E. Liz, M. Pinto, G. Robledo, V. Tkachenko, S. Trofimchuk
We prove that the well-known 3/2 stability condition established for the Wright equation (WE) still holds if the nonlinearity $p(\exp(-x)-1)$ in WE is replaced by a decreasing or unimodal smooth function f with $f'(0)<0$ satisfying the standard negative feedback and below boundedness conditions and having everywhere negative Schwarz derivative.
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Global stability of Wright-type equations with negative Schwarzian
本研究将延迟逻辑斯谛方程的37/24全局稳定性准则推广到具负Schwarzian、递减光滑非线性项且满足负反馈与有界性假设的Wright型方程，通过构造分析界关系并结合区间分析验证，拓展了种群模型稳定性理论的适用范围。
We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation $x'(t)=-μf(x(t-1))$, as the parameter $μ$ passes through the critical values $μ_k$. We give a complete classification of the possible bifurcation sequences. Using this information and the Cooke-transformation, we obtain local estimates and monotonicity properties of the periods of the bifurcating limit cycles along the Hopf-branches. Further, we show how our results relate to the often required property that the nonlinearity has negative Schwarzian derivative.
We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright's equation and the Mackey-Glass type delay differential equations. We present some dichotomy results, which allow us to get easily computable bounds of the global attractor. We also discuss related conjectures, and formulate new open problems.
An old conjecture in delay equations states that Wright's equation \[ y'(t)= - αy(t-1) [ 1+y(t)], α\in \mathbb{R} \] has a unique slowly oscillating periodic solution (SOPS) for every parameter value $α>π/2$. We reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of SOPS of Wright's equation. Using this method, we show that a part of this branch does not have any fold point nor does it undergo any secondary bifurcation, partially answering the new reformulated conjecture.
We consider a family of scalar delay differential equations $x'(t)=f(t,x_t)$, with a nonlinearity $f$ satisfying a negative feedback condition combined with a boundedness condition. We present a global stability criterion for this family, which in particular unifies the celebrated 3/2-conditions given for the Yorke and the Wright type equations. We illustrate our results with some applications.
For equations $ x'(t) = -x(t) + ζf(x(t-h)), x \in \R, f'(0)= -1, ζ> 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $ζ-1 >0$ and $h(ζ-1)^{1/8}$ are less than some constant (independent on $h,ζ$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.
Stochastic Wright's Equation: Existence of Invariant Measures
本文研究随机扰动后的Wright方程，证明其存在至少两个不变测度，包括集中于-1的平凡测度和定义在(-1,∞)上的非平凡测度，关键步骤是证明所有解在概率意义下远离-1。
We prove the existence of a continuous family of positive and generally non-monotone travelling fronts in delayed reaction-diffusion equations $u_t(t,x) = Δu(t,x)- u(t,x) + g(u(t-h,x)) (*)$, when $g \in C^2(R_+,R_+)$ has exactly two fixed points: $x_1= 0$ and $x_2= a >0$. Recently, non-monotonic 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 $h$ grows. For the case of $g$ with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass type equ…
The delayed logistic equation (also known as Hutchinson's equation or Wright's equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes te…
We give bounds for the global attractor of the delay differential equation $x'(t) =-μx(t)+f(x(t-τ))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $μ$ 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 …
Existence of Invariant Probability Measures for Stochastic Differential Equations with Finite Time Delay
本文通过Krylov-Bogoliubov方法研究随机微分方程中有限时间延迟不变概率测度的存在条件，并证明确定性系数满足单侧界时，概率有界的解段也概率有界，从而推得存在不变测度的条件。
We consider equation $u_t(t,x) = Δu(t,x)- u(t,x) + g(u(t-h,x)) (*) $, when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=κ>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(κ)) = [c_*,c^*]$ such that $(*)$ has at least one (possibly, nonmonotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here $c_*>0$ is finite and $c^* \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound $c_*$ is sharp so that there are not wavefro…
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