Diffusive Nicholson's Blowflies Research Articles

Bifurcation analysis for a single population model with advection.

In this paper, the dynamics of a single population model with a general growth function is investigated in an advective environment. We show the existence of a nonconstant positive steady state, and give sufficient conditions for the occurrence of a Hopf bifurcation at the positive steady state. Moreover, the theoretical results are applied to the diffusive Nicholson's blowflies and Mackey-Glass's models with advection and delay, respectively. We numerically show that the population density decreases as the increase of advection rate or death rate, and a delay-induced Hopf bifurcation is more likely to occur with small advection or low mortality rate.

On uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delay

This paper is devoted to study the uniqueness (up to translation) of traveling wave solutions for a general reaction-diffusion equation with spatio-temporal delay . It is shown that the traveling wave solutions with any given admissible speed (including the minimal wave speed ) of this general equation are unique up to translation under certain assumptions. The main result helps us to solve some open problems on the uniqueness of traveling wave solutions of a few well-known models such as the diffusive Nicholson's blowflies model, the diffusive Mackey-Glass' hematopoiesis model, an age-structured population model, an epidemic model and a reaction-diffusion model with a quiescent stage.

The stability and Hopf bifurcation of the diffusive Nicholson’s blowflies model in spatially heterogeneous environment

In this paper, we consider the diffusive Nicholson's blowflies model in spatially heterogeneous environment when the diffusion rate is large. We show that the ratio of the average of the maximum per capita egg production rate to that of the death rate affects the dynamics of the model. The unique positive steady state is locally asymptotically stable if the ratio is less than a critical value. However, when the ratio is greater than the critical value, large time delay can make the unique positive steady state unstable through Hopf bifurcation. Especially, the first Hopf bifurcation value tends to that of the ''average'' DDE model when the diffusion rate tends to infinity. Moreover, we show that the direction of the Hopf bifurcation is forward, and the bifurcating periodic solution from the first Hopf bifurcation value is orbitally asymptotically stable, which improves the earlier result by Wei and Li (Nonlinear. Anal., 60: 1351-1367, 2005).

Stability of Traveling Wavefronts for a Delayed Lattice System with Nonlocal Interaction

In this paper, we mainly investigate exponential stability of traveling wavefronts for delayed $2D$ lattice differential equation with nonlocal interaction. For all non-critical traveling wavefronts with the wave speed $c \gt c_*(\theta)$, where $c_*(\theta) \gt 0$ is the critical wave speed and $\theta$ is the direction of propagation, we prove that these traveling waves are asymptotically stable, when the initial perturbation around the traveling waves decay exponentially at far fields, but can be allowed arbitrarily large in other locations. Our approach adopted in this paper is the weighted energy method and the squeezing technique with the help of Gronwall's inequality. Furthermore, from stability result, we prove the uniqueness (up to shift) of the traveling wavefront. Our results can apply to the discrete diffusive Mackey-Glass model and the dicrete diffusive Nicholson's blowflies model on $2D$ lattices.

An approximation to the minimum traveling wave for the delayed diffusive Nicholson's blowflies equation

In this work, we study the approximation of traveling wave solutions propagated at minumum speeds c0(h) of the delayed Nicholson's blowflies equation: urn:x-wiley:mma:media:mma4401:mma4401-math-0001 In order to do that, we construct a subsolution and a super solution to (∗). Also, through that construction, an alternative proof of the existence of traveling waves moving at minimum speed is given. Our basic hypothesis is that p/δ∈(1,e] and then, the monostability of the reaction term. Copyright © 2017 John Wiley & Sons, Ltd.

Speed Selection and Stability of Wavefronts for Delayed Monostable Reaction-Diffusion Equations

We study the asymptotic stability of traveling fronts and front's velocity selection problem for the time-delayed monostable equation $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\ x \in \mathbb{R},\ t >0$, considered with Lipschitz continuous reaction term $g: \mathbb{R}_+ \to \mathbb{R}_+$. We are also assuming that $g$ is $C^{1,\alpha}$-smooth in some neighbourhood of the equilibria $0$ and $\kappa >0$ to $(*)$. In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of $g$ so that equation $(*)$ can possess pushed traveling fronts. Our first main result says that the non-critical wavefronts of $(*)$ with monotone $g$ are globally nonlinearly stable. In the special and easier case when the Lipschitz constant for $g$ coincides with $g'(0)$, we present a series of results concerning the exponential [asymptotic] stability of non-critical [respectively, critical] fronts for the monostable model $(*)$. As an application, we present a criterion of the absolute global stability of non-critical wavefronts to the diffusive Nicholson's blowflies equation.

Global Attractivity of a Diffusive Nicholson's Blowflies Equation with Multiple Delays

The present paper considers a diffusive Nicholson's blowflies model with multiple delays under a Neumann boundary condition. Delay independent conditions are derived for the global attractivity of the trivial equilibrium and the positive equilibrium, respectively. Two open problems concerning the stability of positive equilibrium and the occurrence of Hopf bifurcation are proposed.

Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space

Partial differential equations with discrete (concentrated)state-dependent delays are studied. The existence and uniqueness ofsolutions with initial data from a wider linear space is provenfirst and then a subset of the space of continuously differentiable(with respect to an appropriate norm) functions is used to constructa dynamical system. This subset is an analogue of thesolution manifold proposed for ordinary equations in [H.-O.Walther, The solution manifold and $C_{1}$-smoothness fordifferential equations with state-dependent delay, J. DifferentialEquations, 195(1), (2003) 46--65]. The existence of a compact global attractor is proven. As far as applications are concerned, we consider the well knownMackey-Glass-type equations with diffusion, theLasota-Wazewska-Czyzewska model, and the delayed diffusiveNicholson's blowflies equation, all with state-dependent delays.

Semi-analytical solutions for the 1- and 2-D diffusive Nicholson's blowflies equation

Semi-analytical solutions are developed for the diffusive Nicholson's blowflies equation. Both one and two-dimensional geometries are considered. The Galerkin method, which assumes a spatial structure for the solution, is used to approximate the governing delay partial differential equation by a system of ordinary differential delay equations. Both steady-state and transient solutions are presented. Semi-analytical results for the stability of the system are derived and the critical parameter value, at which a Hopf bifurcation occurs, is found. Semi-analytical bifurcation diagrams and phase-plane maps are drawn, which show the initial Hopf bifurcation together with a classical period doubling route to chaos. A comparison of the semi-analytical and numerical solutions shows the accuracy and usefulness of the semi-analytical solutions. Also, an asymptotic analysis for the periodic solution near the Hopf bifurcation point is developed, for the one-dimensional geometry. © 2012 The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay

We consider a diffusive Nicholson's blowflies equation with non-local delay and study the stability of the uniform steady states and the possible Hopf bifurcation. By using the upper- and lower solutions method, the global stability of constant steady states is obtained. We also discuss the local stability via analysis of the characteristic equation. Moreover, for a special kernel, the occurrence of Hopf bifurcation near the steady state solution and the stability of bifurcated periodic solutions are given via the centre manifold theory. Based on laboratory data and our theoretical results, we address the influence of various types of vaccinations in controlling the outbreak of blowflies.

Admissible wavefront speeds for a single species reaction-diffusion equation with delay

We consider equation $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x))$(*) , when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=\kappa>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(\kappa)) =$ [ c<SUB>*</SUB>, c <SUP>*</SUP> ] such that (*) has at least one (possibly, non-monotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here c<SUB>*</SUB>$ >0$ is finite and c <SUP>*</SUP> $ \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound c<SUB>*</SUB>is sharp so that there are not wavefronts moving with speed $c < $ c<SUB>*</SUB>. In contrast to reported results, the interval $\mathcal{C}$ can be compact, and we conjecture that some of equations (*) can indeed have an upper bound for propagation speeds of travelling fronts. As particular cases, Eq. (*) includes the diffusive Nicholson's blowflies equation and the Mackey-Glass equation with non-monotone nonlinearity.

Nonmonotone travelling waves in a single species reaction–diffusion equation with delay

We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction–diffusion equations u t ( t , x ) = Δ u ( t , x ) − u ( t , x ) + g ( u ( t − h , x ) ) ( ∗ ) , when g ∈ C 2 ( R + , R + ) has exactly two fixed points: x 1 = 0 and x 2 = K > 0 . Recently, nonmonotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey–Glass-type equations with diffusion fall within this subclass of ( ∗ ) . As an example, we consider the diffusive Nicholson's blowflies equation.

Traveling waves for the diffusive Nicholson's blowflies equation

We consider traveling wave front solutions for the diffusive Nicholson's blowflies equation on the real line. The existence of such solutions is proved using the technique developed by J. Wu and X. Zou (J. Dyn. Differ. Equations 13 (3) (2001)). Some numerical simulation using the iteration formula of Wu and Zou [7] is also provided.

Dynamics of the diffusive Nicholson's blowflies equation with distributed delay

In this paper we study the diffusive Nicholson's blowflies equation. Generalizing previous works, we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible. The linearized stability of the non-zero uniform steady state is studied in detail, mainly by using the principle of the argument. Global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations. Finally, we consider what bifurcations are possible from the non-zero uniform state in the case when it is unstable.

Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays

We consider the diffusive Nicholson's blowflies equation where the time delay is of the distributed kind, incorporated as an integral convolution in time. Of interest is the question of the existence of travelling front solutions and their qualitative form. For small delay, existence of such fronts is proved when the convolution kernel assumes a special form, enabling the use of linear chain techniques. The resulting higher-dimensional system is studied using geometric singular perturbation theory. The method should be applicable to other such kernels as well. For larger delays, numerical simulations show that the main effect is a loss of monotonicity of the wave front.

Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation

For the Dirichlet boundary value problem of the diffusive Nicholson's blowflies equation, it was shown in Ref. [17] that in a certain range of the parameter space, there is a unique positive steady state solution. In this paper, we propose a scheme to compute this steady state numerically. In addition, we describe an iterative procedure to locate the critical values of the delay where a Hopf bifurcation of time periodic solutions takes place near the steady state. Some numerical simulations of both schemes are given.

Dirichlet Problem for the Diffusive Nicholson's Blowflies Equation

The object of this paper is to consider the asymptotic behavior of solutions of the diffusive Nicholson's blowflies equation under Dirichlet boundary condition. A new approach is developed to study the global attractivity of the positive steady state for a reaction diffusion equation with time delay, when monotone or quasi–monotone conditions fail to apply. This approach should be applicable to other Dirichlet problems as well, even though our analysis is tailored to the diffusive Nicholson's blowflies equation.