Positive periodic solutions to the forced non-autonomous Duffing equations

We study the existence and multiplicity of positive solutions to the periodic problem for a forced non-autonomous Duffing equation u ′′ = p ⁢ ( t ) ⁢ u - h ⁢ ( t ) ⁢ | u | λ ⁢ sgn ⁡ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-h(t)\lvert u\rvert^{\lambda}\operatorname{sgn}u+f(t);\quad u(0)=u(\omega),\ u^{\prime}(0)=u^{\prime}(\omega), where p , h , f ∈ L ⁢ ( [ 0 , ω ] ) p,h,f\in L([0,\omega]) and λ > 1 \lambda>1 . The obtained results are compared with the results known for the equations with constant coefficients.

Study on asymptotic behavior of stochastic Lotka–Volterra system in a polluted environment

A three-species non-autonomous stochastic Lotka–Volterra food web system in a polluted environment is proposed, and the existence of positive periodic solutions of this system is established by constructing a proper Lyapunov function. Then the extinction property and its threshold between persistence and extinction are discussed by using Itô’s formula and the strong law of large numbers of martingale, and the sufficient condition of a.s. exponential stability of equilibrium point is obtained. Finally, the conclusions are tested by several numerical simulations.

Dynamic Analysis of a Model for Spruce Budworm Populations with Delay

A class of delayed spruce budworm population model is considered. Compared with previous studies, both autonomous and nonautonomous delayed spruce budworm population models are considered. By using the inequality techniques, continuation theorem, and the construction of suitable Lyapunov functional, we establish a set of easily verifiable sufficient conditions on the permanence, existence, and global attractivity of positive periodic solutions for the considered system. Finally, an example and its numerical simulation are given to illustrate our main results.

On periodic solutions of a discrete Nicholson’s dual system with density-dependent mortality and harvesting terms

In this study, we discuss the existence of positive periodic solutions of a class of discrete density-dependent mortal Nicholson’s dual system with harvesting terms. By means of the continuation coincidence degree theorem, a set of sufficient conditions, which ensure that there exists at least one positive periodic solution, are established. A numerical example with graphical simulation of the model is provided to examine the validity of the main results.

Global Hopf Bifurcation in a Phytoplankton–Zooplankton System with Delay and Diffusion

In this paper, the dynamics of a phytoplankton–zooplankton system with delay and diffusion are investigated. The positivity and persistence are studied by using the comparison theorem and upper and lower solutions method. The stability of steady states and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And the global existence of positive periodic solutions is established by using the global Hopf bifurcation result given by Wu [1996]. Finally, some numerical simulations are carried out to illustrate the analytical results.

Global existence of positive periodic solutions of a general differential equation with neutral type

In this paper, the dynamics of a general differential equation with neutral type are investigated. Under certain assumptions, the stability of positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Krawcewicz et al. Finally, by taking neutral Nicholson’s blowflies model and neutral Mackey–Glass model as two examples, some numerical simulations are carried out to illustrate the analytical results.