1. A Computational Analysis of the Lotka-Volterra Equations Delon Roberts, Dr. Monika Neda Department of Mathematical Sciences, UNLV
INTRODUCTION
The Lotka-Volterra equations(fig 1.) are an intertwined system of nonlinear differential equations.
These equations are essential in mathematical biology and are used to model a predator and prey interacting in an environment.
They have enormous significance because any interaction between a consumer and a consumed may be framed as a “predator-prey” problem: a fox chasing a rabbit, cancer attacking the human body, or two countries going to war.
Figure 1
dx dt represents the change in the prey population.
dy dt represents the change in the predator population.
α represents the proportional increase in the prey
population by reproduction.
β represents the interaction parameter of the prey with its
predator.
ϒ represents the proportional increase in the predator
population.
δ represents the interaction of the predator with its predator
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Figure 2
Figure 4
EXPECTED FINDINGS
We expect to find more robust numerical solutions for systems of nonlinear differential equations that describe natural phenomena from our environment.
These numerical solutions will help us deeply understand the essence of biological processes and then we can use them to predict the future behavior of the phenomena.
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FUTURE RESEARCH
PURPOSE/AIM
Figure 5
Computational analysis of differential equations involving predator-prey interactions but with more parameters such as harvesting.
Computational methods for partial differential equations will also be studied (figure 6).
The creation of stable and computationally robust algorithms for studying coupled biological systems of nonlinear differential equations that cannot be solved by hand.
Modern biology has exceeded the human ability to adequately solve equations by hand. As a consequence, specialists` have no alternative but to program computers to approximate the solutions in the name of technological progress. The creation of algorithms to effectively solve complex mathematical equations is therefore essential.
Figure 3
ACKNOWLEDGEMENTS
We acknowledge our Department of Mathematical Sciences and Office of Undergraduate Research – OUR, at UNLV.
Figure 6
METHODS
The primary programming language we are using is Matlab, but C++ may be employed. Our methodology consists of
studying present numerical techniques(fig.3 and 4.) for solving systems of differential equations.
Improving their stability and robustness.
Applying the more stable and robust techniques to obtain numerical solutions.
REFERENCES
1) ’Elementary Differential Equations And Boundary Value Problems’ Boyce and Deprima 6th Ed.
2)Numerical Analysis’ Burden and Faires, 9th Ed.
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