  The Trojan Y Chromosome Strategy (TYC) is an extremely well investigated
biological control method for controlling invasive populations with an XX-XY
sex determinism. In \cite{GP12, WP14} various dynamical properties of the
system are analyzed, including well posedness, boundedness of solutions, and
conditions for extinction or recovery. These results are derived under the
assumption of positive solutions. In the current manuscript, we show that if
the introduction rate of trojan fish is zero, under certain large data
assumptions, negative solutions are possible for the male population, which in
turn can lead to finite time blow-up in the female and male populations. A
comparable result is established for \emph{any} positive initial condition if
the introduction rate of trojan fish is large enough. Similar finite time
blow-up results are obtained in a spatial temporal TYC model that includes
diffusion. Lastly, we investigate improvements to the TYC modeling construct
that may dampen the mechanisms to the blow-up phenomenon or remove the
negativity of solutions. The results draw into suspect the reliability of
current TYC models under certain situations.
