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Notes on simulating Lotka-Volterra Jeff Achter

Summary: Notes on simulating Lotka-Volterra
Jeff Achter
March 24, 2004
1 Reminder on Poisson processes
1.1 Single events
Suppose that an event A happens at discrete times; that the probability an event happens in a time
interval [t, t + t] is independent of t; and that the chance a single event happens during a time
interval [t, t + t] is t + o(t). Then the time between events is drawn from the probability
distribution with PDF e-t.
(Briefly: Let f (t) be the chance that nothing happens on [0, t]. Then
f (t + t) = f (t)f (t)
= f (t)(1 - t + o(t))
f (t + t) - f (t)
= -t + f (t) o(t)
f (t) = - f (t)
Thus, f (t) = Ce-t for some t. Since the probability of no event happening on [0, 0] is 1, that
constant must be 1.)
Taking expected values reveals that the expected interevent time is 1/, and that the expected


Source: Achter, Jeff - Department of Mathematics, Colorado State University


Collections: Environmental Sciences and Ecology; Mathematics