Local stability implies global stability in some one-dimensional discrete single-species models
Eduardo Liz
Discrete & Continuous Dynamical Systems - B 2007, 7(1): 191-199 doi: 10.3934/dcdsb.2007.7.191
We prove a criterion for the global stability of the positive equilibrium in discrete-time single-species population models of the form $x_{n+1}=x_nF(x_n)$. This allows us to demonstrate analytically (and easily) the conjecture that local stability implies global stability in some well-known models, including the Ricker difference equation and a combination of the models by Hassel and Maynard Smith. Our approach combines the use of linear fractional functions (Möbius transformations) and the Schwarzian derivative.
keywords: discrete-time single-species model Ricker difference equation Schwarzian derivative. global stability
Delayed population models with Allee effects and exploitation
Eduardo Liz Alfonso Ruiz-Herrera
Mathematical Biosciences & Engineering 2015, 12(1): 83-97 doi: 10.3934/mbe.2015.12.83
Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.
keywords: bifurcation. bistability difference equation global stability delay differential equation Allee effect Population model
On the global attractor of delay differential equations with unimodal feedback
Eduardo Liz Gergely Röst
Discrete & Continuous Dynamical Systems - A 2009, 24(4): 1215-1224 doi: 10.3934/dcds.2009.24.1215
We give bounds for the global attractor of the delay differential equation $ \dot 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 delay is sufficiently small, then all solutions 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.
keywords: unimodal feedback delay differential equation global attractor Schwarzian derivative. Nicholson's blowflies equation Mackey-Glass equation
Wright type delay differential equations with negative Schwarzian
Eduardo Liz Manuel Pinto Gonzalo Robledo Sergei Trofimchuk Victor Tkachenko
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 309-321 doi: 10.3934/dcds.2003.9.309
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.
keywords: Schwarz derivative global stability delay differential equations. Wright conjecture 3/2 stability condition
On a generalized Yorke condition for scalar delayed population models
Teresa Faria Eduardo Liz José J. Oliveira Sergei Trofimchuk
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 481-500 doi: 10.3934/dcds.2005.12.481
For a scalar delayed differential equation $\dot x(t)=f(t,x_t)$, we give sufficient conditions for the global attractivity of its zero solution. Some technical assumptions are imposed to insure boundedness of solutions and attractivity of non-oscillatory solutions. For controlling the behaviour of oscillatory solutions, we require a very general condition of Yorke type, together with a 3/2-condition. The results are particularly interesting when applied to scalar differential equations with delays which have served as models in populations dynamics, and can be written in the general form $\dot x(t)=(1+x(t))F(t,x_t)$. Applications to several models are presented, improving known results in the literature.
keywords: Yorke condition Delayed population model global attractivity 3/2- condition.
A new flexible discrete-time model for stable populations
Eduardo Liz
Discrete & Continuous Dynamical Systems - B 2018, 23(6): 2487-2498 doi: 10.3934/dcdsb.2018066

We propose a new discrete dynamical system which provides a flexible model to fit population data. For different values of the three involved parameters, it can represent both globally persistent populations (compensatory or overcompensatory), and populations with Allee effects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how population abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.

keywords: Discrete population model global stability density dependence Allee effects bifurcations of fixed points extinction
Yorke and Wright 3/2-stability theorems from a unified point of view
Eduardo Liz Victor Tkachenko Sergei Trofimchuk
Conference Publications 2003, 2003(Special): 580-589 doi: 10.3934/proc.2003.2003.580
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.
keywords: delay differential equations. 3\2 stability condition global stability

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