Blowflies Equation Research Articles

Dynamic analysis on a delayed nonlinear density-dependent mortality Nicholson's blowflies model

This paper discusses a class of nonlinear density-dependent mortality Nicholson's blowflies equation with multiple pairs of time-varying delays. By utilising differential inequality techniques and the fluctuation lemma, a delay-independent criterion is gained to ensure the global asymptotic stability of the addressed model, which refines some previously known research. Finally, some numerical simulations are listed to show the validity of our methods.

Dirichlet problem for the Nicholson’s blowflies equation with density-dependent diffusion

This paper is concerned with the time delayed Nicholson’s blowflies equation with degenerate diffusion. We prove the existence and uniqueness of the positive steady state solution under the Dirichlet boundary condition and we show the stability of the nontrivial steady state.

Stability analysis of Nicholson’s blowflies equation with two different delays

This paper investigates an autonomous Nicholson’s blowflies equation incorporating two different delays. By using differential inequality techniques and dynamical system approaches, we establish two novel criteria to check the global exponential stability and asymptotical stability on the zero equilibrium point of the addressed equation, respectively. Our research partially answers an open question raised by Berezansky and Braverman (2017). A numerical example with simulations shows that the main theoretical results are correct.

New results on stability of Nicholson’s blowflies equation with multiple pairs of time-varying delays

This paper is concerned with a non-autonomous Nicholson’s blowflies equation with multiple pairs of time-varying delays. By using differential inequality techniques and the fluctuation lemma, we establish two criteria on the global exponential stability and global asymptotical stability on the zero equilibrium of the addressed equation, respectively. The obtained results improve and complement some existing ones. The effectiveness of the obtained results is illustrated by some numerical simulations.

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.

Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state‐dependent delay

In this paper, we study the dynamics of a Nicholson's blowflies equation with state‐dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state‐dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd.

DEPENDENCE OF STABILITY OF NICHOLSON'S BLOWFLIES EQUATION WITH MATURATION STAGE ON PARAMETERS

The stability of Nicholson's blowflies equation with maturation stage is investigated by reducing the number of parameters in the original model. We derive the condition on the stability of the positive equilibrium of the model, and discuss the dependence of the stability on the parameters by analyzing geometrically the dependence of real parts of eigenvalues of the characteristic equation with fewer parameters on the parameters. By restoring parameters, the condition on the stability of the positive equilibrium of the original model are formulated explicitly, and the corresponding regions are depicted for some different cases. The obtained result shows that the parameter determining the maximum reproductive success of the population affects only the size of the positive equilibrium, but plays no role in determining its stability.

Hopf bifurcation and chaos in a delayed Nicholson’s blowflies equation with nonlinear density-dependent mortality rate

The stability and bifurcation of the Nicholson’s blowflies equation with nonlinear density-dependent mortality rate and delay feedback are investigated. First, we study the existence of positive equilibria and the stability of positive equilibria in the absence of delay. Then, how delay influences the stability of positive equilibria is studied. According to our analysis, as delay increases, there are a sequence of critical values where the system undergoes Hopf bifurcations. Using the theory of normal form and center manifold reduction, the direction and stability of Hopf bifurcations can be determined according to the parameters of the system. Finally, numerical simulations are applied to illustrate and expend our theoretical analysis, and delay-induced chaos has been found.

Hybrid control of the Neimark-Sacker bifurcation in a delayed Nicholson's blowflies equation.

In this article, for delayed Nicholson’s blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method.

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.

Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay

This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction–diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction–diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves ϕ(x+ct) exist for all c≥c⁎ and are exponentially stable for all wave speed c>c⁎[13], where c⁎ is called the critical wave speed. In this paper, we prove that the critical traveling waves ϕ(x+c⁎t) (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.

Global Hopf Bifurcation Analysis of a Nicholson’s Blowflies Equation of Neutral Type

We investigate Hopf bifurcations in a delayed Nicholson’s blowflies equation of neutral type, derived from the Gurtin–MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criterion for higher dimensional ordinary differential equations. We show that a branch of slowly varying periodic solutions and a branch of fast oscillating periodic solutions coexist for all large delays.

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies $1 e$, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay $r$ or the wave speed $c$ is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when $e c_*>0$ are exponentially stable, where $c_*>0$ is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed $c>c_*$ and any size of the time-delay $r>0$; however, when $\frac{p}{d}> e^2$ with a smal...

Derivation and computation of discrete-delayand continuous-delay SDEs in mathematical biology

Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.

Nicholson's blowflies equation and a delay independent Aizerman problem

As shown by its title, the aim of this paper is twofold. Firstly, the equation of Nicholson describing population dynamics is interesting in itself since it is a straightforward example of time delay equation where the delay appears naturally in the models as a consequence of the (here biological) phenomenon. On the other hand the nonlinearity occurring in this equation is a sector restricted one and incorporating the stability problems in an absolute stability one makes sense. The main result of the paper is the positive answer to the associated delay independent Aizerman conjecture.

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.