The temporal development of a natural or artificial system can conveniently
be modeled by a semiflow. A semiflow consists of a state space, X, a time-
set, J, and a map, Φ.
The state space X comprehends all possible states of the system: the
amounts or densities of the system parts and, if there are one or several
system structures, their structural distributions.
According to the interests of the authors, this book concentrates on
biological, ecological, and epidemiological systems. For the last, for example,
the state space typically contains the amounts or densities of susceptible and
infective and possibly exposed and removed individuals. For spatial spread,
spatial distributions are included in the state space. If age-structure is
thought to be important, age-distributions are included as well.
Time can be considered as a continuum or in discrete units; the most
common choices for the time set J are the nonnegative reals or the nonneg-
ative integers, R+
= [0, ∞) and Z+ = N {0} = {0, 1,...}. Depending on
the model, the time unit can be a year, month, or day.
The most important ingredient of a semiflow is the semiflow map Φ :
J × X X. Often Φ itself is called the semiflow. If x X is the initial
state of the system (at time 0), then Φ(t, x) is the state at time t. This
interpretation immediately leads to the identity
Φ(0,x) = x, x X.
Further, semiflows are characterized by the semiflow property:
Φ(t + r, x) = Φ(t, Φ(r, x)), r, t J, x X.
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