Math Seminars.
F.Berezovsky
(Howard University)
Ratio-dependent predator-prey and immunological models
Principal results are contained in the paper J.of Math. Biology 43(3).
We present a complete parametric analysis of stability properties and
dynamic regimes of an ODE model in which the functional response is a function
of the ratio of prey and predator abundance. We show the existence of eight
qualitatively different types of system behaviors realized for various
parameter values. In particular, there exist areas of coexistence (which may
be steady or oscillating), areas in which both populations become extinct,
and areas of "conditional coexistence" depending on the initial values.
One of the main mathematical features of ratio-dependent models,
distinguishing this class from other predator-prey models, is that the Origin
is a complicated equilibrium point, whose characteristics crucially determine
the main properties of the model. This is the first demonstration of this
phenomenon in an ecological model. The biological relevance of the
mathematical results is discussed both regarding conservation issues (for
which coexistence is desired) and biological control (for which extinction is
desired).