Volume 20, Issue 7-8 -

In this paper, we consider an N-species Gilpin–Ayala impulsive competition system. By using comparison theorem, Lyapunov functional, and almost periodic functional hull theory of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence, global asymptotical stability, and almost periodic solution of the model. Our results extend some previously known results. The method used in this paper provides a possible method to study the permanence, global asymptotical stability, and almost periodic solution of the models with impulsive perturbations in biological populations.

We investigate the logarithmic (1 + 1) dimensional KdV-like and (2 + 1) dimensional KP-like equations which model many physical processes in the field of soliton theory. In this paper, first, we get the classical Lie point symmetries using the invariance theory. Secondly, we obtain conservation laws of the underlying equations by incorporating the method of multiplier and nonlocal conservation method. A relationship between the obtained symmetries and conservation laws are shown. Then using the generalized double reduction theory for the associated symmetries, reductions are constructed. Finally traveling wave solutions are computed with the aid of the simplest equation method for the logarithmic (2 + 1) -dimensional KP-like equation.

In this article, we propose and analyze a time-delayed susceptible–infected–recovered (SIR) mathematical model with nonlinear incidence rate and nonlinear treatment rate for the control of infectious diseases and epidemics. The incidence rate of infection is considered as Crowley–Martin functional type and the treatment rate is considered as Holling functional type II. The stability of the model is investigated for the disease-free equilibrium (DFE) and endemic equilibrium (EE) points. From the mathematical analysis of the model, we prove that the model is locally asymptotically stable for DFE when the basic reproduction number R0{R_0} is less than unity (R0<1)({R_0} \lt 1) and unstable when R0{R_0} is greater than unity (R0>1)({R_0} \gt 1) for time lag τ≥0 \tau \ge 0. The stability behavior of the model for DFE at R0=1{R_0} = 1 is investigated using Castillo-Chavez and Song theorem, which shows that the model exhibits forward bifurcation at R0=1 {R_0} = 1. We investigate the stability of the EE for time lag τ≥0 \tau \ge 0. We also discussed the Hopf bifurcation of EE numerically. Global stability of the model equilibria is also discussed. Furthermore, the model has been simulated numerically to exemplify analytical studies.

A new class of memristor-based time-delay fractional-order hybrid BAM neural networks has been put forward. The contraction mapping principle has been adopted to verify the existence and uniqueness of the equilibrium point of the addressed neural networks. By virtue of fractional Halanay inequality and fractional comparison principle, not only the dissipativity has been analyzed, but also a globally attractive set of the new model has been formulated clearly. Numerical simulation is presented to illustrate the feasibility and validity of our theoretical results.

In this paper, a three species food chain model has been developed among the interaction of prey, predator and super predator. It is assumed that the predator shows refuge behavior to the super predator. It is also assumed that a certain amount of additional food will be supplied to the super predator. It is considered that the predator population is benefiting partially from the additional food. To get optimal harvesting of super predator the Pontryagin’s maximum principle has been used. It is found that super predator may be extinct if harvesting rate increase. It is observed that as the refuge rate increases, predator population gradually increases, but super predator population decreases. Also, it is found that our proposed system undergoes oscillatory or periodic behavior as the value of refuge rate ( m 1 ), harvesting rate ( E ), the intrinsic growth rate of prey ( r ), carrying capacity of prey ( k ) and conservation rate of prey ( c 1 ) varies for some certain range of these parameters. It is found that this study may be useful for the increase of harvesting of a super predator by supplying the additional food to our proposed system.

Fractional differential equations with delay behaviors occur in fields like physical and biological ones with state-dependent delay or nonconstant delay and has drawn the attention of researchers. The main goal of the present work is to study the existence of mild solutions of neutral differential system along state-dependent delay in Banach space. By employing the fractional theory, noncompact measure and Mönch’s theorem, we investigate the existence results for neutral differential equations of fractional order with state-dependent delay. An illustration of derived results is offered.

In this paper, a new method for the numerical solution of fractional sine-Gordon (SG) equation is presented. Our method consists of two steps, in first step: the main equation is converted to a homogeneous one using interpolation. In second step: two-dimensional approximation of functions by shifted Jacobi polynomials is used to reduce the problem into a system of nonlinear algebraic equations. The archived system is solved by Newton’s iterative method. Our method is stated in general case on rectangular [ a , b ] × [0, T ] which is based upon Jacobi polynomial by parameters ( α , β ). Several test problems are employed and results of numerical experiments are presented and also compared with analytical solutions. Also, we verify the numerical stability of the method, by applying a disturbance in the problem. The obtained results confirm the acceptable accuracy and stability of the presented method.

We study the existence of multiple positive solutions for a nonlinear singular Riemann–Liouville fractional differential equation with sign-changing nonlinearity, subject to Riemann–Stieltjes boundary conditions which contain fractional derivatives. In the proof of our main theorem, we use various height functions of the nonlinearity of equation defined on special bounded sets, and two theorems from the fixed point index theory.

In this paper, we consider a non local boundary value problem of nonlinear fractional Langevin equation with non-instantaneous impulses. Initially, we form a standard framework to originate a formula of solutions to our proposed model and then implement the concept of generalized Ulam–Hyers–Rassias using Diaz–Margolis’s fixed point theorem over a generalized complete metric space.