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Defining a stability boundary for three species competition modelsVan der Hoff, Q, Greeff, JC, Fay, TH 27 November 2008 (has links)
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.
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Stable Coexistence of Three Species in CompetitionCarlsson, Linnéa January 2009 (has links)
<p>This report consider a system describing three competing species with populations <em>x</em>, <em>y</em> and <em>z</em>. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.</p>
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Stable Coexistence of Three Species in CompetitionCarlsson, Linnéa January 2009 (has links)
This report consider a system describing three competing species with populations x, y and z. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.
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