# Structured population models as abstract delay equations by odl20037

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```									    Structured population models as abstract
delay equations
Mats Gyllenberg
Department of Mathematics and Statistics
00014 University of Helsinki
Finland

Age-structured populations have since the work of McKendrick (1926)
and general physiologicallly structured populations since the 1960s (Tsuchiya
et al. 1966; Fredrickson et al. 1967; Bell & Anderson 1967; Sinko & Streifer
1967) beeen modelled by hyperbolic partial diﬀerential equations supple-
mented by a boundary condition describing the birth process. The PDE ap-
proach suﬀers from certain drawbacks: In more complicated nonlinear models
even proving existence and uniqueness may be a formidable task and apart
from age-structured models no satisfactory qualitative theory including such
basic topics as the principle of linearized (in)stability and Hopf bifurcation
exists.
In this talk I will show that general nonlinear structured population mod-
els can be formulated in a way which is very similar in spirit to the the work
on age-structured models by Euler (1760). The model takes the form of an
initial value problem
x(t) = F (xt ), t > 0                      (DE)
x0 (θ) = ϕ(θ),      θ ∈ (−∞, 0],               (IC)
consisting of a delay equation (DE) specifying the rule for extending the
unknown function x from the history given by (IC). Here xt (θ) = x(t + θ).
The unknown function x is a vector
b
x=             ,
I

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where b is the population birth rate vector and I is the environmental con-
dition. If there are only ﬁnitely many states at birth, then b is ﬁnite di-
mensional. In many applications I is inﬁnite-dimensional. I further show
that the initial value problem (DE) & (IC) can be formulated as an abstract
integral equation
t
−1            ⊙∗
u(t) = T0 (t)ϕ + j                T0 (t − s)G(u(s))ds .            (AIE),
0

on the space X = L1 (R− ; Y ) of Bochner integrable Y -valued functions. In
(AIE), T0 is the semigroup deﬁned by translation and extension by zero:

ϕ(t + θ), −∞ < θ ≤ −t,
(T0 (t)ϕ) (θ) =                                       ϕ ∈ X,     t ≥ 0.      (1)
0,      −t < θ ≤ 0,
⊙∗
T0 is also translation and extension by zero but on a bigger space and j is
an embedding of X = L1 (R− ; Y ) into this bigger space.
In the talk I shall present an abstract framework based on adjoint semi-
groups for extending T0 to the bigger space and for deﬁning G when F
is given. I shall also indicate how one proves the principle of linearized
(in)stability and the Hopf bifurcation theorem for (AIE). I illustrate the ab-
stract results by biologically relevant examples.
This talk is based on joint work with Odo Diekmann and Philipp Getto
(Diekmann & Gyllenberg, in the press; Diekmann et al., in the press).

References
Bell, G.I., Anderson, E.C. (1967). Cell growth and division I. A mathe-
matical model with applications to cell volume distributions in mammalian
suspension cultures, Biophysical Journal 7 329–351.
Diekmann, O. and Gyllenberg, M. (2006). Abstract delay equations inspired
by population dynamics, in the press.
Diekmann, O., Getto, Ph., and Gyllenberg, M. (2006). Stability and bifur-
cation analysis of Volterra functional equations in the light of suns and stars,
SIAM Journal on Mathematical Analysis, in the press.
e e                     e
Euler, L. (1760). Recherches g´n´rales sur la mortalit´ et la multiplication du
genre humain, M´moires de l’Acad´mie Royale des Sciences et Belles Lettres
e                 e
XVI 144–164.

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Fredrickson, A.G., Ramkrishna, D., and Tsuchiya, H.M. (1967). Statistics
and dynamics of procaryotic cell populations, Mathematical Biosciences 1
327–374.
McKendrick, A.G. (1926). Applications of Mathematics to Medical Prob-
lems, Proc. Edinb. Math. Soc. 44 98–130.
Sinko, J.W. and Streifer, W. (1967). A new model for age-size structure of
a population, Ecology 48 910–918.
Tsuchiya, H.M., Fredickson, A.G, and Aris, P. (1966). Dynamics of microbial
cell populations, Advan. Chem. Eng. 6 125–198.

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