Asymptotic behaviour of the Darcy-Boussinesq system at large Darcy-Prandtl number
Rana D. Parshad
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1441-1469 doi: 10.3934/dcds.2010.26.1441
We study asymptotic behavior of the Darcy-Boussinesq system at large Darcy-Prandtl number. We prove that the global attractors for this system converge to that of the infinite Darcy-Prandtl number model. We also show the convergence of statistical properties including invariant measures.
keywords: Global Attractor Darcy-Prandtl Number Darcy-Boussinesq system Singular Perturbation. Invariant Measures
On the global attractor of the Trojan Y Chromosome model
Rana D. Parshad Juan B. Gutierrez
Communications on Pure & Applied Analysis 2011, 10(1): 339-359 doi: 10.3934/cpaa.2011.10.339
We consider the Trojan Y Chromosome (TYC) model for eradication of invasive species in population dynamics. We present global estimates for the TYC system in a spatial domain. In this work we prove the existence of a global attractor for the system. We derive uniform estimates to tackle the question of asymptotic compactness of the semi-group for the TYC model in $H^2(\Omega)$. This along with the existence of a bounded absorbing set, which we also derive, demonstrates the existence of a global attractor for the TYC model. The present analysis reveals that extinction of an invasive species is always possible to achieve irrespective of geometric considerations of the domain. This result is valid for TYC systems in which advection is negligible. This theoretical work lays the foundation for experimental studies of the application of the TYC eradication strategy in spatial ecology, since the outcome is in principle guaranteed.
keywords: TYC model population dynamics eradication. invasive species global attractor
A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners
Rana D. Parshad Stephen J. McGregor Michael A. Busa Joseph D. Skufca Erik Bollt
Mathematical Biosciences & Engineering 2012, 9(1): 123-145 doi: 10.3934/mbe.2012.9.123
Control entropy (CE) is a complexity analysis suitable for dynamic, non-stationary conditions which allows the inference of the control effort of a dynamical system generating the signal [4]. These characteristics make CE a highly relevant time varying quantity relevant to the dynamic physiological responses associated with running. Using High Resolution Accelerometry (HRA) signals we evaluate here constraints of running gait, from two different groups of runners, highly trained collegiate and untrained runners. To this end, we further develop the control entropy (CE) statistic to allow for group analysis to examine the non-linear characteristics of movement patterns in highly trained runners with those of untrained runners, to gain insight regarding gaits that are optimal for running. Specifically, CE develops response time series of individuals descriptive of the control effort; a group analysis of these shapes developed here uses Karhunen Loeve Analysis (KL) modes of these time series which are compared between groups by application of a Hotelling $T^{2}$ test to these group response shapes. We find that differences in the shape of the CE response exist within groups, between axes for untrained runners (vertical vs anterior-posterior and mediolateral vs anterior-posterior) and trained runners (mediolateral vs anterior-posterior). Also shape differences exist between groups by axes (vertical vs mediolateral). Further, the CE, as a whole, was higher in each axis in trained vs untrained runners. These results indicate that the approach can provide unique insight regarding the differing constraints on running gait in highly trained and untrained runners when running under dynamic conditions. Further, the final point indicates trained runners are less constrained than untrained runners across all running speeds.
keywords: statistical hypothesis testing Control entropy gait analysis.

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