**Course** Description: | Review of some commonly used mathematical approaches to modeling
ecological populations, including single species approaches
(discrete and continuous time), age/stage structure, species
interactions, spatial structure, harvesting, and management. |

**Course Objectives:** | My aim is to teach students with some useful and commonly used
mathematical approaches in understanding ecological systems. The
kinds of techniques explored can be split along classic lines:
"strategic" (models not based on specific systems but useful in
answering hypothetical questions) and "specific" (those needed
for understanding of or making predictions about particular
systems). The overall motivation for the course will be first
and foremost scientific curiosity, though some applications to
management/conservation will also be discussed.
Students will learn some of the necessary mathematical tools
required to model population dynamics, along with the numerical
skills required for their computer analysis and simulation (the
software associated with the course is Matlab). By the end of
the course, each student will have a sound understanding of the
underlying principles in modelling and their applications to
ecological problems. Equally importantly, they will be able to
formulate, evaluate and analyze models relevant to their own
systems of interest. |

**Topical Outline:** | I. Basic Population Ecology. An introduction to the essential
features of ecological populations and the general questions of
interest, which will be tackled in the course. Predominantly
review material providing an overview of things to come in future
lectures.
II. Individual-Level Considerations. Populations are made up of
individuals - what are the small-scale considerations that
determine large-scale phenomena? Discussion of life-history
models, game theoretic approaches and dynamic models.
III. Population Regulation. The first of these two topics will
present a discussion of simple strategic models framed in
continuous- and discrete-time. Methods used to analyse them:
stability analysis, very simple bifurcation analysis and moment
closure techniques. The second will deal with methods of
exploring population regulation in real data (simple time-series
approaches).
IV. Interacting Species. I will give one lecture on the dynamics
and predictions of generalised Lotka-Volterra models (2nd order
ODEs) as applied to competition/predation/herbivory. The second
lecture will focus on similar interactions in discrete-time and
will perhaps include a detailed discussion of host-parasitoid
systems (2nd order difference equations). There will then
follow two lectures each on host-parasite interactions: micro-
and macro-parasitic infections. These will systematically
explore model construction, framework and predictions contrasted
with findings from data. This section may be used to introduce
students to Monte Carlo simulations, seasonally forced models and
wavelet spectral analysis among others.
V. Structured Populations. Many populations are clearly
comprised of different age or stage classes. Are these
important? The first lecture will deal with matrix models as
applied to ecology with a (brief) discussion of Peron-Frobenius
theory and sensitivity analysis. In the second lecture, I propose
to discuss stage-structure in insect populations, introducing
delay differential equations as applied to host-parasitoid
systems.
VI. Spatial Dynamics. Given the ubiquity of spatial structure in
ecological systems, how do we go about exploring and
understanding its consequences? I will start with a discussion
of space at the level of the individual foraging behaviour and
its implications, mostly focused on host-parasitoids, but
plant-herbivores would also work well here. (This will also link
back to some of the evolutionary issues of the first few
lectures.) In the two lectures that follow, I will then explore
larger-scale metapopulation dynamics from a theoretical
perspective (how I construct a spatial model and what does it
tell me?), proceeded by an analysis of some spatial data.
VII. Applied Ecological Modelling. How can we use models to
determine harvesting of populations that is sustainable? The
first lecture will explore this in matrix-type models and the
second will go into Maximum Sustainable Yield type approaches
from fisheries. The next two lectures will develop models from a
pest management and disease control perspective.
These lectures will be augmented with practical classes, which
may be either computer based or pen-and-paper based. For every 2
hours of lectures, there will be two hours of practicals to
ensure a deep understanding of the materials. |