Mathematical Biology
RESEARCH
EXPERIENCES FOR UNDERGRADUATES AT
Interdisciplinary Training for
Undergraduates in
Biological and Mathematical Sciencesive
(UBM)
Theoretical
Frameworks for Ecological Dynamics Subject to Stoichiometric Constraints
Project
Director: Yang Kuang
Yang Kuang kuang@asu.edu (Principal Investigator current)
James J. Elser j.elser@asu.edu (Co-Principal Investigator
current)
William F. Fagan bfagan@glue.umd.edu (Co-Principal
Investigator current)
John Nagy
john.nagy@sccmail.maricopa.edu (Co-Principal
Investigator current)
Student Participants: Amy
Novotny, Brian Blaugrund, Jonathon Winkler, Alissa Holburn, Daniel
Nelson.
Student Projects: Amy Novotny, Brian Blaugrund,
Jonathon Winkler,
Daniel Nelson.
Differential equations is a central area of
mathematics,
and one of their recent and most exciting applications is in
mathematical
biology. Applications
in mathematical biology concern ecology,
epidemics, genetics, cellular and neural modeling, physiology, and population
dynamics.
Biological stoichiometry is the study of the balance of energy and
multiple chemical elements in living systems. In biological
stoichiometry, the primary focus is on the relative abundance of the
elements carbon (C), nitrogen (N), and especially phosphorus (P) in
organisms and their implications for individual growth, population
dynamics, community structure, ecosysytem processes, and evolutionary
change. It has its roots in the work of Lotka, who in 1925 was one of
the first to consider how thermodynamic laws of physical-chemical
systems structure the living world. Lotka's thinking echoes in concepts
that are now cornerstones of ecology: optimal foraging, resource ratio
competition theory (Tilman 1982), the Redfield ratio in oceanic
biogeochemical cycling (Redfield 1958), and nutrient use efficiency
(Vitousek 1982). Most recently these ideas have come to be actively
applied in study of aquatic food webs, as disparities in elemental
composition between animals and their food can affect animal feeding
behavior, consumer population stability and community organization,
ultimately impinging on trophic dynamics and biogeochemical cycling in
food webs (Reiners 1986, Hessen 1997, Sterner and Elser 2002).
A key to understanding the biological importance of stoichiometric
variation is that phosphorus-rich species appear to have dramatically
higher growth rates than phosphorus-poor species (Elser et al. 1996,
2000). Phosphorus appears critical because in organisms lacking major
mineral storage of P (as in bones), biomass C:N:P ratios depend largely
on the disproportionate demands for P-rich ribosomal RNA in rapidly
growing cells (Hessen and Lyche 1991, Elser et al. 1996, 2000). This
growth rate hypothesis provides
a powerful and effective mechanism
through which cellular allocation and ecological dynamics are connected.
As we detail below, one way of thinking about the ecological
implications of biological stoichiometry is in terms of “food
quality,” where one can interpret a species' quality (as a resource
relative to the stoichiometric demands of consumer species) in terms of
the relative abundance of C (as a rough energy currency) and some
nutrient element (such as N or P). Furthermore, because plant
(resource) species can vary widely in
chemical composition but herbivore (consumer) species cannot, the
ecological implications of stoichiometric variation cannot be
rigorously explored by arbitrarily slashing coefficients of trophic
transfer efficiency in conventional ecological models. Instead, a
new theoretical
framework is required to analyze the dynamics that result from these
stoichiometric constraints and feedbacks. This is the essence of
our DMS project. The groundwork of this framework is the “LKE”
consumer-resource model (Loladze, Kuang and Elser, 2000), which builds
on the early work of Andersen (1997) and reveals unexpectedly rich,
qualitatively novel, and ecologically realistic dynamics not previously
seen in non-stoichiometric predator-prey models.
Here we outline one such framework and some of its implications. We
explicitly assume that resource and consumer species are composed of
two or more essential elements (e.g, C, P) and that resource species
can vary in their elemental content, but that consumer species cannot
(approaches relaxing this strict homeostasis assumption can also be
explored). Using stoichiometric principles, we propose
construction and analysis of particular Lotka-Volterra type models (and
a suite of variants) that incorporate chemical heterogeneity within and
among trophic levels, resulting in, among other things, the potential
for: 1) indirect competition between resource and consumer for a
shared, limiting nutrient; 2) deterministic extinction of consumer
species despite an abundant resource; and, perhaps most startling, 3)
the potential for a large number of consumer species to coexist while
sharing a single prey population (even in the utter absence of spatial
heterogeneity and similar factors known to facilitate competitive
coexistence). In addition, LKE model possesses three important
properties: 1) it has a low dimension of two. 2) it involves a minimum
number of parameters. 3) it embodies very rich and biologically
realistic assumptions.
Potential REU projects
Since the beginning of this project in year 2000, we have expanded and
extended the LKE model into several plausible ecological models
incorporating features such as 1) multi-nutrient dynamics, 2)
multi-species interaction, 3) delayed nutrient recycling, 4) spatial
heterogeneity and 5) age structure. The work on 2) constitutes the main
work in Irakli Loladze’s Ph.D dissertation in 2001 (Thesis title: The
Importance of Being Soichiometric: Population Dynamics from the
Perspective of Chemical Elements. May, 2001). The work on 4)
constitutes the main work in Chris Miller’s Ph.D dissertation in 2002
(Thesis title: Modeling and Analysis of Stoichiometric Two-Patch
Consumer-Resource Systems. December, 2002). Work on 1) is
relatively straightforward and will be reported as part of a research
paper on 2) soon. Significant work is yet to be carried out in the
areas of 3) and 5). To gain insights into these two areas, a large
amount of simulation work and some pertinent experiments will be needed
for guidance and calibration. Such work is ideal for REU projects. In
the following, we list three main possibilities for undergraduate
students who want to explore the exciting stoichiometry framework for
population biology and beyond.
1): Nutrient recycling in food-web models. The
proposed work is in the center of research area 3) mentioned
above. Researchers hypothesize that fast-growing organisms
allocate more resources to the production of ribosomal RNA than do
slow-growing organisms (Elser et al. 1996, 2000). Because RNA is
rich in P, this allocation to RNA can lead to increases in tissue P
content of fast-growing organisms. Thus, if plants grown in
P-rich soils have higher foliar P content, this P-rich food should
result in higher P and RNA contents in herbivores feeding off these
leaves, thus leading to higher growth rates in these herbivores.
Data from terrestrial ecosystems supporting these linkages have emerged
from the IRCEB project (Schade et al. 2003). Thus, the
biogeochemical cycling of P in soils and plants is potentially linked
to the growth of herbivores, and ultimately, to their population
dynamics.
2): Stage-structured population models with stoichiometry. The
second possibility comes from area 5) mentioned above. The models will
keep track of the number of juveniles and adults (meaning reproducing
individuals) in a given population. We will perform several experiments
examining the life table (growth, fecundity schedules, mortality) of
Daphnia growing in various controlled settings (involving high and low
quantities of food of either high or low stoichiometric food
quality). We will then model the resultant population dynamics
using the parameters obtained from the feeding experiments, eventually
comparing output with actual population dynamics in longer-term Daphnia
experiments. For such modeling work, difference and stochastic
equations are appropriate and continuous ordinary and delay
differential equations can be used for approximation and serve as
references. These kinds of difference and stochastic models are easy to
formulate, simulate and are rich in dynamics. In addition, we
will perform bifurcation analysis and data fitting when appropriate and
necessary.
3): Stoichiometrically explicit models of adaptive dynamics. Another
possibility is to explore stoichiometry-based evolutionary population
models. The biological guideline for this effort will be a
modified or expanded Red Queen Principle: dynamical food quality
determines genetic make up which in turn determines growth performance
and population dynamics, but population dynamics change the food
quality as well as its supply and the cycle feeds back on itself. This
demands explicit modeling of resource quantity and quality in the
context of the “struggle for existence”. We will divide our
effort into two contrasting classes: population growth in slow and fast
evolution settings. In view of the rich and realistic dynamics
generated by predator-dependent models, it will be interesting to see
if such dynamics will be maintained and even enriched when we add
stoichiometry and evolutionary constraints.
Finally, we will model the dynamics of populations evolving for long
time scales. There is a heuristic approach and thus a
short cut on this time scale modeling: we explicitly model the
phosphorous (P) storage of individual species. Surplus of P after
filling the storage signals the need of making more copies of rRNA
which in term speeds up growth but dampens or even depletes the supply
of P, which in time may empty the storage and slow down growth.
4): Stoichiometrically explicit models of tumor dynamics. There
is a growing interest in understanding disease dynamics among
mathematical and biological undergraduate students. This often requires
explicit models that keep track of important limiting nutrients since
disease (virus, tumor, bacterial) can often be viewed as an invading
organism that will compete with healthy organism for the limiting
resources. This makes the stoichiometry based population models
standout candidates. In the last two years, together with Professor
John Nagy of the biology department in Scottsdale Community college, we
developed some realistic models for tumor growth (Kuang, Nagy and
Elser 2004, Elser, Nagy and Kuang
2003). This work will be extended in several
directions in the near future and we expect some of these directions
will be appealing to some REU participants to this supplemental work.
An excellent reference book is the survey monograph of Adam and Bellomo
1997.
Project Duration. Summer
research experience for students will last 8
weeks. We expect the students to learn basic ecological stoichiometry
theory (using the timely book of Sterner and Elser 2002), skills of
MATLAB programming and model formulation (using the excellent book of
Meerschaert 1999) in the first two weeks. After that, the students will
team up to plan some simple experiments involving collecting field data
(for research problems in class 1), or growing Daphnia in various
nutrient settings (for research problems in class 2 and 3). Each team
will likely consists of a mathematics student or a student with strong
mathematical background and a biology student. Experiments are expected
to last 3 weeks or so. While carrying out these experiments, students
will be assigned with some reading work. In addition, they will start
to formulate, analyze and simulate (involving some serious adaptation
of existing MATLAB and Maple programs, most of them are written by Yang
Kuang for his classes and research work) the models. Once the
experiments are completed, data processing and fitting will follow,
which is expected to last one week or so. The last two weeks will be
devoted to writing. Each student participant is expected to write up a
report about 20 pages.
Student Selection. Serious
effort will be made in order to attract top
participants from underrepresented groups and two-year colleges. In
fact, We have intensive collaboration on stoichiometry based tumor
models and analysis with Professor John Nagy, of the biology
department of Scottsdale Community College, who continues to work with
us in identifying prospect participants from his department.
Intergration with Existing REU
activities at ASU. REU supplement
students working on this proposal will interact in weekly meetings with
60-80 other undergraduate researchers in the biology department who are
funded through NSF, NIH and Howard Hughes medical Institute. This will
help the integration of these students with the department of biology’s
undergraduate research program. In addition, there is a symposium each
august for presentation of undergraduate summer research project and we
expect our REU students to actively participate.
For related information on mathematical
biology at ASU, read this article in Science magazine's SCIENCE
ONLINE.
Kuang,
Yang