The primary focus of this book is the mathematical theory of persistence.
The theory is designed to provide an answer to such questions as which
species, in a mathematical model of interacting species, will survive over
the long term. In a mathematical model of an epidemic, will the disease
drive a host population to extinction or will the host persist? Can a disease
remain endemic in a population? Persistence theory can give a mathe-
matically rigorous answer to the question of persistence by establishing an
initial-condition-independent positive lower bound for the long-term value
of a component of a dynamical system such as population size or disease
Mathematically speaking, in its simplest formulation for systems of or-
dinary or delay differential equations, and for a suitably prescribed subset
I of components of the system, persistence ensures the existence of 0
such that lim inft→∞ xi(t) , i I provided xi(0) 0, i I. We say
that these components persist uniformly strongly, or, more precisely, that
the system is uniformly strongly ρ-persistent for the persistence function
ρ(x) = mini∈I xi. This persistence function ρ(x) may be viewed as the dis-
tance of state x to a portion of the boundary of the state-space R+,
the states where one or more of species i I are extinct.
The adjective “strong” is often omitted; uniform weak ρ-persistence is
defined similarly but with limit superior in place of limit inferior. The
adjective “uniform” emphasizes that the lower bound is independent of
initial data satisfying the restriction xi(0) 0, i I. Similarly, as in the
definition of Lyapunov stability, the precise value of is unspecified and
usually difficult to estimate. Uniform persistence is a qualitative notion,
not a quantitative one. However, in rare cases, can be related to system
Previous Page Next Page