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DCDS

In this paper we describe some dynamical properties of a Morse decomposition
with a countable number of sets. In particular, we are able to prove that the gradient dynamics on
Morse sets together with a separation assumption is equivalent to the
existence of an ordered Lyapunov function associated to the Morse sets and
also to the existence of a Morse decomposition -that is, the global attractor
can be described as an increasing family of local attractors and their
associated repellers.

DCDS

In this work we study the asymptotic behaviour of the following prey-predator system
\begin{equation*}
\left\{
\begin{split}
&A'=\alpha f(t)A-\beta g(t)A^2-\gamma AP\\
&P'=\delta h(t)P-\lambda m(t)P^2+\mu AP,
\end{split}
\right.
\end{equation*}
where functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are not necessarily bounded above. We also prove the existence of the pullback attractor and the permanence of solutions for any positive initial data and initial time, making a previous study of a logistic equation with unbounded terms, where one of them can be negative for a bounded interval of time. The analysis of a non-autonomous logistic equation with unbounded coefficients is also needed to ensure the permanence of the model.

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