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### Open Access Journals

DCDS-B

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.

MBE

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.

DCDS

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.

DCDS

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.

DCDS

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.

DCDS-B

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.

PROC

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.

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