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We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{n+1}=T_{\alpha }(x_{n-1}, x_{n})=\tau (\alpha \cdot x_{n}+(1-\alpha)\cdot x_{n-1}), $ where $\tau$ is a one-dimensional map on $I=[0, 1]$ and $0 < \alpha < 1$ determines how much memory is being used. $T_{\alpha }$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $\tau $ be the symmetric tent map. We shall prove that for $0 < \alpha < 0.46, $ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $\alpha $ approaches $0.5 $ from below, that is, as we approach a balance between the memory state $x_{n-1}$ and the present state $x_{n}$, the support of the acims become thinner until at $\alpha =0.5$, all points have period 3 or eventually possess period 3. For $% 0.5 < \alpha < 0.75$, we have a global attractor: for all starting points in $U$ except $(0, 0)$, the orbits are attracted to the fixed point $(2/3, 2/3).$ At $%\alpha=0.75, $ we have slightly more complicated periodic behavior.

In the present work, for the first time, we employ Ulam's method to estimate and to predict the existence of the probability density functions of single species populations with chaotic dynamics. In particular, given a chaotic map, we show that Ulam's method generates a sequence of density functions in *L*^{1}-space that may converge weakly to a function in *L*^{1}-space. In such a case we show that the limiting function generates an absolutely continuous (w.r.t. the Lebesgue measure) invariant measure (w.r.t. the given chaotic map) and therefore the limiting function is the probability density function of the chaotic map. This fact can be used to determine the existence and estimate the probability density functions of chaotic biological systems.

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