## Markovian Building Blocks for Individual-Based Modelling

Publication: Research › Ph.D. thesis – Annual report year: 2007

The present thesis consists of a summary report, four research articles, one
technical report and one manuscript. The subject of the thesis is individual-based
stochastic models.
The summary report is composed of three parts and a brief history of some
basic models in population biology. This history is included in order to provide
a reader that has no previous exposure to models in population biology
with a sufficient background to understand some of the biological models that
are mentioned in the thesis. The first part of the rest of the summary is a
description of the dramatic changes in the degree of aggregation of sprat or
herring in the Baltic during the day, with special focus on the dispersion of the
fish from schools at dusk. The next part is a brief introduction to Markovian
arrival processes, a type of stochastic processes with potential applications as
sub-models in population dynamical models. The last part introduces Markov
additive processes as a means of simplifying some individual-based models.
In the first part I present the background to article A and some extra material
that were not included in the final article. The basic observation is that fish
in schools migrate up toward the surface and disperse at dusk and aggregate in
schools close to the bottom at dawn. This creates a periodically varying prey
field to cod. Apart from humans, cod is the main predator of herring and sprat
in the Baltic. In order to evaluate the consequences to cod of this variability
it was necessary to describe this prey field. It was shown that the schools
follow lines of constant light intensity and that they disperse below a critical
light threshold. We propose that the dispersion is due to a random walk when
light levels become sub-critical and provide time-scales for the dispersion of this
type for different school geometries (random or on a regular square grid)—the
time-scales are of the same order as those observed on the echosounder.
The second part is an introduction to Markovian arrival processes (MAPs), this
is the background needed to understand papers C, B, E, and F, given some
previous exposure to Markov chains in continuous time (see e.g. Grimmett and
Stirzaker, 2001)). Markovian arrival processes are very general point processes
that are relatively easy to analyse. They have, so far, been largely unknown
to the ecological modelling community. The article C deals with a functional
response in a heterogeneous environment. The functional response is a model of
the mean ingestion rate of prey per predator as a function of prey and possibly
predator density that appears in most models for populations. A previously proposed
model for prey encounter in heterogeneous environments is reanalyzed, it
is a stochastic process that easily can be implemented as a MAP. In article C we
show that transferring a standard functional response to a heterogeneous environment
does not preserve the functional form, contrary to previous assertions.
In this simple case we provide a time-scale for when the heterogeneous environment
can be assumed to be well-mixed, or close to a Poisson process, for the
predator. It is also shown that in some cases the variability may be more important
than the mean, thus the mean rate does not necessarily provide sufficient
information for the population dynamics. Article B provides the mathematical
apparatus for evaluating any moment of a MAP, and also the means for evaluating
the conditional moments of a transient or terminating MAP. Transient
MAPs are suitable as modelling tools when an important property of the system
is that it can stop. This is the case for the young of many animals, where
most of a large clutch die rather quickly, and yet it is the survivors that are
interesting. The conditional moments can for instance be constructed such that
one can evaluate the mean or the variance of the ingestion rate given that the
animal did not die. Several different methods are used to obtain the formulas,
which is an interesting aspect since some of these methods may be more suitable
in situations where it is problematic to proceed using the standard formalism.
I provide material on how to model periodic MAPs in paper E. These are, or
could be, important since most animals live in a periodic environment and a periodic
system generally have dynamics that are different from the corresponding
system with mean rates. The technical report F concerns how to model
Markovian stomachs. Both aspects can be used in more advanced functional or
numerical responses.
The third part concerns a larger class of Markov processes, to which the above
mentioned MAPs belong. These are the Markov additive processes, which are
bivariate Markov processes (Xt,Nt) where the transition probabilities depend
on the Xt process only. The Xt process is marginally a Markov process, and
the Nt process is a process with conditionally independent increments given the
state of the Xt process. This class is rich enough to provide substantial realism
into individual-based models yet it is so simple that it is not a great extra burden
to solve the partial differential equations (PDEs) that arise for the evaluation
of the moments. They are particularly useful in oceanographic contexts since
here the apparatus for solving the PDEs is usually present due to the need of
solving fluid flow equations. The greatest benefit of the method is due to that it
circumvents the need for statistical evaluation of the individual-based models.
In all three parts further work has been proposed.

Original language | English |
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Publication date | Feb 2007 |

State | Published |
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Name | IMM-PHD-2008-171 |
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