Nicholson's Blowflies Equation Research Articles

Almost periodic positive solutions of two generalized Nicholson's blowflies equations with iterative term

<abstract><p>This article considered two generalized Nicholson's blowflies equations with iteration term and time delay, as well as with immigration, and Nicholson's blowflies equation with iteration term and time delay, as well as harvesting term, respectively. Under appropriate conditions, the existence and uniqueness of almost periodic positive solutions for these two equations were established, respectively, by employing Banach's fixed point theorem. These results were brand new.</p></abstract>

Existence and stability of traveling waves for $p$-Laplacian equations with time delay

We consider the existence and stability of traveling wave solutions for Nicholson's blowflies equation with degenerate $p$-Laplacian diffusion and time delay. Existence results are proved by the monotone iteration method. Furthermore, we prove the existence and regularity of original solution for the time-delayed degenerate Nicholson's blowflies equation via compactness analysis. Finally, we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method.

Threshold convergence results for a nonlocal time-delayed diffusion equation

In this paper we consider the asymptotic behavior for nonlocal dispersion Nicholson blowflies equation ut=D(J⁎u−u)−du+pu(t−τ,x)e−au(t−τ,x) in the whole RN. By the method of Fourier transform, we first derive the decay estimates for the fundamental solutions with time-delay. Then, we obtain the threshold results with optimal convergence rates for the original solution to the constant equilibrium. Namely, when 0<pd<1, the solution u(t,x) globally converges to the equilibrium 0 in the time-exponential form; when pd=1, the solution u(t,x) globally converges to 0 in the time-algebraical form; when 1<pd≤e, the solution u(t,x) globally converges to u+ in the time-exponential form; and when e<pd<e2, it locally converges to u+ in the time-exponential form. This indicates that when the death rate is bigger than the birth rate, the blowflies will disappear in future. While, when the birth rate is bigger than the death rate in a certain range, then the blowflies population will reach an equilibrium after long time. The lower-higher frequency analysis plays a crucial role in the proof.

Basins of attraction and paired Hopf bifurcations for delay differential equations with bistable nonlinearity and delay-dependent coefficient

We consider a class of delay differential equations with bistable nonlinearity, in which the trivial equilibrium may coexist with two positive equilibria. Despite the difficulty caused by delay and bistable nonlinearity, we give a rather complete description on the dynamics including global stability, semi-stability, bistability and Hopf bifurcation. For the case where the stable trivial equilibrium coexists with a stable positive equilibrium, we obtain two delay-dependent intervals as subsets of basins of attraction of two stable equilibria. These subsets are sharp in some sense. Using delay as the bifurcation parameter, we analytically show that the number of local Hopf bifurcation values is finite and these local Hopf bifurcation values are neatly paired. A Nicholson's blowflies equation with Allee effect is used to illustrate our general results. Through this example, we show that delay can induce stability switches, symmetric transitions among multitype bistability and robust phase-transitions for long transients.

Stationary and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

Stationary and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

Global asymptotic stability of a scalar delay Nicholson's blowflies equation in periodic environment

This paper is considered with a scalar delay Nicholson's blowflies equation in periodic environments. By taking advantage of some novel differential inequality techniques and the fluctuation lemma, we set up the sharp condition to characterize the global asymptotic stability of positive periodic solutions on the addressed equation. The obtained results improve and supplement some existing ones in recent literature, and then give a more perfect answer to an open problem proposed by Berezansky et al. in [Appl. Math. Model. 34(2010) 1405–1417]. In particular, two numerical examples are provided to verify the reliability and feasibility of the theoretical findings.

Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

&lt;p style='text-indent:20px;'&gt;The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.&lt;/p&gt;

Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays

&lt;p style='text-indent:20px;'&gt;We consider the classical Nicholson's blowflies model incorporating two distinctive time-varying delays. One of the delays corresponds to the length of the individual's life cycle, and another corresponds to the specific physiological stage when self-limitation feedback takes place. Unlike the classical formulation of Nicholson's blowflies equation where self-regulation appears due to the competition of the productive adults for resources, the self-limitation of our considered model can occur at any developmental stage of an individual during the entire life cycle. We aim to find sharp conditions for the global asymptotic stability of a positive equilibrium. This is a significant challenge even when both delays are held at constant values. Here, we develop an approach to obtain a sharp and explicit criterion in an important situation where the two delays are asymptotically apart. Our approach can be also applied to the non-autonomous Mackey-Glass equation to provide a partial solution to an open problem about the global dynamics.&lt;/p&gt;

Bifurcation analysis in delayed Nicholson blowflies equation with delayed harvest

In this paper, we investigate a delayed Nicholson equation with delay harvesting term which was proposed in open problems and conjectures formulated by Berezansky et al. (Appl. Math. Model. 34: 1405–1417, 2010). The stability switching curves by taking two delays as parameters are obtained via the method introduced by An et al.(J. Differ. Equ. 266: 7073–7100, 2019). The existence of Hopf singularity on a two-parameter plane is determined by the varying direction of two parameters. Furthermore, the normal form near the Hopf singularity is derived via applying the center manifolds theory and normal forms method of FDEs. Finally, some numerical simulations are carried out to illustrate the theoretical conclusions.

Asymptotic behavior of solutions for time-delayed nonlocal dispersion equations with Dirichlet boundary

This paper is concerned with the asymptotic behavior of the solutions for Nicholson's blowflies equation with nonlocal dispersion subjected to Dirichlet boundary condition. We first prove the existence and uniqueness of the solution for the initial boundary value problem and its non-trivial steady state. Then we give a threshold result on global stability of equilibria: when , the solution time-exponentially converges to the constant equilibrium 0 for any large initial data; when , the solution time-asymptotically converges to its positive steady-state for any large initial data, once , where D>0 is the diffusion coefficient, is the death rate, p>0 is the birth rate and is the principal eigenvalue for the nonlocal characteristic equation. The adopted approach is the energy method and the monotonic technique.

About one method of stability investigation for nonlinear stochastic delay differential equations

AbstractStability of equilibria of a nonlinear delay differential equation is investigated under stochastic perturbations. The use of Stiltjes integrals allows to consider both discrete and distributed delays in the investigated equation. Obtained conditions of stability in probability are formulated in terms of linear matrix inequalities. The proposed method of investigation can be applied to a number of very popular in research mathematical models such as Mackey–Glass model, neoclassical growth model and e‐commerce model, Nicholson's blowflies equation, mosquito and glassy‐winged sharpshooter populations, SIR epidemic model, social epidemic models, and many similar other mathematical models which can be considered as particular cases of the investigated nonlinear equation.

Positive periodic solutions for revisited Nicholson's blowflies equation with iterative harvesting term

Nicholson's blowflies differential equation is one of the famous equations in the field of population dynamics. The first steps of the apparition of this equation began in the 1950s with the Nicholson experiments on the sheep blowfly Lucilia cuprina, the hot topic in Australia at that time and since then it has drawn more and more attention of many researchers. The main purpose of the current study is to investigate the existence of periodic positive solutions for Nicholson's blowflies differential equation with an iterative harvesting function by using Schauder's fixed point theorem. Finally, an example is employed to illustrate the effectiveness of our results.

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.

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 &amp; 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 &amp; 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.

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.