chiral quark condensate $-\left\langle { \bar \psi \psi} \right\rangle_L$ is indistinguishable from that at $L=\infty$) using SWC is $L_0\approx 500$ fm which is much larger than the results obtained using APBC or PBC. More importantly, $L_0\approx 500$ fm is also much large than the typical size of the quark-gluon plasma produced by the relativistic heavy ion collisions. This means that the finite volume effects play a very important role in Relativistic Heavy Ion Collisions. In addition, we also found that when $L\leq 2$ fm, even at zero temperature the chiral symmetry is effectively restored. Furthermore, to quantitatively reflect the finite volume effects on the QCD chiral phase transition, we introduce a new vacuum susceptibility, $\chi_{1/L}(T)=-\frac{\partial \left\langle { \bar \psi \psi} \right\rangle}{\partial (1/L)}$. With this new vacuum susceptibility, it is very interesting to find $\chi_{1/L}(T=0)=\chi_{1/L}(T=1/L)$ for SWC.