Physiologically Structured Population Dynamics:
A Modeling Perspective
B. W. Kooi
F. D. L. Kelpin
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Comments on Theoretical Biology, 8: 1–44, 2003
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DOI:
Physiologically Structured Population Dynamics:
A Modeling Perspective
B. W. Kooi
F. D. L. Kelpin
Department of Theoretical Biology, Institute of Ecological
Science, Faculty of Earth and Life Sciences,
Vrije Universiteit, Amsterdam,
The Netherlands
5
In classical population dynamic models all the individuals in a population
are assumed to be identical or only population averages are considered.
The state of the population is therefore the number of individuals or the
total amount of biomass in the population. The mathematical model is a
first-order ordinary differential equation (ODE) that specifies the time
derivative of this population state. Subsequently, the dynamics of an ecosystem where populations interact with each other and with their abiotic
environment is usually described by a system of ODEs. The classical
models are sometimes called unstructured population models, as opposed
to structured population models, which take into account the differences
between individuals and characterize an individual by some state. The
development of this state is modeled from birth till death, often using a
first-order ODE that specifies the time derivative of the individual state.
The model is complemented with models for the birth and death of individuals. The structured population model is derived straight forwardly from
the individual model using balance laws. The state of a population is no
longer a single number; it is the distribution of the individuals over
their possible states. If the number of individuals is large, this distribution
is continuous instead of discrete. If, in addition, the distribution is a density with respect to the individual state, the population model can be written as a partial differential equation (PDE). The first structured
Address correspondence to B. W. Kooi, Department of Theoretical Biology, Institute of
Ecological Science, Faculty of Earth and Life Sciences, Vrije Universiteit, De Boelelnaan 1087,
1081 HV Amsterdam, The Netherlands. E-mail: kooi@bio.vu.nl
We thank Stijntje Kelpin-Vlaskamp for improving the English.
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B. W. Kooi and F. D. L. Kelpin
population models use age for the state of the individuals. For many
organisms, however, a difference in age alone does not explain the differences in individual behavior and other state variables such as size or
energy reserves are more suitable. This has led to physiologically structured population models (PSPs). The life cycle of the individuals may consist of a number of life stages, such as egg, juvenile, and adult. The
distribution of the number of individuals with respect to the state variable
may be irregular. The introduction of these two extra components, namely,
several life stages and nonsmooth distributions, gave mathematical difficulties in the PDE formulation. Recently, an alternative cumulative formulation in terms of renewal integral equations has been proposed that deals
with these difficulties. In this review we describe the various model formulations from a modeling perspective. The dynamics of a population of
worms that propagate by division into two unequal daughters serves as
an example. We give an overview of numerical methods for structured
population models.
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Keywords: age-structured, discrete-time model, continuous-time model, physiologically
structured population model, size-structured
A population is the collection of individuals belonging to one species living
in a spatial region which is either closed or has known immigration and emigration fluxes of individuals across its borders. A description of the dynamics
of a population is therefore essentially a description of the dynamics of all of
its individuals. There is always individual variability, which is usually incorporated in the model as a random variable.
For instance, individuals may be affected differently by environmental stochasticity or mutations, and by random event behavioral aspects, such as
motion in space in search of mates or food. As a consequence, also
intra- and interspecific encounters, which are important in the predator–
prey interplay, are stochastic. If the number of individuals is limited, the
individual-based modeling (IBM) (DeAngelis and Gross 1992) approach can
be used, where each individual is followed in time. Birth, growth, death but
also kinds of behavior, such as defence and mating, can be incorporated in a
straight forward manner. Upon birth, the newborn individual is assigned
parameter values that are random perturbations of those of its parent(s).
These parameters describe the development of the individual. Due to the
stochastic effects, each run will give different outcomes. Therefore, the
results of one simulation run are of limited interest but several simulations
can be done. The final results then are statistics of these runs, for instance,
mean values of population statistics, or the extinction probability.
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Physiologically Structured Population Dynamics
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The prospects of applying the IBM approach in ecology is in the possibility of applying numerical simulations. In general, no mathematical tools are
available to obtain analytical solutions nor to do mathematical analysis.
This type of population models can, however, sometimes be analyzed
mathematically with the theory of branching processes (Jagers 1975; Arino
and Kimmel 1989; Jagers 1991; Arino and Kimmel 1993). This theory was
applied to the analysis of models for cell proliferation by fission. Jagers
(1991) describes how the classical Galton–Watson branching process,
where the population evolves from generation to generation by individuals
getting an independent and identically distributed (IID) number of children,
yields a dichotomy between extinction or unlimited growth.
This dichotomy results from the fact that the population grows, on average, geometrically with a net reproductive rate equal to the mean number
of offspring. If the expected number of offspring is one or less than one,
the population dies out. If it is larger than one, the population grows
beyond all bounds. This means that the population cannot stabilize. If we
want stabilization to occur, a description of the reproduction of individuals
does not suffice and the interaction with their environment must be included
in the model.
More elaborate models with different types of individuals can be modeled
using multipoint processes. The dynamics of these populations can be studies
by individual-based simulations. However, such a simulation is not necessary
since the properties of the population can be analyzed by straightforward generalizations of the analysis for the single-type process (see Caswell and John
1992) where a semelparous organism (they die after first reproduction) with
fixed generation length is considered.
The Galton–Watson branching process models only the branching of the
family trees in the population. No information is specified about the times
at which the branches are formed. However, the age of the individuals and
the ages at child bearing, can be included. Then a mother’s age as well as
her type (i-state) determine the starting platform from which the newborn’s
life is to evolve. This form of dependence between two next generations
leads to a Markov model for the population growth. A superposition of the
resulting point processes, essentially Markov renewal theory, provides information such as the average population growth rate (see Jagers, 1991).
The life cycle of some species can be divided into a number of stages, thus
splitting the population into classes. If the number of individuals in each class
can be written as a linear combination of the class sizes some time step ago,
the vector containing all class sizes can be multiplied by a so-called population matrix to yield the next population state. The elements in this matrix are
transition probabilities from one class into the other, so the matrix is nonnegative. In the special case where the classes are age classes the matrix is
called the Leslie matrix and the elements are fecundities on the first row
and survivabilities on the subdiagonal. If the matrix is constant, the model
is linear and the Perron–Frobenius theorem provides conditions for the
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B. W. Kooi and F. D. L. Kelpin
convergence to a population state that is the eigenvector belonging to the
dominant eigenvalue.
Once it has reached this state, the population grows geometrically. Each
time step, the number of individuals in each class is multiplied with the dominant eigenvalue. This yields a trichotomy of the following three possibilities.
If the dominant eigenvalue is less than one, the population goes extinct. If it is
larger than one, the population grows without bounds. If it equals one, the
population stabilizes.
The entries in the matrix can depend on the population size or explicitly on
time. Then the matrix equations are nonlinear with possibly positive bounded
asymptotic behavior. The mathematical analysis of these matrix models is
worked out in Caswell (1989; 1997), Tuljapurkar (1997), and Cushing
(1997).
Stage-structured population models emphasize the stage structure of the
populations. The development of the population is described by a number
of life-cycle stages, such as eggs, larvae, pupae, and adults. The rate of
change of the stage size must equal the difference between total recruitment
and loss rates for the stage. Each term in these equations is related to stagerate processes such as recruitment, death, and maturation, for each stage. The
parameters in these equations are stage-dependent mean value rates, for
instance, death and birth rates. These balance equations form the dynamic
equations for each stage where the number of individuals in the different
stages are the population state variables, called p-state variables. The resulting
population dynamic model is generally described by delay differential equations (DDEs). This type of model is worked out in Nisbet (1997). Thieme
(2003) discusses a stage-structured population model where the definition
of a density-dependent transition rate from one stage into the other enables
a formulation in terms of ODEs, which are easier to handle than DDEs.
In the more general physiologically structured population models, each
individual is described by its so-called i-state, which contains a number of
physiological traits such as age, energy, or size. If the number of individuals
is sufficiently large, a deterministic model can be used (see Metz and de Roos
1992). The individual model is described by individual rates, such as birth,
growth, and death rates that describe the change of the momentary i-states
of the individual. The deterministic ODE model for the individual together
with balance laws directly leads to a PDE model formulation at the population level.
In special cases the PDE model can be reduced to an equivalent DDE or an
ODE for a p-state variable such as the total number or biomass of all individuals that make up the population. The later case presents unstructured
models that average over the i-state distribution in the population without
loss of information. It is a special case of a technique referred to as the
linear chain trick. Then the model is reduced to a number of coupled
ODEs being a closed, finite-dimensional system for moments of the biomass
weighted with powers of size (see Metz and Diekmann 1989).
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Physiologically Structured Population Dynamics
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The interplay with the environment is of paramount importance for the
techniques that can be used to analyze the resulting model. If the environment
is assumed to be constant, the model is linear and mathematical techniques
are available to prove existence, uniqueness, and positivity of a solution.
As we mentioned earlier, linear models show a trichotomy between extinction, stabilization, and unlimited growth of the population. Stabilization
only occurs for specific parameter values. If the population has a time-dependent
food source or suffers from a nonconstant predation pressure, the food or the
predator has to be modeled explicitly as well. The resulting nonlinear model
is autonomous. If the environment changes explicitly with time (seasonal or
daily fluctuations) the resulting model is nonautonomous. Generally, in these
last two cases the model is nonlinear and consequently more difficult to analyze. Often a middle course is adopted in which the vital rates, such as birth
rate, growth rate, and death rate, are density dependent; that is, they depend
on the population size (number of individuals or total mass), thus implicitly
modeling the effect of the environment on the individual life history.
These models are often called semilinear. In nonlinear and semilinear
models, biologically realistic positive bounded solutions can occur for large
parameter ranges.
Age has often been chosen to describe the state of an individual; in addition, the birth and death rate depend solely on age. For continuous-time problems without any interplay with the environment the resulting model is a
linear first-order PDE for the density function. Webb (1985) analyzes nonlinear versions of this PDE model. Properties such as the existence
and uniqueness of the solution are studied and, with respect to long-term
dynamic behavior, existence of a so-called equilibrium or stable age
distribution is important.
For many biological systems, age is not appropriate to characterize the
state of an individual. Other i-state variables such as size, energy, or mass
have been proposed leading to a class of models called physiologically
structured population models (PSPs), which are formulated and analyzed
in Metz and Diekmann (1986) and de Roos (1997). If the environment is
constant it is, however, always possible to use age as the individual
state variable.
Traditionally, the PDE was used. Thieme (1988) works out an example
that shows that PDE model formulations can be ill-posed. Diekmann et al.
(1998; 2001) propose a cumulative approach where a description of the life
cycle of each individual is the starting point for the derivation of the population model formulation. Here ingredients from the theory of multitype
branching processes are used. Mathematically this leads to renewal integral
equations (RIEs). In the 2001 paper, the nonlinear case is discussed, where
interplay with a time-dependent environment is allowed.
There is considerable literature on the dynamics of a population consisting
of individuals that propagate by binary fissions. Division can be seen as a
birth and death at the same time. The mother dies but two or more daughters
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B. W. Kooi and F. D. L. Kelpin
are born at the same time. Division can occur at different i-states. This situation corresponds to a stochastic life length of the cell. Bell and Anderson
(1967), Sinko and Streifer (1971), Metz and Diekmann (1986) assume that
the number of individuals is large and that there is a population density
and analyze the deterministic PDE formulation. Diekmann et al. (1993)
assume that the population size is large, but allow for measures and analyze
a cell cycle model using the theory based on renewal equations describing the
cumulative effect of divisions.
Models for the cell cycle of the majority types of cells including bacteria,
unicellular eukaryotic organisms, and mammalian cells have been developed
and described in Lasota and Mackey (1984) and in Tyson and Hannsgen
(1985; 1986). They propose and analyze a model where cells divide into
two equal daughter cells. The random division time of the daughters is correlated to that of the mother. In Metz and Diekmann (1986) and Diekmann
et al. (1993), the division time of the daughters is also random, but with
the same distribution for all cells. In both formulations there is no stable
age-distribution when the cells grow exponentially.
Lasota and Mackey (1984) discuss cell cycle models in terms of Markov
operators. These operators connect the density function of the age or mass
distribution at birth over generations. The age-dependent cell generation
time case is worked out in Tyrcha (2001). It is assumed that the mass
dynamics obeys a first-order autonomic ODE. The generation time is the
sum of a random variable plus a constant time after the initiation of the
DNA synthesis that triggers the cell to transverse through subsequent
phases of the cell cycle that have approximately a constant length. This
approach gives alternative proof of the fact that there is no stable mass distribution of the cells that grow exponentially.
Many sciences study the link between what is happening at the singleparticle scale and what can be observed among large populations of particles.
We mention population balance models (PBMs) which have been used in
chemical engineering since the 1960s (see Fredrickson, 1991; Ramkrishna
2000; Fredrickson and Mantzaris 2002; and references therein).
In this review we restrict our attention to ecology, in particular to one population that lives in a spatially homogeneous environment. Only physiological
traits, such as age, size, and energy reserves, serve as individual state variables.
We discuss various model formulations, starting with age-structured
models and proceeding to PSP models. The interplay between the choice
of the mathematical model formulation and the features of the biological
system is emphasized.
Generally, no closed-form solutions can be derived for the transient or
short-term dynamics and numerical techniques have to be used to approximate the solution. We describe the most popular numerical methods.
We elaborate one example in great detail in which the various model
formulations as well as their mathematical and numerical analysis are
elucidated. We consider the dynamics of a population of worms which
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Physiologically Structured Population Dynamics
7
propagate by division into two unequal daughters. We restrict ourselves to the
case where there is an abundant food supply. In this type of continuous-time
problems it is sometimes advantageous to look for an equivalent
discrete-time model formulation. For long-term population dynamics the
ratio of the interdivision times of the two sisters is crucial. Convergence to a
stable age or size distribution depends on whether this ratio is a rational or
irrational number.
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STRUCTURED POPULATION MODELS
All individuals in a population share the same species-specific behavior.
This behavior generally depends not only on environmental conditions but
also on the internal state of the individual. Unstructured population models
assume that all the individuals are identical. Structured population models,
however, allow for differences between individuals. Each individual has its
own internal i-state x. This i-state may vary in time. It necessarily contains
all intrinsic information needed to define individual behavior for given
environmental conditions. The model for the individual defines the individual’s behavior and the development of its i-state as a function of its environment E and its internal i-state x.
In the previous paragraph, the precise meaning of the word ‘‘behavior’’
was deliberately not specified. Whole ranges of models can be thought up
to describe one particular population, depending on what is considered to
be its relevant behavior. These models differ in their level of detail, but
also in the mechanisms that they focus on. There is not one single correct
model for a population. The choice of model, and as a result of that also
the contents of the i-state, depend on the model’s intended use.
First, we discuss age-structured models, which have one single i-state variable, the age a of an individual. They are a subgroup of physiologically structured population models (PSP models), in which the i-state variables
represent physiological properties of the individual.
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Age-Structured Population Models
Consider the following age-structured population model. Individuals are
born with age 0 and may live up to a maximum age o. The survival function
F ðaÞ specifies the probability that an individual survives to age a. So F is a
nonincreasing function with F ð0Þ ¼ 1 and F ðoÞ ¼ 0. The maternity function bðaÞ is defined such that bðaÞ da is the number of newborns mothered
by an individual from age a to age a þ da. We look at several classical
continuous-time and discrete-time formulations of this model, where we
partly follow Kot (2001).
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Continuous-Time Renewal Integral Equation
Lotka’s model (Sharpe and Lotka 1911) is a continuous-time model that
tracks the population’s birth rate. Let bðtÞ dt be the number of births during
time interval t to t þ dt. Then Lotka’s model reads
bðtÞ ¼
Z
290
t
bðt aÞF ðaÞbðaÞ da þ GðtÞ
ð1Þ
0
where GðtÞ, the contribution of all individuals already present at t ¼ 0, is
defined by
GðtÞ ¼
Z
ot
nð0; aÞ
0
F ða þ tÞ
bða þ tÞ da
F ðaÞ
ð2Þ
Here nð0; aÞ equals the age density distribution of the initial population; that
is, nð0; aÞ da equals the number of individuals of age a to a þ da at time 0.
Notice that GðtÞ ¼ 0 for t > o, in which case the resulting integral equation,
Eq. (1), is homogeneous.
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Discrete-Time Renewal Equation
The time t and also age a are now integers. Mathematically the renewal
equation yields a difference equation of the form
bt ¼
t
X
bta F a ba þ Gt
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ð3Þ
a¼1
where
Gt ¼
ot
X
a¼1
n0;a
F aþt
b
F a aþt
ð4Þ
The functions bt and Gt have the same meaning as in the continuous-time
case except now at fixed census time intervals. An example of a renewal
equation is the Fibonacci (1202) series
bt ¼ bt1 þ bt2
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ð5Þ
where the number of newborn rabbits bt is the sum of the number of births
one (bt1 ) and two time intervals ago (bt2 ). In Eq. (3) we have F a ¼ 1
for all a (no mortality) and b1 ¼ 1, b2 ¼ 1, and ba ¼ 0 for a > 2 (one newborn at two preceding time points).
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Physiologically Structured Population Dynamics
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Discrete-Time Leslie Matrix
Here the state is described by the age distribution in discrete points in time
and age. The classical Leslie matrix reads
1 0F
n0 ðtÞ
0
P0
B n1 ðtÞ C B
B
C B
B
..
C B
B
C¼B 0
B
.
C B .
B
..
A @ .
@
.
.
no1 ðtÞ
0
0
..
P1
.
.. ..
.
.
0
0
..
.
F1
0
0
Po2
1
Fo1 0 n0 ðt 1Þ 1
0 C
CB n1 ðt 1Þ C
C
.. CB
..
C
B
. C
C
B
.
CB
C
..
C@
A
.
0 A
no1 ðt 1Þ
0
315
ð6Þ
We give the Fibonacci series as a example. Then o ¼ 2 and F0 ¼ F1 ¼
P0 ¼ 1, leading to
n0 ðtÞ
n1 ðtÞ
¼
1
1
1
0
n0 ðt 1Þ
n1 ðt 1Þ
ð7Þ
which is equivalent to the two equations
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n0 ðtÞ ¼ n0 ðt 1Þ þ n1 ðt 1Þ
ð8Þ
n1 ðtÞ ¼ n0 ðt 1Þ
ð9Þ
Substitution of n1 ðtÞ of the second equation in the first equation gives the
familiar Eq. (5), now for n0 : n0 ðtÞ ¼ n0 ðt 1Þ þ n0 ðt 2Þ. Indeed, the
term n0 ðtÞ is associated with the birth bt rate of the previous paragraph.
This is the general procedure to derive from a higher order difference equation a set of first-order difference equations, similar to the procedure in ordinary
differential equations.
Let NðtÞ denote the total number of individuals at time t. Then we have
NðtÞ ¼ n0 ðtÞ þ n1 ðtÞ ¼ n0 ðt þ 1Þ; that is, the number of individuals at time
t equals the number of individuals that will be born at t þ 1. Indeed, in
this biological interpretation both classes reproduce at the same time and
give birth to one child. This simple example shows that the number of individuals in the population is just the summation of the number of individuals
born and that survive (all survive in this example) whereby the summation is
over all possible instants individuals were born (two in this example).
Continuous-Time Partial Differential Equation
Let nðt; aÞ denote the population age-distribution function and hence
nðt; aÞ da the number of individuals of age a to a þ da at time t. The
McKendrick–von Foerster equation (McKendrick, 1926; von Foerster
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B. W. Kooi and F. D. L. Kelpin
1959) reads
340
@nðt; aÞ @nðt; aÞ
þ
¼ dðt; aÞnðt; aÞ
@t
@a
ð10Þ
where dðt; aÞ is the per capita mortality rate. The boundary condition at age
zero is
Z o
nðt; 0Þ ¼
nðt; aÞbðt; aÞ da
ð11Þ
0
where bðt; aÞ is the per capita birth rate. Observe that we should multiply the
left-hand side with da=dt but since it is dimensionless and equals one, it is
generally omitted. The initial condition completes the mathematical formulation and reads
ð12Þ
nð0; aÞ ¼ n0 ðaÞ
345
The resulting model is a first-order linear PDE and mathematical theory
exists dealing with the most important issues such as existence and asymptotic
age distribution. To that end the problem is formulated in the framework of
the theory of semigroups of linear operators (see Webb 1985).
In order to derive a link with Lotka’s model [Eq. (1)]; we assume that the
birth rate and death rate depend only on age, that is, b ¼ bðaÞ and d ¼ dðaÞ.
The survivorship F ðaÞ is linked to the death rate by dF =da ¼ dðaÞ where
F ð0Þ ¼ 1. With bðtÞ ¼ nðt; 0Þ we have for t > o, nðt; aÞ ¼ bðt aÞF ðaÞ.
Substitution into Eq. (11) yields the Lotka equation, Eq. (1). Similarly, the
equivalence can be shown for t o.
The case of asynchronous exponential growth, where the number of individuals grows or dies out exponentially but their distribution over the i-state
space stabilizes, is often observed, for instance, in models where all individuals at all time contribute to reproduction and all individuals interact
through one food resource. This can be seen as follows. Suppose there is
exponential growth, hence bðtÞ ¼ expfrtg. Substitution into Eq. 1 where
t > o gives the characteristic equation, called the Euler–Lotka equation
Z o
F ðaÞbðaÞ expfrag da ¼ 1
ð13Þ
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0
It has exactly one real root r ¼ m, the population growth rate.
In Figure 1 the solution nðt; aÞ of Eq. (10) is shown where the initial age
distribution, seen on the left above the line t ¼ 0 (the age axis), is uniform.
This unnatural initial condition leads to a discontinuity of the density
which propagates along the line t ¼ a. For t > o this discontinuity disappears
and eventually the distribution converges to the stable age distribution, seen
on the Rright. The rates bðaÞ and dðaÞ are such that the population size
o
NðtÞ ¼ 0 nðt; aÞ da blows-up as is clear from Figure 2 where the size NðtÞ
is plotted as function of the time t.
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Physiologically Structured Population Dynamics
11
FIGURE 1 Solution of the linear McKendrick–von
Foerster equation, Eqs. (10), (11),
0 0a<a
(12), with a ¼ 0:5, o ¼ 1, bðaÞ ¼
, dðaÞ ¼ 1, and n0 ðaÞ ¼ 1.
5 aa1
FIGURE 2 Total number of individuals NðtÞ for the solution of the linear
McKendrick–von Foerster equation plotted in Figure 1.
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B. W. Kooi and F. D. L. Kelpin
McKendrick (1926) turned his attention to a single population and considered the linear case, in which the birth and mortality terms were both linear
functions of the state variable: the instantaneous distribution of the population. Later on, Gurtin and MacCamy (1974) worked on the nonlinear case,
in which the birth rate bðNÞ and the mortality rate dðNÞ depend on the
dynamic states of the populations involved. Due to this density dependence
there is a stable age distribution and the total number of individual stabilizes.
This is illustrated in Figures 3 and 4 where the solutions for the age distribution nðt; aÞ and total number of individuals NðtÞ is shown where death is density dependent similar to the Lotka–Volterra model.
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Reduction to Unstructured Model Formulation
In special cases a set of ODEs or DDEs can be constructed which is
equivalent to the PDE formulation. Here we give two examples.
Reduction to an Ordinary Differential Equation. Suppose that both
the birth rate and mortality rate do not depend on age but on the total number
of individuals in the population,
FIGURE 3 Solution of the nonlinear McKendrick–von Foerster equation, Eqs. (10),
0 0a<a
(11), (12), with a ¼ 0:5, o ¼ 1, bðaÞ ¼
, dðt; aÞ ¼ N ðtÞ 1, and
5 aa1
n0 ðaÞ ¼ 1.
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Physiologically Structured Population Dynamics
13
FIGURE 4 Total number of individuals N ðtÞ for the solution of the nonlinear
McKendrick-von Foerster equation plotted in Figure 3.
NðtÞ ¼
Z
o
nðt; aÞ da
ð14Þ
0
then the PDE Eq. (10) becomes
@nðt; aÞ @nðt; aÞ
þ
¼ dðNÞnðt; aÞ
@t
@a
ð15Þ
with
nðt; 0Þ ¼
Z
o
nðt; aÞbðNÞ da ¼ bðNÞN
and nð0; aÞ ¼ n0 ðaÞ
ð16Þ
0
Integration by parts of the second term of Eq. (15), with nðt; oÞ ¼ 0 and the
boundary conditions of Eq. (16) gives
dN
¼ ½bðNÞ dðNÞN
dt
and
Nð0Þ ¼ N0
ð17Þ
Hence, if the two vital rates that are defined at the individual level, namely,
the birth and death rates, do not depend on age, the age structure is superfluous and we can simply use an ODE formulation. It is, however, not possible to recover the age distribution at all times. This is the price we have
to pay for dealing with a much simpler model formulation that gives only
the time dependence of the total number of individuals.
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B. W. Kooi and F. D. L. Kelpin
Reduction to a Delay Differential Equation. Now we assume that
there are two life stages, juvenile and adult. The individual mature at age a.
The birth rate and mortality rate functions are only positive in the age interval
ða; oÞ. Thus, we assume that the juveniles do not die. We perform a reduction
procedure similar to the one used in the previous section. There are two agedistribution functions, and their dynamics are described each by a PDE:
@nJ ðt; aÞ @nJ ðt; aÞ
þ
¼ 0;
@t
@a
@nA ðt; aÞ @nA ðt; aÞ
þ
¼ dðNA ÞnA ðt; aÞ
@t
@a
410
ð18Þ
with boundary conditions
nJ ðt; 0Þ ¼ bðNA Þ
Z
o
nðt; aÞ da
nJ ðt; aÞ ¼ nA ðt; aÞ
nA ðt; oÞ ¼ 0
ð19Þ
a
Ra
R o Let NJ ¼ 0 nJ ðt; aÞ da denote the number of juveniles and NA ¼
a nA ðt; aÞ da the number of adults. Integration of Eq. (18) using the boundary conditions of Eq. (19) gives
dNA
¼ bðNA ÞNA ðt aÞ dðNA ÞNA ðtÞ
dt
ð20Þ
with initial function NA ðtÞ ¼ NA0 for a t 0. In a similar way we obtain
a DDE for the number of juveniles NJ ðtÞ
dNJ
¼ bðNA ÞNA ðtÞ bðNA ÞNA ðt aÞ
dt
420
ð21Þ
As this involves NA ðtÞ, one first has to calculate the number of adults NA ðtÞ
and afterward the number of juveniles NJ ðtÞ. This last equation shows that the
change in the number of juveniles equals the difference in population birth
rate at t (the newborn) minus the population birth rate at t a (the individuals
that mature at t and were born a ago).
425
Physiologically Structured Population Model
In Kooijman (2000) it is argued that age is in many cases not an appropriate state variable to describe the growth and development of a population. As
an example, consumption of food depends on sizes and the waterflea
Daphnia magna typically has a length of 2.5mm at the onset of reproduction.
Size differences are also essential in models of cannibalistic populations
(Diekmann et al. 1986; van den Bosch et al. 1988; Kirkilionis et al. 2001;
Claessen et al. 2000; Claessen and Dieckmann 2002) where large adults eat
small juveniles. For bacteria there is a similar situation. The DNA duplication
is triggered upon exceeding a fixed cell size and DNA duplication lasts
430
435
Physiologically Structured Population Dynamics
15
a fixed time period (see Donachie, 1968). Hence the division of an individual
does not occur at a specific age, but size is also involved.
Furthermore, many species go through different life stages. In Kooijman
(2000) three stages in the life history of individuals are recognized: egg, juvenile, and adult. Eggs neither eat nor reproduce, juveniles eat but do not reproduce, and adults both eat and reproduce.
In the PSP model formulation, age is therefore replaced or augmented with
other state variables such as size=mass or energy content. The type of information stored in the i-state varies from model to model, but in many cases x is
an n-dimensional vector. Each number in this vector then represents a physiological trait of the individual.
Recently, models have been proposed where the i-state is a twodimensional vector. Droop (1973) introduced the concept of cell nutrient quota
(internal nutrient density) besides biomass in a model for phytoplankton
growth. In this model, growth is decoupled from nutrient uptake. Phytoplankton cells store nutrients and the growth rate of a cell depends on the
amount of stored nutrient. Hallam et al. (1990) describe a model in which an
organism consists of storage: lipids, the major source of energy, and a
structural component—nonlipid dry material consisting mostly of proteins
and carbohydrates. In Persson et al. (1998) the components are called the
reversible and irreversible mass. In reversible mass they include energy
reserves such as fat, muscle tissue, and gonads, and in irreversible mass they
include compounds like bones and organs, which cannot be starved away by
the consumer. The Dynamic Energy Budget (DEB) model (Kooijman 2000),
deals with storage materials (reserves) and structural biomass. Here the
storage materials are carbohydrates and lipids but also a part of ribosomal
RNA and of the proteins (see Hanegraaf et al. 2000). The storage materials
have a dual function as a reserve pool for energy and elementary compounds.
These storage materials are continuously used and replenished, while
structural materials are permanent and require maintenance. The reserves
adapt to the external food availability, that is, to the environment. If this food
availability is constant or varies slowly compared to the adaptation process,
the energy reserves will be in or near their equilibrium state. Individuals will
not be spread out all over the i-state space, but concentrated on or near the
line through these equilibrium values for energy reserves.
Given a PSP model, the set of all possible i-states is called the i-state space
O. If the individual goes through different life stages, O may be divided into
subsets Oi (e.g., OE ; OJ ; OA for the egg, juvenile, and adult stage). The
boundaries of these subsets mark points where the individuals move from
one life stage to the next, such as where they hatch or mature.
As we saw in the age-structured model formulation, the p-state of the
population can be a number or mass density that depends on the
i-state variables. This leads to a PDE formulation for the population with
boundary and initial conditions similar to the age-structured case described in
the previous section.
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16
B. W. Kooi and F. D. L. Kelpin
In Diekmann et al. (1998) the mathematical difficulties that may arise in
this formulation are explained in biological terms as follows. In order to keep
models parameter scarce, one wants to allow for discontinuities with respect
to i-state of rates (think of waterflies that start to reproduce upon reaching a
critical size; see Thieme 1988). Now consider a situation in which the i-state
of some individual moves in O for an extended period of time along a line of
discontinuity of the rate of offspring production, that is, the line that separates
OJ and OA . Then the model is not acceptable as a model and one should not
expect that existence and uniqueness of solutions at the p-level holds.
Whether this phenomenon actually occurs or not in a specific model, is
hidden in the rates. It is the combined global effect of the rates that makes
the difference between the model being ill- or well-posed.
In this section we first introduce the model ingredients. Then we show
how in one big step the behavior of all the individuals in a population is
lifted to the population level. We start with the classical formulation of
PSP models (Metz and Diekmann 1986), which uses densities and leads to
a partial differential equation with boundary and initial conditions. Then
we continue with the more recently developed cumulative formulation leading to a renewal integral equation.
485
490
495
500
i-State Formulation
If the individuals go through life stages, the population is split into subpopulations that contain the individuals in each life stage. We now describe the
dynamics for one single population or subpopulation. Later on, boundary
conditions will connect the subpopulations at the borders of their respective
subspaces Oi . In the PDE formulation, an i-state space O (or Oi ) always is a
subset of Rn , for some n. The i-states in different life stages may have different interpretations and their dimension may vary from one stage to another.
Typical instantaneous vital rates that describe the temporal change of the
i-state variable are the growth, birth, and death rates, which will occur as
per capita rates at the population level.
Growth Rate. Since we assume that the environment is homogeneous in
space, the spatial coordinates of an individual are irrelevant for its behavior.
The coordinates of its i-state x, however, follow a trajectory through O.
The growth rate gðE; xÞ 2 Rn defines a vector field on O that determines
the speed and direction of this movement through i-state space. This
implies that individual growth is deterministic. Note that entries of g do
not refer to the physiological ‘‘growth’’ rate if the corresponding i-state
is not size. If gi represents growth it is sometimes allowed to have
gi ðE; xÞ < 0 >, that is, a shrinking individual. This can lead to hard
mathematical problems and therefore it is generally assumed that
gi ðE; xÞ > 0; that is, the direction of the movement is never reversed and
the movement never stops.
505
510
515
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Physiologically Structured Population Dynamics
17
Death Rate. Death removes an individual from the population. Death is
modeled by specifying the death rate dðE; xÞ 0. The environmental condition that influences the death rate generally is predation pressure. Emigration or harvest of individuals from the region under consideration can also
be modeled using the death rate.
525
Birth Rate. Different reproduction modes exist. Some of these can be
modeled using an individual birth rate bðE; xb ; xÞ 0. This is the rate at
which an individual with state x produces offspring with state at birth xb .
Often the support of this function is some OA O, the subset of adults.
530
Influence on the Environment=Feeding=Predation. The environment
EðtÞ can represent food consumed by the population. When food is abundant
the growth rate does not depend on the food (gðxÞ). On the other hand, when
food is limiting, growth depends on the food availability (gðEðtÞ; xÞ). In the
latter case the amount of food available depends on the population size since
this determines how much food is consumed. This feedback interplay is
typical for predator–prey, or similar trophic, interactions in ecological
communities. In a similar way the population can be consumed by another
population in which case the death rate stands for natural death rate plus
predation rate (dðEðtÞ; xÞ).
If food is assumed to be an abiotic nutrient which is supplied to the region
where the population lives in, the dynamics of this nutrient is generally
described by an ODE. In case of interplay with other populations they can
be described by one or more ODEs or even PDE.
The growth process is new with respect to age-structured population
models. That is, when gðEðtÞ; xÞ ¼ 1, where x is one-dimensional, we get
essentially an age-structured population model. As mentioned earlier, the
environment sometimes implicitly modeled as density-dependent birth and
death rates. Murphy (1983) studies a model with a density-dependent
growth rate.
535
540
545
550
p-State Formulation
The state of the population (the p-state) is now represented by a continuous number density function nðxÞ. The integral of this density function over
an area of i-states equals the total number of individuals
in that area. To be a
R
bit more precise, for any measurable set o O, o nðxÞ dx equals the number
of individuals in the population with i-states in o. To express the fact that this
p-state changes in time, we will denote it as nðt; xÞ, where x is a vector if the
i-state space is multidimensional.
Field Equation. The p-state can be shown (see for instance Metz and
Diekmann, 1986, 14), to satisfy the PDE
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18
B. W. Kooi and F. D. L. Kelpin
@nðt; xÞ X @
þ
ðgðEðtÞ; xÞnðt; xÞÞ ¼ dðEðtÞ; xÞnðt; xÞ
@t
@xi
i
ð22Þ
or shorter
565
@nðt; xÞ
þ H ðgðEðtÞ; xÞnðt; xÞÞ ¼ dðEðtÞ; xÞnðt; xÞ
@t
ð23Þ
This basically is a transport equation with velocity g, but with a nonzero
right-hand side, which represents the death of individuals.
Boundary Conditions. Individuals may be born on @O, the border of O.
Only part of this border, @ þ O, is of interest. This is because the growth vector
gðE; xÞ should point inward. Otherwise newborn individuals will leave O,
which they are not supposed to do. In mathematical terms: Birth only occurs
in i-states xb 2 @ þ O, where the inner product of g with the inward-pointing
normal n is positive. Then the boundary condition reads
nðxb Þ gðEðtÞ; xb Þnðt; xb Þ ¼
Z
bðEðtÞ; xb ; xÞ nðt; xÞ dx
xb 2 @ þ O
570
ð24Þ
O
This is sometimes also called the renewal equation.
Initial Conditions. Initial conditions complete the mathematical formulation:
nð0; xÞ ¼ n0 ðxÞ x 2 O0
ð25Þ
where O0 O is the support of the density n.
In Murphy (1983) the PDE and its boundary and initial conditions are
transformed from numbers density into biomass density. The resulting PDE
formulation is the same as Eqs. (23), (24), and (25), but the interpretations
of some coefficients differ. We remark that in general this transformation
is not needed.
If the dynamics of the environment, for instance, nutrients or predators,
are described by an ODE, the initial amount of these substances completes
the initial conditions and that ODE is solved together with the PDE for the
population. Just as in density-dependent age-structured models, also in PSP
models the birth, growth, and death rates can be assumed to be density dependent in order to avoid modeling the food or predator explicitly.
For populations that propagate by binary fission birth rate and death rate
(division rate) are coupled. If division is into two equal daughters the resulting PDE reads:
@nðt; a; vÞ @nðt; a; vÞ @gðt; a; vÞnðt; a; vÞ
þ
þ
@t
@a
@v
¼ ½bðt; a; vÞ þ dðt; a; vÞnðt; a; vÞ þ 2½2bðt; a; 2vÞnðt; a; 2vÞ ð26Þ
580
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Physiologically Structured Population Dynamics
19
where bðt; a; vÞ is now the per capita division rate and dðt; a; vÞ the death rate,
that is, here the disappearance of individuals by causes other than division.
The terms of the left-hand side are the same as before, but those of the
right-hand side are sink and source terms associated with division. If an individual with size v divides, the first term, two newborns appear at size v from
an individual with size 2v.
Characteristics. The formulation in Eqs. (23), (24), and (25), is based on
the Eulerian approach where a description is given of the flow of individuals
around a fixed point in the i-state space O 2 Rn . The Lagrangian approach
follows a frame moving with the individuals along their trajectories in the istate space O. These trajectories correspond to the so-called characteristics of
the PDE. Characteristics are curves in the space O Rþ . The following
equation defines the characteristic, parameterized by a (which can be age but
is not necessarily so), that is, tðaÞ and xðaÞ.
dt
¼1
da
dx
¼ gðEðtÞ; xÞ 2 Rn
da
600
605
ð27Þ
The PDE Eq. (23) can be rewritten as
@nðt; xÞ
þ Hnðt; xÞ gðEðtÞ; xÞ ¼ nðt; xÞ½H gðEðtÞ; xÞ þ dðEðtÞ; xÞ ð28Þ
@t
where we used
H ½gðEðtÞ; xÞnðt; xÞ ¼ Hnðt; xÞ gðEðtÞ; xÞ þ nðt; xÞ½H gðEðtÞ; xÞ
ð29Þ
Along the characteristic of the PDE Eq. (23), that is, tðaÞ and xðaÞ satisfy Eq.
(27), we obtain with the chain rule the result
615
dn @nðt; xÞ dt
dx
¼
þ Hnðt; xÞ ¼ nðt; xÞ½H gðEðtÞ; xÞ þ dðEðtÞ; xÞ ð30Þ
da
@t da
da
The initial conditions tð0Þ; xð0Þ are now formulated. Initially we have
tð0Þ ¼ 0 and xð0Þ ¼ x, substituted in Eq. (25). For t > 0 the initial values
are related to the appearance rate of the newborn individuals, which is
fixed by the boundary condition for the PDE [Eq. (16)] where tð0Þ ¼ t and
xð0Þ ¼ xb 2 Oþ . Loosely speaking, the PDE problem with boundary and
initial conditions is rewritten as an initial value problem for a set of infinite
number of ODEs.
In the age-structured case we have gðxÞ ¼ 1 and the solution of Eq. (27)
gives characteristic curves that are straight lines in the ða; tÞ plane with
slope 1.
If the growth rate gðxÞ depends on the i-state x and not on time (for
instance, in the case of constant food availability), the characteristic curves
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20
B. W. Kooi and F. D. L. Kelpin
tðaÞ and xðaÞ can be calculated by solving the ODE Eq. (27) without knowing
the density distribution nðt; xÞ. This is in the linear case. However, in the nonlinear case where gðEðtÞ; xÞ depends on time or on the state of the population
itself with density dependent i-state rates, the characteristic curves and the
density nðtðaÞ; xðaÞÞ have to be calculated simultaneously.
If newborns only start at the boundary xb 2 @ þ O it is possible to transform
a size-structured population density nðt; xÞ to an age-structured density
mðt; aÞ. This technique is known as the Murphy trick (Murphy 1983).
Cumulative and RIE Formulation. A more general, so-called
‘‘cumulative’’ formulation exists in terms of renewal integral equations
(Diekmann et al., 1998; 2001). Here the ingredients that serve to describe the
processes at the i-level are not rates, such as the birth rate, but quantities at
the generation level where the whole life history of an individual is taken into
account, such as the lifetime expected number of offspring.
The RIE formulation can be split into different steps. First, the environmental conditions are assumed to be known. With these conditions as an
input, individual development, reproduction, and survival can be modeled
in a manner that resembles multitype branching processes. From this description for one individual, an expression is derived for its entire ‘‘clan.’’ This
clan contains an individual plus all its descendants. The definition of the
dynamics of the population now comes down to summing up all clans of
the individuals in an initial population. Finally, the population influences
its environment. This output follows from the population dynamics. It
should of course equal the input that was initially assumed.
Only the fixed-point problem of closing the feedback loop between the
environment and the population is nonlinear. This explains why the cumulative formulation is successfully used for the continuation of equilibria, where
the environment is constant (see Kirkilionis et al. 2001 and Diekmann et al.
2003).
i-State Development and Survival. Diekmann et al. (2001) give the
cumulative formulation for the populations from the previous subsection,
which we copy here for reference. The input from the environment is a
function I that is defined on a time interval ½0; ‘ðIÞ. So ‘ðIÞ is the length of
the input.
Two operations on inputs are of importance here, the restriction operator
rðsÞ, which restricts the input to the time interval ½0; s, and a shift yðsÞ
which is defined by ðyðsÞIÞðtÞ ¼ Iðt þ sÞ; 0 t < ‘ðIÞ s. This means
the input is shifted s forward in time.
For a given environmental input I, one can construct the trajectory of an
individual with initial i-state xb . Diekmann et al. (2001) define XI ðx0 Þ; as
the i-state of an individual at time ‘ðIÞ; given that it had i-state x0 at time
zero, it experienced input I and it survived.
For populations with instantaneous growth and death rates, the function
t 7! XrðtÞI ðx0 Þ is the solution of the initial value ODE problem
630
635
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645
650
655
660
665
670
Physiologically Structured Population Dynamics
d
xðtÞ ¼ gðIðtÞ; xðtÞÞ xð0Þ ¼ x0
dt
21
ð31aÞ
The survival probability F I ðx0 Þ of this individual equals
F I ðx0 Þ ¼ expf
Z
‘ðIÞ
dðIðtÞ; XrðsÞI ðx0 ÞÞ dsg
ð32Þ
0
The development measure uI ðxb ; oÞ combines this information. It specifies the probability that, after experiencing input I , an individual with initial
state xb is still alive and has an i-state in o O.
680
F I ðx0 Þ if xI ðx0 Þ 2 o
0
otherwise
¼ dxI ðx0 Þ ðoÞF I ðx0 Þ
uI ðxb ; oÞ ¼
ð33Þ
ð34Þ
where d denotes a Dirac delta measure.
In general, though, u need not be confined to one single point. Stochastic
movement through i-state space can therefore be modeled using this more
general cumulative formulation. This is not possible using the classical
PDE, which requires an instantaneous growth rate g and therefore assumes
that growth is deterministic.
Reproduction Kernel. The lifetime reproduction measure LI ðxb ; oÞ
equals the expected number of offspring with states of birth in o. The formulation is the same whether individuals are born in internal points of O or
on the boundary Oþ
LI ðxb ; oÞ ¼
Z Z
o
685
690
‘ðIÞ
bðIðtÞ; xb ; XrðtÞI ðx0 ÞÞ F rðtÞI ðx0 Þ dxb dt
ð35Þ
0
Then a next generation operator is defined, which is the expected number of
offspring but now for each distribution of individuals. This leads to a generation expansion, that is, the iteration of the reproduction rule to specify the
expected total offspring of the entire clan. In a similar way a next state operator is defined that is the distribution of the individuals for a specific
distribution of individuals at time zero. For the definition and mathematical
treatment of these two operators the reader is referred to (Diekmann et al.
(2001).
Combining the two ingredients, namely, the reproduction together with
i-state development and survival, specifies the structured population model
by adding the contributions of all individuals.
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B. W. Kooi and F. D. L. Kelpin
NUMERICAL TECHNIQUES FOR PHYSIOLOGICALLY
STRUCTURED POPULATION MODELS
705
Extensive descriptions of numerical methods for the time integration of
structured population models are given in the literature. Finite difference
techniques are derived from the mathematical model by replacing derivatives
by differential quotients. With other techniques biological interpretation is
taken into account with developing the numerical procedure.
In full discretization schemes, time and i-state space are discretized simultaneously. In classical finite difference schemes (Richtmeyer and Morton
1967), derivatives are replaced by differential quotients based on Taylor
series expansion in grid points. For the nonlinear density-dependent models
the classical Lax–Wendroff method is used in Sulsky (1993) for the agestructured model and in Sulsky (1994) for size-structured models.
In Sulsky (1993; 1994) a fixed grid is used and the resulting scheme is an
adapted version of the classical Lax–Wendroff method. For age-structured
models the support for the density distributions is known for the initial conditions and subsequently for all times in the time interval of interest. For sizestructured populations, the support of the size distribution is not known
beforehand and even several magnitudes in the size of the individuals with
in the population may occur. Also, the support of the size density may
increase with time. This leads to wasted work over much of the grid at
early times in the calculation when this density function is zero in a large
portion of the computational domain.
In the box method (Fairweather and López-Marcos 1991; Angulo and
López-Marcos 2002), the starting point is the balance law for a cell that is
between x and x þ Dx, and t and t þ Dt:
Z xþDx
nðt þ Dt; xÞ nðt; xÞ dx
x
þ
Z
715
720
725
tþDt
gðEðtÞ; x þ DxÞnðt; x þ DxÞ gðEðtÞ; xÞnðt; xÞ dt
t
¼
710
Z
tþDt
Z
xþDx
dðEðtÞ; xÞnðt; xÞ dx dt
t
ð36Þ
x
The difference scheme is obtained by discretization of the integral terms in
this equation. Another implicit scheme is proposed and analyzed in Ackleh
and Ito (1997).
In semidiscretization schemes, initially only the i-state space is discretized. In Gurney and Nisbet (1998) a number of these schemes (upwind difference scheme and central difference scheme) are discussed. The resulting
set of ODEs is then further discretized in time using an ODE solver. Similar
to the full discretization scheme, these methods work with a discrete representation of the density function.
735
Physiologically Structured Population Dynamics
23
Perhaps the simplest scheme for the age-structured model formulation,
which is a full discretization scheme as well as a semidiscretization
scheme, is an explicit finite difference scheme on a regular grid. As the
grid size in both time and age direction are taken equal the method is also
equivalent to a time-integration along the characteristics with equidistant
time steps. At each time step the number of newborns with age zero is
obtained as a sum of adults weighted with the reproduction rate. This
simple scheme was used with the calculation of the results presented in
Figures 1–4.
For age- and size-structured PDE models a similar approximation scheme
is proposed in Ito et al. (1991), Milner and Rabbiolo (1992), Angulo and
López-Marcos (2000, 2002), and Kostova (2002). This procedure comprises
three steps. First a grid is constructed such that each point belongs to the
same characteristic. This is done by numerical integration of the individual
growth equation, Eq. (31). In principle any ODE solver can be used. Solving
the ODEs of Eq. (30) with initial points on the characteristic curve gives the
solution for the next time step. Again, any ODE solver can be used. The final
step is to calculate the population birth rate occurring in the boundary condition, which is obtained by employing a numerical quadrature for the integral
term in Eq. (16). If the environment is constant the characteristic curves are
fixed in time; that is, all individuals born at points in @ þ O follow trajectories
that originate in these points. If, on the other hand, the birth and death rates
are density dependent, a similar approach can be used where the last two
steps are more complicated.
Kostova (2002) gives a short overview of existing numerical methods with
proven global error estimates and describes an explicit method of third order.
In Angulo and López-Marcos (2002) the efficiency of three numerical
methods—the box method, the Lax–Wendroff, and a characteristics
scheme—are assessed by considering five problems with diverse degrees
of freedom. They conclude that for an autonomous problem no best
method exists and that all specific features involved in the numerical integration have to be considered in order to choose a specific numerical scheme for
a particular problem. For nonautonomous problems the box method was more
efficient than the Lax–Wendroff scheme in four of the five test problems.
These schemes are related to the Eulerian approach. The escalator boxcar
train (EBT) method (de Roos 1988) uses the Lagrangian approach. This
method can be used for a broad class of PSP models.
The EBT method (de Roos 1988) uses a semidiscretization scheme. In
order to discretize the i-state space, individuals with similar i-states are
grouped into so-called cohorts. Each cohort is represented by a delta peak
placed in the average i-state of the individuals in the cohort. The height of
the peak equals the number of individuals in the cohort. The p-state is
approximated by the sum of these delta peaks.
Figure 5 gives an example of the discretization of a one-dimensional
i-state space where the p-state is represented by a continuous density
740
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760
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770
775
780
24
B. W. Kooi and F. D. L. Kelpin
FIGURE 5 An i-state discretization of a continuous density function nð0; xÞ into n
cohorts. Individuals with i-states in the sameRrange Oj form a cohort. This cohort is
represented by a delta peak with height lj ¼ Oj nð0; xÞ dx (the number of individuals
R
in the cohort), placed at Zj ¼ l1j Oj xnð0; xÞ dx (the average i-state of all individuals in
the cohort).
function. The method also works, however, for a higher dimensional i-state
space and for p-states given by a measure on the i-state space.
Each individual follows a trajectory through the i-state space, which is a
solution of the ODE that describes the individual development. The righthand side of the ODE is a function of the individual’s i-state and the environment. Initially, cohorts consist of individuals with nearly identical i-states.
If the solutions of the development ODE do not diverge, the individuals in a
cohort will stay close in the time interval of interest. As a consequence, the
trajectory starting in the initial average i-state of a cohort will at all times be a
good approximation for the i-states of the individuals in that cohort. The
death rate on this trajectory will also be a good approximation for the
death rates of all individuals in the cohort, as death rates too depend only
on i-state and environment.
A treatment of classical type boundary conditions with continuous reproduction by all adult individuals can be found in de Roos (1988). The introduction of one or several special boundary cohorts preserves the order of
the numerical integration technique. These boundary cohorts are filled with
785
790
795
800
Physiologically Structured Population Dynamics
25
newborn individuals. Once they are large enough they are transformed into
ordinary cohorts.
Due to the continuous introduction of new cohorts, the total amount of
cohorts may grow so large that the numerical solution of the system is
slowed down significantly. If this happens, we can reduce the number of
cohorts by merging cohorts with nearly the same i-state or by removing
cohorts with a sufficiently small number of individuals.
Since all finite-difference methods are based on Taylor-series expansion, it
is assumed that the frequency distribution as well as the birth and death rate
distribution functions are sufficiently smooth. For a model with pulsed reproduction, that is, a model where individuals are born in a specific period of the
year or when a certain threshold for the mothers is satisfied, we successfully
used the simple forward Euler scheme in Kelpin et al. (2000). We showed
that for a nontrivial test problem, an age-structured population model with
pulsed reproduction, the calculated results obtained with a first-order forward
Euler scheme agree well with those obtained with the EBT method.
In Ackleh and Deng (2000) a (monotone) approximation, based on an
upper and lower solutions technique, is developed for the nonautonomous
size-structured model.
805
810
815
820
WORMS
Naidids occur in a wide range of aquatic habitats such as rivers, lakes and
estuaries, sewage filterbeds, slow sand filters of water works, and drinkingwater distribution systems. In sewage treatment plants the number of
worms present is inversely related to sludge disposal; however, their presence
is still unpredictable and uncontrollable (see Ratsak, 2001). Naidids are often
important detrivores in organic polluted river systems and therefore may be
an important element in the transfer of energy from the heterotrophic bacteria
associated with organic matter to higher trophic levels such as fish.
The length of Nais elinguis is generally less than 12 mm and the mean
diameter is 0.15 mm. It is a hermaphrodite, under normal conditions it
propagates by division into an anterior and a somewhat smaller posterior
part. It was observed that both the anterior and the posterior worm divide
when they reach the same particular volume. Individual growth of two sisters
with constant, abundant food availability is shown in Figure 6. The naidids
were collected from a sewage treatment plant and fed with activated
sludge particles from the same plant. The anterior worm initially grows
faster than her posterior sister but the difference diminishes and is negligible
when they divide.
Ratsak et al. (1993) give a model for the feeding, growth, and division of
worms, based on DEB theory (Kooijman 2000). The model gives an explanation for the initial growth spurt of anterior worms after division. While food
flows from mouth to anus its nutritional value diminishes as nutrients are
825
830
835
840
26
B. W. Kooi and F. D. L. Kelpin
FIGURE 6 Growth curves of sister naidids. The upper and lower curves represent the
fit of the DEB model to the measured data of anterior () and posterior naidid (u),
respectively; see Ratsak et al. (1993). Observe that the worms grow in length only,
so length is directly proportional to volume. The length at division is vd ¼ 8:41 mm.
assimilated into the energy reserves of the worm. These energy reserves are
used for maintenance and growth. Close to the mouth, the food still contains
many nutrients and the energy reserves are richer. After division, the anterior
worm therefore has richer energy reserves and initially grows faster than her
posterior sister.
We first give the mathematical model for the individual, in dimensionless
form. Then we show how a series of time-scale assumptions greatly reduce
the complexity of this model.
845
850
Individual Model Formulation
Naturally, the i-state of a worm in this model contains its length or
volume, but we use, according to the (DEB) model (Kooijman 2000), its
volume v and its energy density e. Ratsak et al. (1993) parameterize the
worm along its length using a parameter x 2 ½0; 1, which equals 1 at the
mouth and 0 at the anus. A function eðt; xÞ is the energy reserve density at
time t and at position x. A severe drawback is that the i-state becomes infinite
dimensional, since it includes the function eð; xÞ: ½0; 1 ! R. The population
dynamics can no longer be described in the familiar PDE formulation.
The ingestion through the mouth of the worm equals Im f ðtÞv, where Im is
the maximum ingestion rate and f ðtÞ is the scaled functional response. A
more rigorous version of the model would also model the digestion of food
inside the gut. Here we simply assume that the nutritional value of the
food in the gut of the worm decreases linearly from 1 at the mouth (x ¼ 1)
to 0 at the anus (x ¼ 0). Then the energy reserve density at location x is
assumed to follow
855
860
865
Physiologically Structured Population Dynamics
@eðt; xÞ
¼ nð2f ðtÞx eðt; xÞÞ
@t
27
ð37Þ
The growth rate of a worm is computed by integrating the local growth rate
along its length:
dv
¼v
dt
Q1
Z
0
1
neðt; xÞ mg
dx
eðt; xÞ þ g
870
ð38Þ
When a worm’s volume reaches v ¼ vd , it divides. The volumes of the two
daughters are a fixed fraction of that of the mother, not necessarily one half.
The anterior daughter has volume va ¼ ð1 xd Þvd ; the posterior daughter has
volume vp ¼ xd vd ; see also Figure 6. So just prior to the division the range of
the anterior part is from 1 to xd , and of the posterior part from xd to 0. At
division we redefine the variable x, such that it still has the value 1 at the
anterior end and 0 at the posterior end of the daughters. Thus, for the anterior
part position x of the daughter corresponds with position xð1 xd Þ þ xd of
the mother. For the posterior part, position x of the daughter corresponds
with x xd of the mother. Observe that the distribution of the nutritional
value for both anterior and posterior worm changes abruptly at division.
Let us assume that the energy density is at equilibrium at the moment of
division eðtb ; xÞ ¼ 2f ðtb Þx. Then the initial condition for the anterior and posterior daughters, respectively, is
ea ðtb ; xÞ ¼ 2f ðtb Þ½xd þ x ð1 xd Þ
ð39Þ
ep ðtb ; xÞ ¼ 2f ðtb Þxd x
ð40Þ
875
880
885
and
Hence, all individuals belonging to the same time have an unique state at
birth, that is the same volume vi and energy density distribution ei ðtb ; xÞ,
i 2 fa; pg. That is, the mother decides when a daughter is born but passes
not her i-state information. The growth rate does not depend on the x
explicitly, only on the integral over a function that depends on the energy
density distribution over x. For all x we have
Z t
n½2f ðtÞx ei ðt; xÞ dt
ð41Þ
ei ðt; xÞ ¼ ei ðtb ; xÞ þ
890
tb
Substitution of this expression in Eq. (38) gives the instantaneous growth
rate, which depends on the history of the environment f ðtÞ and the energy
reserves since birth.
In the next sections we describe various model formulations for the population dynamics of the worm where mortality is assumed to be zero. We
900
28
B. W. Kooi and F. D. L. Kelpin
assume that food supply is abundant, that is, the environment is timeindependent and f ¼ 1. This allows for an age-dependent model formulation.
We start with a continuous-time formulation and we continue with a
discrete-time formulation.
Continuous-time Age-Dependent Model Formulation
905
We assume that there are two-populations with densities ma ðt; aÞ defined
on ½0; 1Þ ½0; ta and mp ðt; aÞ on ½0; 1Þ ½0; tp with interdivision times ta
and tp . The PDEs read
@
@
m ðt; aÞ ¼ m ðt; aÞ
@t a
@a a
@
@
m ðt; aÞ ¼ m ðt; aÞ
@t p
@a p
ð42Þ
We arrive at a boundary condition for this coupled PDEs system, Eq. (42), at
a ¼ aa ¼ ta and a ¼ ap ¼ tp :
ma ðt; 0Þ ¼ ma ðt; ta Þ þ mp ðt; tp Þ
ð43Þ
mp ðt; 0Þ ¼ mp ðt; tp Þ þ ma ðt; ta Þ
which describe division. The interdivision times ta and tp are implicitly given
by
Z
ta
Z
1
nea ðt; xÞ mg
dx dt
ea ðt; xÞ þ g
0
0
Z tp Z 1
nep ðt; xÞ mg
dx dt
lnfvd g ¼ lnfvp g þ
ep ðt; xÞ þ g
0
0
lnfvd g ¼ lnfva g þ
ð44aÞ
ð44bÞ
where ea ðt; xÞ and ep ðt; xÞ are given by Eq. (39) and Eq. (40) with constant
food f ¼ 1.
In this continuous-time model we introduce the number of individuals in
the populations:
Na ðtÞ ¼
Z
ta
ma ðt; aÞ da
and Np ðtÞ ¼
0
Z
tp
mp ðt; aÞ da
ð45Þ
0
Integration by parts of Eq. (42) and the use of the boundary conditions Eq.
(43) yields
dNa
¼ mp ðt; tp Þ
dt
and
dNp
¼ ma ðt; ta Þ
dt
ð46Þ
910
Physiologically Structured Population Dynamics
29
These results have a clear biological interpretation; when an anterior individual divides an extra new posterior worm is born, and vice versa, when an
posterior individual divides an extra new anterior worm is born.
Let T ðtÞ and Ai ðaÞ be defined such that mi ðt; aÞ ¼ T ðtÞAi ðaÞ. Then
separation of variables yields that the two functions are determined by
dT j
¼ cj T j
dt
T j ð0Þ ¼ 1
925
ð47Þ
and
930
dAaj
da
¼ cj Aaj
and
dApj
da
¼ cj Apj
ð48Þ
with boundary conditions
Aaj ð0Þ ¼ Apj ð0Þ ¼ Aaj ðta Þ þ Apj ðtp Þ
ð49Þ
This gives T j ¼ expfcj tg and
Aaj ðaÞ ¼ K expfcj ag
and Apj ðaÞ ¼ K expfcj ag
ð50Þ
where K is a normalization factor. Substitution of these results into Eq. (49)
yields the Euler–Lotka equation
expfcj ta g þ expfcj tp g ¼ 1
ð51Þ
The roots cj give the following solutions of the PDE, Eq. (42):
ma ðt; aÞ ¼ mj expfcj tg expfcj ag and mp ðt; aÞ ¼ mj expfcj tg expfcj ag,
where mj is a constant fixed by the initial conditions.
The principle of superposition of these solutions can, at least formally, be
applied to describe the transient solution of the linear, PDEs Eq. (42). Then
the Fourier coefficients (the mj terms) are fixed by the initial age distribution,
which is assumed to be sufficiently smooth. In practice it turned out that for a
reasonable approximation many terms are needed and this restricts
applicability for practical purposes. This completes the analysis of the
short-term dynamics.
We assume that the age distributions for both populations are the stable
distributions belonging to the real root c0 of the Euler–Lotka equation,
Eq. (51), Aa0 ðaÞ and Ap0 ðaÞ. Then both populations grow exponentially, so
T ðtÞ ¼ expfc0 tg or, if Nw ðtÞ ¼ Na ðtÞ þ Np ðtÞ denotes the total number of
worms,
dNa
¼ c0 N a
dt
and
dNw
¼ c0 Nw
dt
ð52Þ
940
945
950
30
B. W. Kooi and F. D. L. Kelpin
With Eq. (46) and Eq. (51) we get
c0 Na ¼ T 0 ðtÞAp0 ðtp Þ
955
and c0 Nw ¼ T 0 ðtÞAp0 ð0Þ
ð53Þ
Then, with Eq. (50) and Eq. (51) we obtain the result
t =t
Na ðtÞ
Na ðtÞ a p
þ
¼1
Nw ðtÞ
Nw ðtÞ
ð54Þ
Under the condition that for j 6¼ 0, Re cj < c0 , Eq. (54) holds true asymptotically for t ! 1. We have Re cj ¼ c0 if and only if the ratio ta =tp is
rational, that is, there are multiple poles denoted by j with Re cj ¼ c0 ,
j 6¼ 0. We elaborate this case later with the discrete-time formalism. The
case where ta =tp is irrational is dealt with in the next section where a renewal
equation for the cumulative birth rate is derived.
Continuous-Time RIE Model Formulation
965
El Houssif (2001) derives a renewal equation for the cumulative birth rate.
The Renewal Theorem is used to derive an expression for the asymptotic
behavior of the cumulative birth rate and for the asymptotic composition
of the population. He obtains Eq. (54) but now for irrational ratios ta =tp .
As in the last part of the previous section, so is food assumed to be constant.
In El Houssif (2001), mortality was taken into account. For the sake of simplicity we assume mortality zero.
Let BðtÞ denote the cumulative number of individuals that passed through
size va during the time interval ½0; tÞ. Then the renewal equation reads for
t > tp , that is, when all individuals belonging to the initial population have
divided at least once,
BðtÞ ¼ Bðt ta Þ þ Bðt tp Þ þ GðtÞ
ð55Þ
The term GðtÞ is the contribution of the population at time t ¼ 0 to BðtÞ. It is
the sum of three contributions given by
GðtÞ ¼
Z
va
H½t tp þ ta þ aw ðvÞnp ð0; vÞ ðdvÞ
vp
þ
Z
vd 960
Hðt tp þ aw ðvÞÞ þ Hðt 2tp þ ta þ aw ðvÞÞ nw ð0; vÞ ðdvÞ ð56Þ
va
where H is the (Heaviside) step function defined by HðxÞ ¼ 0 for x < 0 and
HðxÞ ¼ 1 for x > 0. The population size and composition at time 0 is
described by a Borel measure nw ð0; vÞ ¼ np ð0; vÞ for vp < v < va and
970
975
Physiologically Structured Population Dynamics
31
nw ð0; vÞ ¼ na ð0; vÞ þ np ð0; vÞ for va < v < vd . The function aw ðvÞ is the time
it takes for an individual to grow from vp to v. Thus tp ¼ aw ðvd Þ and
ta ¼ aw ðvd Þ aw ðva Þ. The first term is the contribution of all posterior
worms with size vp < v > va , the second term all those who are anterior
worms after division, and the third term those who are posterior worms
after division.
The explicit expression for the solution of Eq. (55) reads
BðtÞ ¼ GðtÞ þ
1 X
k¼n X
n
n¼1 k¼0
k
Gðt kðtp ta Þ nta Þ
Z
990
ð57Þ
where GðtÞ ¼ 0 for t < 0.
The renewal theorem (Feller 1971 and Jagers 1975, section 5.2) is used to
deduce the asymptotic behavior for BðtÞ for t ! 1.
Let GðsÞ be a real function then the Laplace–Stieltjes transform ^f ðaÞ is
defined as
^ ðaÞ ¼
G
985
995
1
expfasgG ðdsÞ
ð58Þ
0
The characteristic equation Eq. (51) is now defined using this Laplace–
Stieltjes transform technique,
^0 ðaÞ ¼ expfata g þ expfatp g ¼ 1
m
ð59Þ
There exists a unique real root a1 that is positive.
Two cases are distinguished. In the lattice case the ratio of the interdivision times, ta =tp , is rational, that is ta =tp ¼ l=k with l, k 2 N where for the
greatest common divisor (gcd) of l and k we have gcdðl; kÞ ¼ 1. In the nonlattice case this ratio is irrational.
For both cases El Houssif (2001) derives the asymptotic behavior of BðtÞ
where t ! 1. He formulates expressions for Na ðtÞ and Np ðtÞ in terms of BðtÞ
and thereafter he derives the asymptotic behavior for Na ðtÞ and Np ðtÞ given in
Eq. (54) for the nonlattice case whether the death rate is positive or zero, but
in the lattice case only when the death rate is zero, and it is conjectured that
the relation holds also for the lattice case when the death rate is positive.
In the next section we make a link with a discrete-time formulation where
we derive the asymptotic behavior given in Eq. (54) in the lattice case where
death rate is zero.
Discrete-Time Model Formulation
We assume that ta ¼ l=ktp for k ¼ 1; 2; . . . ; 1 and l ¼ 1; 2; . . . ; k 1,
while gcdðk; lÞ ¼ 1, hence the lattice case defined in the previous section.
1005
1010
1015
32
B. W. Kooi and F. D. L. Kelpin
The number of anterior daughters after the i-th division is denoted by Na ðita Þ,
the number of posterior ones by Np ðita Þ, and the total number of individuals
by Nw ðita Þ.
1020
Discrete-Time Renewal Equation
The number of anterior daughters after the i-th division follows the generalized Fibonacci series
Na ðita Þ ¼ Na ðði lÞta Þ þ Na ðði kÞta Þ for i k
ð60Þ
with, for instance, one of Na ð0Þ, Na ðta Þ; . . . ; Na ððk 1Þta Þ equal to 1. This
follows from the following two relationships:
Np ðita Þ ¼ Np ðði lÞta Þ þ Na ðði lÞta Þ
ð61Þ
Na ðita Þ ¼ Na ðði kÞta Þ þ Np ðði kÞta Þ
ð62Þ
1025
and
This implies for the number of posterior daughters
Np ðita Þ ¼ Na ðði þ k lÞta Þ
1030
ð63Þ
The Fibonacci series, Eq. (60), converges to a geometrical one with
Na ðita Þ ¼ aNa ðði 1Þta Þ, with a given by the characteristic equation
ak akl 1 ¼ 0
or
1
1
¼1
ak al
ð64Þ
With c0 ¼ ta1 ln a Eq. (64) is equivalent to Eq. (51) being the characteristic
equation for the continuous-time case. In Kooi and Boer (1995) this approach
is worked out in detail; here we continue with a population matrix formulation.
1035
Discrete-Time Leslie Matrix
We continue with a description of an approach using life cycle graphs and
Leslie matrices (see Caswell 1989). We define life stages here as follows. At
time il ta the length interval ½1=kvd ; vd is divided into k subintervals each
representing a life stage. The number of individuals at time il ta in life stage
j with volume j=kvd v ðj þ 1Þ=kvd is denoted as njp ðil ta Þ. The anterior
worms live only in life stages k l þ 1 j k. The duration of one time
step is now tla . The number of anterior and posterior worms in the same life
stage are lumped together with nkw ðil ta Þ ¼ nkp ðil ta Þ þ nka ðil ta Þ as shown
in Figure 7 where the life-cycle graph (see Caswell 1989) for a one-population and two-population formalism is given.
1040
1045
Physiologically Structured Population Dynamics
33
FIGURE 7 Life-cycle graphs for the one- and two-population formalism.
As an example of the two-population formalism we consider the special
case where l ¼ 1 and k ¼ 2. The interdivision time of the posterior worm
tp is a multiple (two times) of the interdivision time of the anterior worm
ta . This problem is directly related to the Fibonacci series by the following
population matrix:
0
1 0
0
n1p ðita Þ
B 2
C @
@ np ðita Þ A ¼ 1
0
n2a ðita Þ
1
0
1
1
10 1
1
np ðði 1Þta Þ
B
C
0 A@ n2p ðði 1Þta Þ A
1
n2a ðði 1Þta Þ
1050
ð65Þ
Now Na ðita Þ ¼ n2a ðita Þ and Np ðita Þ ¼ n1p ðita Þ þ n2p ðita Þ and this leads directly
to the Fibonacci series, Eq. (60). The asymptotic behavior can be obtained
here straightforwardly by the study of the population matrix in Eq. (7)
where n0 ðita Þ ¼ n2a ðita Þ þ n2p ðita Þ and n1 ðita Þ ¼ n1p ðita Þ.
In the one-population formalism the population projection matrix for the
general case reads
1055
1060
34
B. W. Kooi and F. D. L. Kelpin
1 0
i
nlþ1
0
w ðl ta Þ
C B1
B
.
C B
B
..
C
B
B nk ð i t Þ C B
0
B w l a C B
.
C¼B
B 1 i
B
B np ðl ta Þ C B ..
C
B
.
..
C B
B
A @ ..
@
.
0
nlp ðil ta Þ
0
0
..
.
..
.
..
.
1
0 0
.. .. ..
.
.
.
.. .. ..
.
.
.
..
. 1
0
0
1
10 lþ1 i1 1
nw ð l ta Þ
1
C
B
..
0C
C
B
.
C
B
.. C
C
C
k i1
. CB
n
ð
t
Þ
C
B
a
w
l
.. C
C
B
1 i1
n
ð
t
Þ
C
B
p
a
.C
l
CB
C
CB
.
C
..
0 A@
A
l i1
0
np ð l t a Þ
ð66Þ
The nonzero elements in the population matrix are two elements of the first
row that model the division and the subdiagonal, which models survival to
the next class.
The Perron–Frobenius theorem applies (see Caswell 1989, 58, 60),
because the graph is irreducible (i.e., there is a path in the graph from
every node to every node) and the graph is primitive (i.e., it is irreducible
and gcdðl; kÞ ¼ 1).
The characteristic equation for this matrix equation is the same as Eq.
(64). The column eigenvector corresponding to the eigenvalue a reads
ak1 ; ak2 ; . . . ; a; 1ðTÞ
k1
X
ai1 ¼
i¼kl
k1
X
Np ¼
i¼0
ai1
k
akl
1
1 a1
¼
1 a1
1 a1
1 ak1 akl
1
¼
1 a1 1 a1
1070
ð67Þ
The existence of a real eigenvalue larger than 1 follows from the evaluation of the function ð1 al Þakl . So, there is one real dominant eigenvalue
a1 > 1, a root of Eq. (64) with largest real part. This multiplier a1 relates
to the asymptotic population growth rate m as m ¼ ta1 ln a1 .
Kooi and Boer (1995) show that for the population projection matrix given
in Eq. (66) there is a real eigenvalue with greater absolute value than any
other eigenvalue. The simpler case l ¼ 1, which is the interdivision time of
the posterior worm tp , is a multiple of the interdivision time of the anterior
worm ta , dealt with in Ratsak et al. (1993).
Frauenthal (1986) proves, using spectral decomposition, that if one starts
with a strictly positive vector and repeatedly applies the linear transformation
defined in Eq. (66), the resulting vector converges to the column eigenvector,
Eq. (67), corresponding to the dominant real eigenvalue.
The relationship Eq. (54) is obtained with
Na ¼
1065
and
ð68Þ
where a1 is again the positive real dominant eigenvalue. In combination with
the characteristic Eq. (64) for the dominant eigenvalue a1 > 1, this implies
1075
1080
1085
Physiologically Structured Population Dynamics
Na
1
¼ kl
Np a1
or
Na
1
1
¼
¼ k
kl
Nw ð1 þ a1 Þ a1
35
ð69Þ
Consequently, we have the relationship Eq. (54) as time goes to infinity.
1090
Comparison of the Discrete-Time and the Continuous-Time Model
For worms living in abundant food environments that divide into two possibly unequal daughters, the anterior and posterior ones with interdivision
times ta and tp , the asymptotic population growth rate m is given by Eq.
(51) with m ¼ c0 . This is also the characteristic equation Eq. (64), with
m ¼ ta1 ln a. Hence this equation holds whether the ratio of the interdivision
times ta and tp is rational or irrational. Asymptotically for the fraction of
anterior worms, that is, Na divided by the total number of worms Nw , relationship Eq. (54) holds.
The results obtained with the discrete-time model can be used to clarify
results of the continuous-time model. Consider the case of division in a
fixed partition, where ta =tp ¼ 7=19. In Figure 8 we show the trajectories of
the worms when we start out with one cohort of worms. The scaled age of
the posterior individual a=tp is shown as a function of that of the sister, the
anterior individual, a=ta . That is, 7 revolutions along the center line of a
torus representing the growth of the anterior individual are accompanied by
FIGURE 8 The increase in the scaled age of the posterior worm, a=tp , as a function
of the scaled age of the anterior worm, a=ta . Dotted line for rational ratio of interdivision times ta =tp ¼ 7=19 and solid line for irrational ratio ta =tp ¼ 1 ðln 2Þ=ðln 3Þ.
1095
1100
1105
36
B. W. Kooi and F. D. L. Kelpin
19 revolutions along the torus. Then the line is continued along the path travelled at time 19ta ¼ 7tp ago, showing periodicity. We conclude that the age
distribution does not converge to a stable age distribution. The asymptotic
behavior is cyclic.
When the ratio of the interdivision times is rational, both the continuoustime and the discrete-time model apply. We lump the number of individuals
into the life stages introduced earlier, so at time il ta in life stage j we have,
R j=kvd
np ðil ta ; vÞ dv. The relationship between the growth rate
njp ðil ta Þ ¼ ðj1Þ=kv
d
m ¼ c0 and the dominant real eigenvalue a1 is: expfmta g ¼ a1 .
Convergence to the eigenvector of Eq. (67) means that the distributions of
the total numbers in the life stage converge. So, within a life stage there is no
convergence, and this is known from the solution of the continuous-time
model, but the distribution of the total numbers of individuals within the
life stages converges to an asymptotic distribution.
If the ratio is irrational the dimension of the matrix becomes infinite.
Therefore we study a series ln =kn with kn ! 1 for n ! 1 such that this
series of rational numbers converges to this irrational number. Then the
length of the life stages converges to zero, and therefore we have effectively
convergence to a length distribution. However, the rate of convergence
becomes small because the absolute magnitude of the eigenvalues of the
second largest one is almost equal to the largest real eigenvalue. Hence,
there is convergence, but the rate becomes infinitely small.
In Figure 8 we show the trajectories also for the case
ta =tp ¼ 1 ln 2=ðln 3Þ and thus irrational. We do not have periodicity now
as we had before in the rational case (where ta =tp ¼ 7=19 in Figure 8).
1110
1115
1120
1125
1130
CONCLUDING REMARKS
The following observation can be made. First unstructured population
models were replaced by simple structured ones where age served as istate variable. Mathematical techniques were developed to prove existence,
permanence, and stability properties of the ultimate age or size distribution.
Then the model for the individual underlying the structured population model
became more complex for the need to get more biological realism. Later
other i-state variables then age were used and growth depending on the
food availability was taken into account, as well as different life stages.
Because food is consumed by the population but on the other hand growth
depends on the food availability, generally the food dynamics has to be modeled as well. The individual model was described by individual rates in deterministic ODEs, and balance laws directly lead to a PDE model formulation at
the population level. The mathematical techniques were adapted to cope with
these more complex models and these techniques seemed initially to be successful. It was possible to take food dynamics into account and to formulate
1135
1140
1145
Physiologically Structured Population Dynamics
Q2
37
the interaction via a feedback. However, rather subtle problems arose and
very different mathematical techniques had to be developed to tackle these
problems. The step for i-state to p-state is now postponed by dealing with
cumulative quantities at the generation level. Stochastic movement through
i-state space can therefore be modeled. This approach is partly based on
techniques from branching processes.
We have mentioned that the growth rate is generally assumed strictly positive, gðE; xÞ > 0. Then the direction of the movement along the characteristics neither reverses nor stops. Growth occurs when available internal energy
reserves or external food is sufficient to meet maintenance costs which are
needed to maintain the state of living. Consequently, the individual dies
when gðE; xÞ ¼ 0; see Kooijman (2000), Muller and Nisbet (2000), and
Hanegraaf and Kooi (2002). In Kooi and Kooijman (1999) we studied the
dynamics of a rotifer population consuming algae in a chemostat. During a
transient there was insufficient food to support life of the whole population
for a relatively short period. When an unstructured model formulation is
used where all individuals are identical, death of one member would directly
lead to extinction of the whole population. In Kooi and Kooijman (1999) we
showed that when food becomes scarce for a relatively short period, some
unlucky individuals with insufficient energy reserves will die, leaving more
food for survivors. Later when food recovered the population recovers as
well and continuously persists.
In Gyllenberg et al. (1997) and Persson et al. (1998), populations with both
continuous, namely, death, and discrete, namely, reproduction, elements are
investigated. In Gyllenberg et al. (1997) the dynamics of the population with
pulsed reproduction are formulated by a difference equation for one state
variable, namely, the population size. The influence of the environment is
taken into account as a density-dependent within-season mortality. This formulation is appropriate for simple ecological systems of seasonally breeding
populations with non-overlapping generations.
In the worms case study, the age at which reproduction occurs determines
the projection time interval in the Leslie matrix formulation. In Kooi et al.
(2001) a similar model formulation was used to analyze the asymptotic
dynamics of a waterflea Daphnia magna population with iteroparous
propagation (they reproduce more than once and die immediately after last
reproduction). In that model all individuals are born with a species specific
biovolume and are clones of the parent. In the fixed period between
consecutive reproductive events the adult density-dependent mortality is a
continuous process. Natural death occurs when adults reach the species
specific maximum age immediately after their last reproductive event.
With the Leslie matrix model formulation it was possible to clarify longterm behavior results obtained with numerical integration of the continuoustime model.
Cushing (1998) derives the continuous-time PDE model from the discretetime matrix model by the limiting process of infinitesimally refining the
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classes and the census time interval. This approach shows in the case of agestructured populations the relationship between the McKendrick–von Foerster formulation and the Leslie matrix formulation. The relationships between
the various discrete- and continuous-time models for the dynamics of PSP
models are elucidated in de Roos et al. (1992), where a numerical scheme
for the solution of physiologically structured populations is derived from
the Leslie matrix model. In that formulation there is one life stage and therefore the time step is equal to the fixed time period between two consecutive
reproduction events.
Most models presented and analyzed in the literature deal with one isolated structured population or one in interaction with unstructured resources
or predators. The modeling of the trophic interactions is a very important
issue when we want to deal with multispecies systems such as communities
and biosystems, where in the latter case the interaction with a physical environment is explicitly modeled. In principle, each population can be described
by a PDE together with boundary and initial conditions. For several agestructured species, discrete- and continuous-time models are proposed in
Cushing (1998). With size-structured populations, modeling interactions
among species, predator=prey as well as competition interactions are even
more troublesome. Numerical difficulties with simulations of spatially inhomogeneous populations with physiologically distinguishable individuals were
discussed in Gurney et al. (2001).
In the worms case, the division into an anterior and a posterior worm disturbs the equilibrium state of the energy reserves, as the anterior daughter
gets a bigger part of the reserves than the posterior one. It is assumed that
the energy dynamics are fast enough to smooth out the disturbance caused
by one division before either of the daughter worms divides. In other
words, both daughter worms reach the equilibrium value for energy reserves
before they in turn divide. This means that all worms have the same i-state
when they divide. All anterior worms are born identical and therefore the
population of anterior worms is concentrated on one line in i-state space.
The same is true for the posterior worms, so there are only two possible
states of birth.
In community ecology, predator=prey interaction, host–parasitoid interaction, inducible defense, competition, mutualism, symbiosis, nutrient recycling, invasion, succession, persistence, stability, and so on play an
important role in theory building. These theoretical concepts are used in
the study of ecological issues such as the stability–diversity of ecosystems
debate, the maximum length of food chains, and occurrence of succession.
Unstructured ODE population models have been used frequently to address
these issues whereby the short- and long-term dynamic behavior is studied.
Spatially structured population models (Durrett and Levin 1994, Gurney
et al. 2001), in which the i-state contains the spatial coordinates of an individual, are not discussed in this article but are important in many applications.
Structured population models can also be combined with adaptive dynamics
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39
theory to study evolutionary processes. In Claessen and Dieckmann (2002) an
age-structured model for predators is used and resource polymorphisms and
adaptive speciation are studied. For general structured metapopulation
models, fitness is defined in Gyllenberg and Metz (2001). The theory of
structured populations can be applied to metapopulations straightforwardly.
There is analogy between local populations or dispersers and individuals,
and between metapopulation and population. Birth is now colonization of
empty patches by dispersers and death is extinction of a patch population
(Gyllenberg and Metz 2001). Thieme (2003) links age-structured models
with endemic disease models.
There are a limited number of papers dealing with the bifurcation analyses
of (PSP) models. We mention the work of Cushing (1989) and Kirkilionis
et al. (2001). A computer package BASE (Kirkilionis et al. 2001) is available
for continuation of steady states of PSP models. For finite dimensional maps
and ODE systems, computer packages are available. We mention AUTO
(Doedel et al. 1997) and CONTENT (Kuznetsov and Levitin 1997;
Kuznetsov 1998). These packages are in these days heavily used for the
numerical bifurcation analysis of unstructured ecosystem (food chain and
food web) models. Although these packages can be used for continuation
of bifurcation points of PSP models in principle, this is far from easy in
practise. The availability of a bifurcation package for PSP would stimulate
potential users to apply these models for populations as part of food chain
or food web ecosystems.
It would be interesting to have the efficiency of the three numerical
methods—the box method, the Lax–Wendroff, and a characteristics
scheme—be compared with that of the EBT method and those based
on upper and lower solutions techniques. Especially when the support
of the density function is time dependent, the EBT method works fine,
while methods using fixed grids in the i-space are less efficient in those
situations because this grid has to span the largest support in the time interval
of interest. In the case where the population density vanishes almost
everywhere in the i-state space, integration along the characteristics and
the EBT algorithm follows cohorts of individuals through the i-state
space and will work fine. Another possibility is to reduce the dimension of
the i-state space and to transform to a size or an age-structured population
with normal population densities using the so-called Murphy trick (Murphy
1983).
Sometimes mathematical techniques can be used to reduce the PSP model
to equivalent unstructured ODEs or to stage-structured model DDEs (see
Nisbet 1997). In Kooi and Kooijman (1999) and Bocharov and Hadeler
(2000), conservation laws play an important role in the derivation of the
reduced model. When this applies, assumptions often made implicitly
during the construction of unstructured models are made explicit and the
parameters of the reduced model can be expressed as lumped parameters
of the structured model.
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The aggregation technique is an analytical tool which has been used for
unstructured models and is at present applied to structured population
models. Arino et al. (2000) study a model of an age-structured population
with two time scales. The slow scale corresponds to the demographic process
and the fast one describes the migration process between different spatial
patches. Singular perturbation technique is used to show that the solutions
of the system can be approximated by means of the solutions of a simpler
(unstructured) problem.
In some papers PDE formulations were already used in case studies
addressing the issues mentioned above, but in general this is a tremendous
task. In many studies the PSP model formulation has proven to be essential
for realistic population modeling; we mention the study of cannibalism.
Mathematical and=or biological insight is needed to know beforehand
under which conditions the use of a structured instead of an unstructured
population model is essential to have realism that is required for the purpose
for which the mathematical model is used.
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