The monotonicity of the linear complementarity problem (LCP) is discussed in this paper. Both the monotone property about the single element of the solution and the monotone property of the whole solution are presented. In order to illustrate the results, some corresponding numerical experiments are provided.

The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon $x$ alone and are shown to be less important. The systems have both Liouvillian and non–Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetric in their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper. The relation of certain Riccati type systems is considered with respect to Abel differential equations.

In this paper, the existence of viscous solutions of a hyperbolic equilibrium law system derived from the nonlinear entropy moment closure of a dynamic equation is established. In addition, by using the natural entropy of the system, some higher order estimates of some viscosity solutions are obtained.

In this paper, a Lyapunov-type inequality for a linear differential equation involving right Riemann-Liouville and left Caputo fractional derivatives under Sturm-Liouville boundary conditions is established. Furthermore, the existence of solutions for the corresponding nonlinear differential equation is obtained by fixed point theorems.

In this paper, we investigate the Novikov equation with weak dissipation terms. First, we give the local well-posedness and the blow-up scenario. Then, we discuss the global existence of the solutions under certain conditions. After that, on condition that the compactly supported initial data keeps its sign, we prove the infinite propagation speed of our solutions, and establish the large time behavior. Finally, we also elaborate the persistence property of our solutions in weighted Sobolev space.

Existing numerical schemes, maybe high-order accurate, are obtained on uniformly spaced meshes and challenges to achieve high accuracy in the presence of singular perturbation parameter, and nonlinearity remains left on nonuniformly spaced meshes. A new scheme is proposed for nonlinear 2D parabolic partial differential equations (PDEs) that attain fourth-order accuracy in $xy$-space and second-order exact in the temporal direction for uniform and nonuniform mesh step-size. The method proclaims a compact character using nine-point single-cell finite-difference discretization on a nonuniformly spaced spatial mesh point. A description of splitting compact operator form to the convection-dominated equation is obtained for implementing alternating direction implicit scheme. The procedure is examined for consistency and stability. The scheme is applied to linear and nonlinear 2D parabolic equations: convection-diffusion equations, Burger’s-Huxley, Burger’s-Fisher and coupled Burger’s equation. The technique yields the tridiagonal matrix and computed by the Thomas algorithm. Numerical simulations with linear and nonlinear problems corroborate the theoretical outcome.

The dynamics of a class of abstract delay differential equations are investigated. We prove that a sequence of Hopf bifurcations occur at the origin equilibrium as the delay increases. By using the theory of normal form and centre manifold, the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived. Then, the existence of the global Hopf bifurcation of the system is discussed by applying the global Hopf bifurcation theorem of general functional differential equation.

The need for efficient statistical models has increased with the flow of new data, which makes distribution theory a particularly interesting and attractive field. Here, we provide a thorough study of the applications of the Lindley distribution and its diverse generalizations. More precisely, we review some special applications in various areas, such as time series analysis, stress strength analysis, acceptance sampling plans and data analysis. We also conduct a comparative study between the Lindley distribution and some of its generalizations by using four real-life data sets.

A diffusive predator-prey system with Holling functional response is considered. Firstly, existence of positive equilibrium of this reaction diffusion model under Neumann boundary condition is obtained. Meanwhile, the existence conditions for Turing instability and Hopf bifurcations of a system with Holling II functional response are established. Next, the existence of the hydra effect is demonstrated, when the system is undergoing non-homogeneous steady-state solutions. Finally, numerical simulations are illustrated to support our theory results.

Considering the individual difference, this paper deals with an infection-age structured epidemic model coupling within-host and between-host for environmentally-driven infectious disease. The full system with two time scales, the cellular level and population level, is first separated into the isolated fast and slow systems. For the isolated fast and slow systems, combined with the within-host and between-host reproduction numbers, $R_{w0}$ and $R_{b0},$ we give the complete global dynamics by using Lyapunov function respectively. Our results indicate that when there is no virus in environment the disease can be not only controlled, but also eliminated. However, when there is always virus in environment the disease is only controlled but not eliminated. Furthermore, the coupled slow system has complex dynamics with multiple positive equilibria and backward bifurcation. The virus contaminated environment plays a critical role on backward bifurcation. When the initial environmental virus is below some threshold the disease will be eliminated, when it is above the threshold the disease will develop an endemic disease. Some numerical simulations are performed to illustrate these results. The age structured model is more general, and this work includes some previous results.

In this paper, we pay attention to the number of limit cycles for a class of piecewise smooth near-Hamiltonian systems. By using the expression of the first order Melnikov function and some known results about Chebyshev systems, we study upper bound of the number of limit cycles in Hopf bifurcation and Poincaré bifurcation respectively.

We consider a stochastically forced epidemic model with medical-resource constraints. In the deterministic case, the model can exhibit two type bistability phenomena, i.e., bistability between an endemic equilibrium or an interior limit cycle and the disease-free equilibrium, which means that whether the disease can persist in the population is sensitive to the initial values of the model. In the stochastic case, the phenomena of noise-induced state transitions between two stochastic attractors occur. Namely, under the random disturbances, the stochastic trajectory near the endemic equilibrium or the interior limit cycle will approach to the disease-free equilibrium. Besides, based on the stochastic sensitivity function method, we analyze the dispersion of random states in stochastic attractors and construct the confidence domains (confidence ellipse or confidence band) to estimate the threshold value of the intensity for noise caused transition from the endemic to disease eradication.

To understand the influence of fear effect on population dynamics, especially for the populations with obvious stage structure characteristics, we propose and investigate a diffusive prey-predator model with stage structure in predators. First, we discuss the existence and stability of equilibrium of the model in the absence of diffusion. Then, we obtain the critical conditions for Hopf and Turing bifurcations. Some numerical simulations are also carried out to verify our theoretical results, which indicate that the fear can induce the prey population to show five pattern structures: cold-spot pattern, mixed pattern with cold spots and stripes, stripes pattern, hot-spot pattern, mixed pattern with hot spots and stripes. These findings imply that the fear effect induced by the mature predators plays an important role in the spatial distribution of species.