Preface

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

prevalence.

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+,

n

namely

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

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