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John Alford
Sam Houston State University
Mathematics and Biology
 Biology
has inspired mathematics for
hundreds of years
– Thomas Malthus, An Essay on the
Principle of Populations (1798)
 population
– Gregor Mendel, Experiments in Plant
Hybridization (1865)
 genetic
Mathematical Biology Classic Papers
Volterra, V.
– Fluctuation in the abundance of a species
considered mathematically, Nature (1926)
Kermack, W.O. and McKendrick, A.G.
– Contributions to the mathematical theory of
epidemics, Proc. Roy. Soc. (1927)
Fisher, R.A.
– The wave of advance of advantageous genes,
Ann. Eugenics (1937)
Mathematical Biology Classic Papers
Turing, A.M.
– The chemical basis of morphogenesis, Phil.
Trans. Roy. Soc. Lond. (1952)
Hodgkin, A.L. and Huxley, A.F.
– A quantitative description of membrane
current and its application to conduction and
excitation in nerve, J. Physiology (1952)
Rosenzweig, M.L. and MacArthur, R.H.
– Graphical representation and stability
conditions of predator-prey interactions,
American Naturalist (1963)
Mathematics and Biology
Increases in computing power and
decreases in computing costs has
revolutionized the biological, medical, and
environmental sciences
– gene sequencing data
– GPS data
– etc.
More data has resulted in a greater need
for mathematical and statistical models to
interpret it.
Invasive Species
red imported fire ant
Invasive Species
zebra mussels
Invasive Species
Definition: plants, animals, pathogens and
other organisms that are non-native to an
ecosystem, and which may cause economic
or environmental harm or adversely affect
human health.
Invasive Species
Impact adversely upon biodiversity,
including decline or elimination of native
species - through competition, predation,
or transmission of pathogens - and the
disruption of local ecosystems and
ecosystem functions
Invasive Species
Since the 17th century, invasive species
have contributed to nearly 40% of all
animal extinctions for which the cause is
Invasive Species Control
– The National Invasive Species Information Center
– Pulling Together Initiative, a Texas-sized partnership to
manage non-native invasive plants
– AERF (aquatic ecosystem restoration foundation)
– Aquatic Plant Control Research Program (APCRP) US
Army Corps of Engineers (see APIS link for simulation
Invasive Species
Invasion Stages
– arrival
 natural
or human transportation of a species to a new
– establishment
 the
forming of a self-sustaining population in the new
– integration
 invading
species forms ecological links with other
species in the region
Vermeij, G.J., An agenda for invasion biology,
Biological Conservation, Biological
Conservation, 1996
Invasive Species Control
Fire ant being attacked by a phorid fly
Invasive Species Control
Parasitic flies turn fire ants into zombies
By Bill Hanna, Fort Worth Star-Telegram Bill Hanna, Fort Worth Startelegram – Tue May 12, 2:09 pm ET
It sounds like something out of science fiction: zombie fire ants. But
it's all too real. Fire ants wander aimlessly away from the mound.
Eventually their heads fall off, and they die. "It's a tool — they're not
going to completely wipe out the fire ant, but it's a way to control
their population," said Scott Ludwig , an integrated pest management
specialist with the AgriLife Extension Service in Overton , in East
Texas .
The tool is the tiny phorid fly, native to a region of South America
where the fire ants in Texas originated. Researchers have learned that
there are as many as 23 phorid species along with pathogens that
attack fire ants to keep their population and movements under
The flies "dive-bomb" the fire ants and lay eggs. The maggot that
hatches inside the ant eats away at the brain, and the ant starts
exhibiting what some might say is zombie-like behavior.
Invasive Species Control
 Stop
the arrival (educate the public)
– The Ballad of Aquatic Invasive Species
by Scott Gatzke
Invasive Species Control
 Once
arrival has occurred, the
control options include
– eradication
– suppresion (reducing invading
population densities)
– slowing the spread
Invasive Aquatic Weeds
 Invasive
aquatic weeds
– Hydrilla verticillata
– water hyacinth
– giant salvinia
free-floating fern
– many, many more
Invasive Aquatic Weed: Hydrilla
Invasive Aquatic Weed: Water Hyacinth
Invasive Aquatic Weed: Giant Salvinia
The photo on the right was taken 41 days
after the photo on the left
Invasive Aquatic Weeds
 Negative
– drinking water quality
– flood control
– recreational boating
– habitats for other (native) plants
– fish and wildlife habitat
– etc.
Invasive Aquatic Weeds: Control
$100,000,000+ per year spent to control
aquatic weeds*
Common methods of control
– mechanical
– chemical
– biological
*Rockwell, H.W., Summary of the literature on the
economic impact of aquatic weeds (2003)
Invasive Aquatic Weeds: Biological Control
 Suppression:
classical biological
control of invasive plants
– collect natural enemies (parasites,
predators, or pathogens) of an invasive
plant from its country of origin
– release natural enemies into regions of
– suppress the population density of the
invading plant
Invasive Aquatic Weeds: Biological Control
Examples of Biological Control Agents
– herbivores
leaf-mining flies
grass carp
– pathogens
Invasive Aquatic Weeds: Biological Control
 Four
host-specific insect biocontrol
agents have been introduced in
North America to Hydrilla
 The leaf mining fly H. pakistanae has
been the most successful of the four
(Doyle et. al., 2002)
Invasive Aquatic Weeds: Biological Control
H. pakistanae (species of leaf mining fly)
Invasive Aquatic Weeds: Biological Control
H. pakistanae (species of leaf mining fly)
– small (2 mm in length)
– Not strong flyers (appear to hop from one leaf
to another)
– lays eggs on a plant (each female produces
several hundred eggs during her reproductive
– eggs hatch in 3-4 days and larvae tunnel or
mine plant leaves (9 to 12 leaves during 3
larval stages)
Invasive Aquatic Weeds: Biological Control
H. pakistanae (species of leaf mining fly)
– pupae are formed within a puparium (pupal
stage lasts 6-15 days) and attach to stems
– adult flies emerge from the puparium and float
to the surface in an air bubble
– total development time is 20-35 days (depends
on temperature and nutritional content of the
– total number of generations per growing
season is variable (as high as 7)
Invasive Aquatic Weeds: Biological Control
H. pakistanae (species of leaf mining fly)
– Hydrilla plants that are damaged by H.
pakistanae appear brown
– total photosynthetic area of the plant is
reduced which reduces growth and vigor
– damaged hydrilla are less competitive
with other plants
Invasive Aquatic Weeds: Biological Control
 3,000,000+
flies have been released
in US (including Lake Conroe and
Huntsville state park lake) and have
established at most of the release
sites (Doyle et. al., 2002)
Invasive Aquatic Weeds: Biological Control
 In
general, there are a number of
factors which may suppress the
density of a biological control agent
such as H. pakistanae
– abiotic: cold winters, heavy rainfall
– biotic: natural enemies, poor plant
 This
is an active area of research in
(Wheeler G.S. and Center T.D., 2001)
Invasive Aquatic Weeds: Biological Control
The LURE Project
 Generally
– Learn some fundamental principles of
nonlinear dynamical system
– Learn some fundamental principles of
mathematical ecology
(more on these later)
The LURE Project
 Specifically
– Formulate a dynamical systems model
(or models) for the biological control of
aquatic weeds by insect herbivory
– Analyze the model (or models) to
investigate how biotic and/or abiotic
factors influence success or failure of
the biocontrol agent
The LURE Project
 Problem
–Formulate and analyze a
mathematical model to
describe the population
dynamics of an aquatic weed
and an insect herbivore
The LURE Project
 Problem
–Formulate and analyze a
mathematical model to
describe the population
dynamics of two aquatic weeds
competing for resources
The LURE Project
 Problem
–Formulate and analyze a
mathematical model to describe
the population dynamics of two
aquatic weeds competing for
resources and an insect herbivore
which selectively eats one type of
weed but not the other
The LURE Project
– Increasing plant quality by manipulating
nutrient content (fertilizer) may also increase
the level of control (damage to plant) by the
insect; but there exists a point of diminishing
return for which too much fertilizer results in
the plants outgrowing the damage from the
insects (Cuda, 2008; Coetzee, 2007).
Determine if there is an optimal amount of
fertilizer that will promote insect herbivory but
not decreases in insect density.
The LURE Project
 Problem
– Investigate the role that temperature
has in the population dynamics of
aquatic weeds and insect herbivores.
The LURE Project
Reading List (Journal Articles)
– Coetzee J.A. et. al., Impact of nutrients and
herbivory by Eccritotarsus catarinensis on the
biological control of water haycinth, Eichhornia
crassipes, Aquatic Botany, 2007
– Cuda, J.P. et. al., Recent advances in biological
control of submersed aquatic weeds, Journal of
Aquatic Plant Management, 2008
– Doyle R. et. al., Impact of herbivory by H.
pakistanae on growth and photosynthetic
potential of Hydrilla, Biological Control, 2002
The LURE Project
Reading List (Journal Articles)
– Doyle R. et. al., Separate and interactive
effects of competition and herbivory on the
growth, expansion, and tuber formation of
Hydrilla verticillata, Biological Control, 2007
– Wheeler G.S. and Center T.D., The influence of
hydrilla leaf quality on larval growth and
development of the biological control agent
Hydrellia pakistanae, Biological Control, 1996
– Van, T.K. et. Al., Competitive interactions
between Hydrilla and Vallisneria as influenced
by insect herbivory, Biological Control, 1998
The LURE Project
 Reading
List (Journal Articles)
– Wilson, J.R. et. al., Determinants and
patterns of population growth in water
hyacinth, Aquatic Botany, 2005
– Shukla V.P., Modeling the dyamics of
wetland macrophytes: Keoladeo
National Park wetland, India, Ecological
Modelling, 1998
The LURE Project
Reading List (Journal Articles)
– Edelstein-Keshet L. and Rausher M.D., The effects of
inducible plant defenses on herbivore populations 1.
mobile herbivores in continuous time, The American
Naturalist, 1989
– Lewis M.A., Spatial coupling of plant and herbivore
dynamics: the contribution of herbivore dispersal to
transient and persistent waves of damage, Theoretical
Population Biology, 1992
– Noy-Meir I., Stability of grazing systems: an application of
predator-prey graphs, Journal of Ecology, 1975