Steady states

The symbol I denotes a constant input defined on [0, to). (Slightly abusing notation we do not distinguish between the function and the value it takes.) A steady state is a measure m on i such that

where

Since T(t) := Tp(t)j is a semigroup of positive linear operators and m has to be positive, (29) amounts to the condition that the spectral radius is an eigenvalue and is equal to one. (For future reference we observe that, whenever there is a spectral gap,

T(t)m ^ cm as t ^tt exponentially in the weak*-sense, for any positive initial measure m. Here c = c(m) is a positive real number.)

The defining relations (29)-(30) are not suitable for "finding" steady states. For that purpose, the generation perspective is much more suitable. In particular one can concentrate on newborn individuals and the offspring they are expected to produce, with due attention to the state-at-birth of the offspring.

In the simple case of one possible state-at-birth, a first steady state condition is that the basic reproduction ratio, the expected number of offspring, equals one:

This is a condition on I. If dim I = 1 this is one equation in one unknown. Very often Ro is monotone in I which then immediately yields uniqueness. More generally we should, in the notation of (26), have

with b a positive measure on the set of possible birth states. Written out in detail (32) reads f Ap{^)i(x,w)b(dx) = b(w) (33)

J nb for all measurable subsets w of . And if is a nice subset of Rk for some k and b has a density f we may rewrite this as f eLP^)~I(x)f = f (0 , e € a • (34)

Equation (32) is a linear eigenvalue problem: the dominant eigenvalue of a positive operator should be one. This is, just as (31) but now more implicitly, a condition on the parameter I. If this condition is satisfied and the eigenvalue is algebraically simple (a sufficient condition being the irreducibil-ity of the positive operator) then the eigenvector b is determined uniquely modulo a positive multiplicative constant, to be denoted by c below.

Returning to the case of a fixed state-at-birth, we note that (10)-(12) simplify considerably when the input is constant. For given I we define x and

and next we note that

Let c denote the steady p-birth rate. Then uP(a)I (xb,w)da = cj F (a)&x(a)(w)da (38)

and consequently (30) can be written as f ^

Beware that F and x depend on I.

Theorem 2.6.1 m is a steady state, i. e., (29)-(30) hold, iff m is given by (38), with x and F defined by (35)-(36), where I and c are such that (31) (with R0 (I) given by (37)) and (39) hold.

For the proof see Part III. Note that (31) and (39) are 1 +dimI equations in as many unknowns, viz., c and I. Also note that (37) is defined completely in terms of solutions of ODE, since we may supplement (35)-(36) with

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