  Several applications of molecular communications (MC) feature an alarm-prompt
behavior for which the prevalent Shannon capacity may not be the appropriate
performance metric. The identification capacity as an alternative measure for
such systems has been motivated and established in the literature. In this
paper, we study deterministic identification (DI) for the discrete-time
\emph{Poisson} channel (DTPC) with inter-symbol interference (ISI) where the
transmitter is restricted to an average and a peak molecule release rate
constraint. Such a channel serves as a model for diffusive MC systems featuring
long channel impulse responses and employing molecule counting receivers. We
derive lower and upper bounds on the DI capacity of the DTPC with ISI when the
number of ISI channel taps $K$ may grow with the codeword length $n$ (e.g., due
to increasing symbol rate). As a key finding, we establish that for
deterministic encoding, the codebook size scales as $2^{(n\log n)R}$ assuming
that the number of ISI channel taps scales as $K = 2^{\kappa \log n}$, where
$R$ is the coding rate and $\kappa$ is the ISI rate. Moreover, we show that
optimizing $\kappa$ leads to an effective identification rate [bits/s] that
scales linearly with $n$, which is in contrast to the typical transmission rate
[bits/s] that is independent of $n$.
