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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Prior research in the field of boundary layer turbulence has identified streamwise-elongated large-scale turbulence structures in both low speed compressible and high speed (M=2.0) flow. No experimental work has been done in any flow of M> or =3 in an attempt to identify the presence or quantify the behavior of these structures, nor has any study of favorable pressure gradient or surface roughness element effects on these structures been conducted. This research used high-resolution Particle Imaging Velocimetry in a M = 4.9 blow-down wind tunnel accompanied by a series of data analysis in order to identify the existence of streamwise-elongated large-scale turbulence structures in a hypersonic boundary layer. Furthermore, this study identified physical and statistical behavior which suggests that increasing favorable pressure gradient had a substantial impact on both the structural coherence and relative intensity of these turbulent structures at all boundary layer heights tested. This experiment also identified similar effects on these structures in the lower half of the boundary layer as a result of the introduction of surface roughness elements. Finally, several trends were identified between the averaged turbulence statistics and the behavior of the large-scale streamwise-elongated turbulence structures present in this study.

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