Dpdt cp1 p mp c mp cp2

for which p = 0 is always a steady state. There is another steady state when p = (c — m)/c, which makes biological sense only if c > m. It is easy to see that p = 0 is an unstable steady state and that p = (c — m)/c is stable. We thus conclude that as c declines towards m, the two steady states collide and the unstable steady state at p = 0 becomes stable. The bifurcation picture in this case is simple and I suggest that you try to draw it. A more complicated version of the Levins model involves the "rescue effect,'' in which patches can go extinct in some time interval and be colonized in that same time interval. The equation for the dynamics of patches then becomes dp/dt = cp(1 — p — [mp/(1 + Ap)] where A measures the size of the rescue. By simultaneously sketching y = cp(1 — p and y = mp/(1 + Ap) you should convince yourself that there may be one, two, or three steady states of this system, depending upon the parameter values. There are, of course, plenty more complicated versions of the Levins model with various applications (Lin 2003). One of my personal favorite examples of situations corresponding to the cusp catastrophe involves work on models of the tuna-dolphin purse seine fishery that Colin Clark and I did when I was a graduate student, working for Colin as a research assistant (Clark and Mangel 1979). In this paper, we develop models for the tuna purse seine fishery and ask the question "What information does fishery related data give us about the status of the stock?''. Incases where a cusp bifurcation may occur, the answer can be ''very little.'' The recently published book of Bazykin (1998) makes his important work generally available (for many years, the only English versions were preprints of translated papers). This book contains a fully complete description of phase planes for predator-prey, competitive, and mutualistic systems, and a good amount of work on three species systems. It is well worth looking at. Scheffer eta/. (2001) and Scheffer and Carpenter (2003) discuss the role of the cusp catastrophe in ecosystem dynamics.

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