  In mathematical modeling, several different functional forms can often be
used to fit a data set equally well, especially if the data is sparse. In such
cases, these mathematically different but similar looking functional forms are
typically considered interchangeable. Recent work, however, shows that similar
functional responses may nonetheless result in significantly different
bifurcation points for the Rosenzweig-MacArthur predator-prey system. Since the
bifurcation behaviours include destabilising oscillations, predicting the
occurrence of such behaviours is clearly important. Ecologically, different
bifurcation behaviours mean that different predictions may be obtained from the
models. These predictions can range from stable coexistence to the extinction
of both species, so obtaining more accurate predictions is also clearly
important for conservationists. Mathematically, this difference in bifurcation
structure given similar functional responses is called structural sensitivity.
We extend the existing work to find that the Leslie-Gower-May predator-prey
system is also structurally sensitive to the functional response. Using the
Rosenzweig-MacArthur and Leslie-Gower-May models, we then aim to determine if
there is some way to obtain a functional description of data such that our
model is not structurally sensitive. We first add stochasticity to the
functional responses and find that better similarity of the resulting
bifurcation structures is achieved. Then, we analyze the functional responses
using two different methods to determine which part of each function
contributes most to the observed bifurcation behaviour. We find that prey
densities around the coexistence steady state are most important in defining
the functional response. Lastly, we propose a procedure for ecologists and
mathematical modelers to increase the accuracy of model predictions in
predator-prey systems.
