Permanence and periodic solutions for a class of delay Nicholson’s blowflies models

This paper covers the dynamic behaviors for a class of Nicholson’s blowflies models with a nonlinear density-dependent mortality term. By using inequality analyze technique and coincidence degree theory, some sufficient conditions are determined that guarantee the permanence of the model and the existence of positive periodic solutions. Moreover, we give an example to illustrate our main results.     

Positive periodic solutions of delayed Nicholson’s blowflies models with a nonlinear density-dependent mortality term

This paper is concerned with the periodic solutions for a new fishery model with a nonlinear density-dependent mortality term, which is a generalization of the classical Nicholson’s blowflies model. By using coincidence degree theory, some criteria are obtained to guarantee that the existence of positive periodic solutions of the model. Moreover, an example and its numerical simulation are given to illustrate our main results.     

On positive periodic solution for the delay Nicholson’s blowflies model with a harvesting term

By using Krasnoselskii’s fixed point theorem, we obtain a sufficient condition for the existence of positive periodic solutions for the delay Nicholson’s blowflies model with a harvesting term. Three examples and numerical simulation are given to illustrate the effectiveness of our result.     

Positive almost periodic solution for a class of Nicholson’s blowflies model with a linear harvesting term

In this paper, we study the problem of positive almost periodic solutions for the generalized Nicholson’s blowflies model with a linear harvesting term and multiple time-varying delays. By applying the fixed point theorem and the Lyapunov functional method, we establish some criteria to ensure that the solutions of this model converge locally exponentially to a positive almost periodic solution. Moreover, we give an example to illustrate our main results.     

Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for a class of Nicholson’s blowflies model with patch structure and multiple linear harvesting terms. Under appropriate conditions, we establish some criteria to ensure that the solutions of this system converge locally exponentially to a positive almost periodic solution. Moreover, we give some examples and numerical simulations to illustrate our main results.     

On the behavior of the solutions to periodic linear delay differential and difference equations

Some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given. These results are obtained by the use of two distinct roots of the corresponding (so called) characteristic equation.     

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.     

Integral averaging technique and oscillation of certain even order delay differential equations

Using the generalized Riccati technique and the averaging technique, new oscillation criteria for certain even order delay differential equation of the form

     

A Volterra integral formulation for determining the periodic solutions of some autonomous, nonlinear, third-order ordinary differential equations

A Volterra integral formulation based on the introduction of a term proportional to the velocity times the square of the (unknown) frequency of oscillation, a new independent variable equal to the original one times the (unknown) frequency of oscillation, the method of variation of parameters and series expansions of both the solution and the frequency of oscillation, is used to determine the periodic solutions to three nonlinear, autonomous, third-order, ordinary differential equations. It is shown that the first term of the series expansion of the frequency of oscillation coincides with that determined from a first-order harmonic balance procedure, whereas the two-term approximation to the frequency of oscillation is shown to be more accurate than that of a parameter perturbation procedure for the second equation considered in this paper. For the third equation, it is shown that the two-term approximation presented in this paper is more accurate than the corresponding one for one of the parameter perturbation methods, and for initial velocities less than one, for the other parameter perturbation approach.     

Oscillation properties of nonlinear impulsive delay differential equations and applications to population models

Comparison theorem and explicit sufficient conditions are obtained for oscillation and nonoscillation of solutions of nonlinear impulsive delay differential equations which can be utilized to population dynamic models. Our results in this paper generalize and improve several known results.     

Two unsolved problems in the stability theory of stochastic differential equations with delay

Two unsolved problems of the stability theory for stochastic differential equations with delay are offered for consideration.     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

On stability of some linear and nonlinear delay differential equations

New explicit conditions of exponential stability are obtained for the nonautonomous equation with several delays

     

Periodic solutions of some fourth-order nonlinear differential equations

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence of T-periodic solutions for fourth-order nonlinear differential equations. Two illustrative examples are provided to demonstrate that the results in this paper hold under weaker conditions than the existing ones, and are more effective.