Stochastic Nicholson’s blowflies delayed differential equations

In this paper, we consider a class of stochastic Nicholson’s blowflies delayed differential equations. Firstly, we obtain the existence and uniqueness of the global positive solution with nonnegative initial conditions. Then the ultimate boundedness in mean of solution is derived under the same condition. Moreover, we estimate the sample Lyapunov exponent of the solution, which is less than a positive constant. In the end, an example with its numerical simulations is carried out to validate the analytical results.     

Stochastic Nicholson’s blowflies delay differential equation with regime switching

In this paper, we investigate the global existence of almost surely positive solution to a stochastic Nicholson’s blowflies delay differential equation with regime switching, and give the estimation of the path. The results presented in this paper extend some corresponding results in Wang et al. Stochastic Nicholson’s blowflies delayed differential equations, Appl. Math. Lett. 87 (2019) 20–26 .     

Bifurcation analysis in a diffusive Segel–Jackson model

Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling predator–prey interaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns.     

Dynamics of Nicholson’s blowflies models with a nonlinear density-dependent mortality

This paper is concerned with a asymptotically almost periodic Nicholson’s blowflies model involving a nonlinear density-dependent mortality term.By combining Lyapunov function method with differential inequality techniques, some novel sufficient conditions are derived to guarantee the asymptotically almost periodicity of the addressed model. Especially, a numerical example with graphical illustration is present to show the correctness of the theoretical results.     

Exponential convergence for a class of Nicholson’s blowflies model with multiple time-varying delays

This paper is concerned with the generalized Nicholson’s blowflies model with multiple time-varying delays which is defined on the nonnegative function space. By using some differential inequalities, some new results are established to ensure that all solutions of the model converge exponentially to zero equilibrium point. Moreover, an example is given to illustrate our main results.     

Bifurcation analysis on a diffusive Holling–Tanner predator–prey model

This paper is concerned with a diffusive Holling–Tanner predator–prey model subject to homogeneous Neumann boundary condition. By choosing the ratio of intrinsic growth rates of predator to prey λ as the bifurcation parameter, we find that spatially homogeneous and non-homogeneous Hopf bifurcation occur at the positive constant steady state as λ varies. The steady state bifurcation of simple and double eigenvalues are intensively investigated. The techniques of space decomposition and the implicit function theorem are adopted to deal with the case of double eigenvalues. Our results show that this model can exhibit spatially non-homogeneous periodic and stationary patterns induced by the parameter λ. Numerical simulations are presented to illustrate our theoretical results.     

Convergence for a class of non-autonomous Nicholson’s blowflies model with time-varying coefficients and delays

This paper is concerned with a class of the generalized non-autonomous Nicholson’s blowflies model with time-varying coefficients and delays which is defined on the nonnegative function space. Under proper conditions, we employ a novel proof to establish some criteria to ensure that all solutions of this model converge to an equilibrium. Moreover, we give an example to illustrate our main result.     

Convergence and permanence of a delayed Nicholson’s Blowflies model with feedback control

In this paper, we study a generalized Nicholson??s Blowflies model with feedback control and multiple time-varying delays. Under proper conditions, we employ a novel proof to establish some criteria to guarantee the global exponential convergence and permanence of this model. Moreover, we give two examples to illustrate our main results.     

Global dynamics of Nicholsonʼs blowflies equation revisited: Onset and termination of nonlinear oscillations

We revisit Nicholson?s blowflies model with natural death rate incorporated into the delay feedback. We consider the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from a positive equilibrium. We show that the model has only a finite number of Hopf bifurcation values and we describe how branches of Hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals. The bifurcation analysis and the Matlab package DDE-BIFTOOL developed by Engelborghs et al. guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions.     

Global stability of a class of Nicholson’s blowflies model with patch structure and multiple time-varying delays

This paper is concerned with a class of the generalized Nicholson’s blowflies model with patch structure and multiple time-varying delays, which is defined on the nonnegative function space. Under proper conditions, we employ a novel proof to establish some criteria of the global stability of positive equilibrium for this model. Moreover, we give an example to illustrate our main result.     

Bifurcation analysis in a discrete differential-algebraic predator–prey system

The discrete-time predator–prey biological economic system obtained by Euler method is investigated. Some conditions for the system to undergo flip bifurcation and Neimark–Sacker bifurcation are derived by using new normal form of differential-algebraic system, center mainfold theorem and bifurcation theory. Numerical simulations are given to show the effectiveness of our results and also to exhibit period-doubling bifurcation in orbits of period 2, 4, 8 and chaotic sets. The results obtained here reveal far richer dynamics in discrete differential-algebraic biological economic system. The contents are interesting in mathematics and biology.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.     

一类具无穷时滞 Nicholson’s blowflies模型的正概周期解(英文)

通过利用锥上的不动点定理,本文主要研究具无穷时滞Nicholson’s blowflies模型的正概周期解的存在唯一性.从而得到此正概周期解存在唯一性和指数收敛的充分条件.最后给出一个例子说明本文结果的可行性.     

Multiple bifurcation analysis in a NDDE arising from van der Pol’s equation with extended delay feedback

Van der Pol’s equation with extended delay feedback control is considered, which is equivalent to a system of neutral differential–difference equations (NDDEs). Fold bifurcation and Hopf bifurcation in this NDDE are studied by the formal adjoint theory, the center manifold theorem and the normal form method. These methods are also first employed in studying the Bogdanov–Takens singularity of NDDE. Bifurcation sets theoretically indicate the existence of a homoclinic orbit and the coexistence of three periodic solutions, which are all illustrated by the numerical methods. The coexistence of three stable periodic solutions and the existence of stable torus near the Hopf–fold and Hopf–Hopf bifurcations are also illustrated, respectively.     

Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response

In this paper, a diffusive predator–prey system with Holling III functional response and nonconstant death rate subject to Neumann boundary condition is considered. We study the stability of equilibria, and Turing instability of the positive equilibrium. We also perform a detailed Hopf bifurcation analysis to PDE system, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. In addition, some numerical simulations are carried out.