Nicholson's Blowflies Equation Research Articles

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.

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.

Remark on critical speed of traveling wavefronts for Nicholson's blowflies equation with diffusion

This note is devoted to the study on the traveling wavefronts to the Nicholson's blowflies equation with diffusion, a time-delayed reaction-diffusion equation. For the critical speed of traveling waves, we give a detailed analysis on its location and asymptotic behavior with respect to the mature age.

On travelling wavefronts of Nicholson's blowflies equation with diffusion

This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.

Traveling wavefronts for time-delayed reaction–diffusion equation: (I) Local nonlinearity

In this paper, we study a class of time-delayed reaction–diffusion equation with local nonlinearity for the birth rate. For all wavefronts with the speed c > c ∗ , where c ∗ > 0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x → − ∞ , but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei, So, Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594] for the speed c > 2 D m ( ε p − d m ) with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c > c ∗ with sufficiently small delay time r ≈ 0 . The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction–diffusion population model, Quart. J. Mech. Appl. Math. 58 (2005) 257–268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c > c ∗ and for an arbitrary time-delay r > 0 .

On the global attractor of delay differential equations with unimodal feedback

We give bounds for the global attractor of the delay differential equation $x'(t) =-\mu x(t)+f(x(t-\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the d5A5Aelay is sufficiently small, then all solution enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.

Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity

This paper is concerned with the existence of traveling wave fronts for delayed non-local diffusion systems without quasimonotonicity, which can not be answered by the known results. By using exponential order, upper–lower solutions and Schauder's fixed point theorem, we reduce the existence of monotone traveling wave fronts to the existence of upper–lower solutions without the requirement of monotonicity. To illustrate our results, we establish the existence of traveling wave fronts for two examples which are the delayed non-local diffusion version of the Nicholson's blowflies equation and the Belousov–Zhabotinskii model. These results imply that the traveling wave fronts of the delayed non-local diffusion systems without quasimonotonicity are persistent if the delay is small.

Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case

In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson's equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities.

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.

Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback

The dynamics generated by the delay differential equation with unimodal feedback is studied. The existence of the global attractor is shown and bounds of the attractor are given. We find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f ′ is negative with respect to a positive equilibrium, so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In particular, the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established. Numerical examples using Nicholson's blowflies equation and the Mackey–Glass equation are provided to illustrate the main results.

Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis

Known Nicholson's blowflies equation (which is one of the most important models in ecology) with stochastic perturbations is considered. Stability of the positive (nontrivial) point of equilibrium of this equation and also a capability of its discrete analogue to preserve stability properties of the original differential equation are studied. For this purpose, the considered equation is centered around the positive equilibrium and linearized. Asymptotic mean square stability of the linear part of the considered equation is used to verify stability in probability of nonlinear origin equation. From known previous results connected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals construction, necessary and sufficient condition of stability in the mean square sense in the continuous case and necessary and sufficient conditions for the discrete case are deduced. Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories. The trajectories of stable and unstable solutions of considered equations are simulated numerically in the deterministic and the stochastic cases for different values of the parameters and of the initial data. Numerous graphical illustrations of stability regions and solution trajectories are plotted.

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.

Stability and bifurcation of numerical discretization Nicholson blowflies equation with delay

A kind of discrete system according to Nicholson's blowflies equation with a finite delay is obtained by the Euler forward method, and the dynamics of this discrete system are investigated. Applying the theory of normal form and center manifold, we not only discuss the linear stability of the equilibrium and the existence of the local Hopf bifurcations, but also give the explicit algorithm for determining the direction of bifurcation and stability of the periodic solution of bifurcation.

Hopf bifurcation analysis in a delayed Nicholson blowflies equation

The dynamics of a Nicholson's blowflies equation with a finite delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799), and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney (J. Differential Equations 106 (1994) 27).

Global stability in difference equations satisfying the generalized Yorke condition

We present several conditions sufficient for global stability of the zero solution of nonautonomous difference equation x n + 1 = q x n + f n ( x n , … , x n − k ) , n ∈ Z , when the nonlinearities f n satisfy a sort of negative feedback condition. Moreover, for every k ∈ N , we indicate q k such that one of our stability conditions is sharp if q ∈ ( 0 , q k ] . We apply our results to discrete versions of Nicholson's blowflies equation, the Mackey–Glass equations, and the Wazewska and Lasota equation.

Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion

This paper considers the nonlinear stability oftravelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L2 norm, ifa solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate ofconvergence is also estimated.