a b s t r a c t
A periodic steady state is a familiar phenomenon in many areas of theoretical biology and provides a
satisfying explanation for those animal communities in which populations are observed to oscillate in a
reproducible periodic manner. In this paper we explore models of three competing species described by
symmetric and asymmetric May–Leonard models, and specifically investigate criteria for the existence
of periodic steady states for an adapted May–Leonard model:
x˙ = r(1 − x − ˛y − ˇz)x
y˙ = (1 − ˇx − y − ˛z)y
z˙ = (1 − ˛x − ˇy − z)z.
Using the Routh–Hurwitz conditions, six inequalities that ensure the stability of the system are
identified. These inequalities are solved simultaneously, using numerical methods in order to generate
three-dimensional phase portraits to illustrate the steady states. Then the “stability boundary” is
defined as the almost linear boundary between stability and instability. All the mathematics discussed
is suitable for advanced undergraduate mathematics or applied mathematics students, offering them
the opportunity to incorporate a computer algebra system such as Mathematica, DERIVE or Matlab in
their investigations. The adapted May–Leonard model provides a practical application of steady states,
stability and possible limit cycles of a nonlinear system.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:tut/oai:encore.tut.ac.za:d1001763 |
| Date | 27 November 2008 |
| Creators | Van der Hoff, Q, Greeff, JC, Fay, TH |
| Publisher | Elsevier |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | |
| Rights | © 2009 Elsevier B.V. |
| Relation | Ecological Modelling |
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