MPAs (Marine Protected Areas) are regions where fishing is
restricted or prohibited. As a fisheries management tool, their purpose is
to protect overharvested species and their habitats.
Assume that the fish population consists of a predator and a prey. One
system of nonlinear ODEs describes the predator-prey interaction in the MPA
where fishing is prohibited. A similar but different system describes the
interaction of predator and prey in the Fishing Grounds where fishing is
allowed. Now assume that both predator and prey can move between MPA and
Fishing Ground, in a way that mimics Fick's albeit in an ODE context. How
does the resulting coupled model, consisting of 4 nonlinear ODEs behave?
In a recent Biomathematics Seminar I discussed some features of such a
Then Jan Medlock (OSU, Department of Biomedical Sciences) asked me what
would happen if the MPA dynamics and Fishing Ground dynamics are described
by the more classical (neutrally stable, but not structurally stable)
Lotka-Volterra predator-prey dynamics. In this seminar I will report on some
progress regarding Jan's question. I will show that the coupling tends to
stabilize the system. In the special case that the prey is immobilized, but
the predator can move, I will outline a proof showing global stability based
on the construction of a Lyapunov function, in conjunction with Lasalle's
No background on MPAs, Lyapunov functions or Lasalle's invariance principle
is required for this talk. The talk should be accessible to undergraduates
who had some exposure to differential equations.