Modeling food web dynamics using variations of the LotkaVolterra models
Ian J. Anderson
Professor: Ronald M. Coleman
Bio260: Advanced Ecology
Dr. Coleman
Ian
INTRODUCTION
Global biodiversity has been drastically declining due to
human overexploitation and habitat destruction for decades.
To better understand some of the possible effects diversity
losses can have on a biological community, a greater
understanding of biotic interactions and the role they play in
structuring communities are needed. One such method for
achieving this goal is the construction of various ecological
models that can help describe some of the interactions within
natural ecosystems. With the construction of these
mathematical models, researchers are better able to predict,
and possibly mitigate, fluctuations within these systems due to
human influences.
OBJECTIVE
ABSTRACT
The methodology of modeling interactions between
organisms, and manipulating the interactions using a
computational software program, is an invaluable tool
in ecological research. In this project, my goal was to
first expand the Lotka-Volterra predator-prey equations
to describe a simple three-level food chain. Then I
combined the predator-prey and competition models to
describe interactions I have been observing in my own
vernal pool research. Mathematica was then used to
display the various models, and allow the manipulation
of parameter values to depict the response of
organisms to dynamic conditions.
The goal of this project was to manipulate various
mathematical models to describe complex food web
dynamics, and visualize the relationships between organisms
using Mathematica, a computational software program.
algae t , zooplankton t , carnivore t
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Lotka-Volterra predator-prey
equations
METHODS
The first goal of the project was to
manipulate the Lotka-Volterra Predator-Prey
model to illustrate a three species food chain
(Fig.1). The new model was then incorporated
into Mathematica so the interactions between
species could be visualized, and each
parameter could be manipulated (Fig. 2).
The last goal of the project was to combine
both Lotka-Volterra models (predator-prey and
competition) to more accurately illustrate a
dynamic food web of interacting species within
an aquatic phase of a vernal pool (Fig. 3). This
model was also visualized by writing a program
using Mathematica (Fig. 4). Each of the
parameters within the model can be
manipulated, but start at values that illustrate
interactions that have been observed in my own
research.
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• Alterations to model a 3-species chain:
• 1st level:
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dx₁/dt = rx₁ – αx₁x₂
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• 2nd level:
Figure 5: Series of vernal pool mesocosms to demonstrate the
negative impact of tadpoles on macrophyte growth and abundance.
Top photo has no tadpoles. The middle photo has 0.14 tadpoles/L.
The bottom photo has 0.28 tadpoles/L.
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dx₂/dt = βx₁x₂ – qx₂ – αx₂x₃
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• 3rd level:
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dx₃/dt = βx₂x₃ – qx₃
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Figure 1: Manipulated Lotka-Volterra predator-prey equations to model a three
species food chain. Model describes a system where level three feeds upon the
second level which feeds upon the first level.
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14
t
DISCUSSION
Figure 2: Graph shows the interaction of a three species food chain.
Overall, the manipulations of the various Lotka-Volterra
models were successful in modeling complex
interactions between groups of organisms. Variations in
the strength of these interactions could be explored by
manipulating the parameters of the equations within
Mathematica. Ultimately, this feature gives us a better
understanding of the complexities of interactions
between organisms, and possibly the ability to predict
variation in abundances due to biotic fluctuations. As
biodiversity continues to decrease due to anthropogenic
influences, this method of modeling natural systems will
no doubt play a major role in ecological research.
algae t , macrophytes t , tadpoles t , ostracods t
Combining the Lotka-Volterra models
• Algae:
dx₁/dt = rx₁ ((K₁ - x₁ - α₁₂x₂)/K₁) - αx₁x₃ - αx₁x₄
• Macrophytes:
dx₂/dt = rx₂ ((K₂ - x₂ - α₂₁x₁)/K₂) - αx₂x₃
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• Tadpoles:
dx₃/dt = rx₃ ((K₃ - x₃ - α₃₄x₄)/K₃) + βx₃x₁ - qx₃
• Ostracods:
dx₄/dt = rx₄ ((K₄ - x₄ - α₄₃x₃)/K₄) + βx₄x₁ - qx₄
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0.002
Figure 3: Combined Lotka-Volterra predator-prey and competition models to
illustrate the interactions between four groups of vernal pool organisms.
0.004
0.006
Figure 4: Graph shows the interaction of a four-level food web.
0.008
0.010
t