Rana D. Parshad

Lower extremity fatigue increases complexity of postural control during a single-legged stance

BackgroundNon-linear approaches to assessment of postural control can provide insight that compliment linear approaches. Control entropy (CE) is a recently developed statistical tool from non-linear dynamical systems used to assess the complexity of non-stationary signals. We have previously used CE of high resolution accelerometry in running to show decreased complexity with exhaustive exercise. The purpose of this study was to determine if complexity of postural control decreases following fatiguing exercise using CE.MethodsTen subjects (5 M/5 F; 25 ± 3 yr; 169.4 ± 11.7 cm; 79.0 ± 16.9 kg) consented to participation approved by Western Oregon University IRB and completed two trials separated by 2-7 days. Trials consisted of two single-legged balance tests separated by two Wingate anaerobic tests (WAnT; PreFat/PostFat), or rest period (PreRest/PostRest). Balance tests consisted of a series of five single-legged stances, separated by 30 s rest, performed while standing on the dominant leg for 15-s with the participant crossing the arms over the chest and flexing the non-dominant knee to 90 degrees. High resolution accelerometers (HRA) were fixed superficial to L3/L4 at the approximate center of mass (COM). Triaxial signals from the HRA were streamed in real time at 625 Hz. COM accelerations were recorded in gs for vertical (VT), medial/lateral (ML), and anterior/posterior (AP) axes. A newly developed statistic (R-test) was applied to group response shapes generated by Karhunen Loeve (KL) transform modes resulting from Control Entropy (CE) analysis.ResultsR-tests showed a significant mean vector difference (p < .05) within conditions, between axes in all cases, except PostFat, indicating the shape of the complexity response was different in these cases. R-test between conditions, within axis, differences were only present in PostFat for AP vs. PreFat (p < .05). T-tests showed a significantly higher overall CE PostFat in VT and ML compared to PreFat and PostRest (p < .0001). PostFat CE was also higher than PostRest in AP (p < .0001).ConclusionsThese data indicate that fatiguing exercise eliminates the differential complexity response between axes, but increases complexity in all axes compared to the non-fatigued condition. This has implications with regard to the effects of fatigue on strategies of the control system to maintain postural control.

Turing patterns and long-time behavior in a three-species food-chain model

We consider a spatially explicit three-species food chain model, describing generalist top predator-specialist middle predator-prey dynamics. We investigate the long-time dynamics of the model and show the existence of a finite dimensional global attractor in the product space, L(2)(Ω). We perform linear stability analysis and show that the model exhibits the phenomenon of Turing instability, as well as diffusion induced chaos. Various Turing patterns such as stripe patterns, mesh patterns, spot patterns, labyrinth patterns and weaving patterns are obtained, via numerical simulations in 1d as well as in 2d. The Turing and non-Turing space, in terms of model parameters, is also explored. Finally, we use methods from nonlinear time series analysis to reconstruct a low dimensional chaotic attractor of the model, and estimate its fractal dimension. This provides a lower bound, for the fractal dimension of the attractor, of the spatially explicit model.

Finite Time Blowup in a Realistic Food-Chain Model

We investigate a realistic three-species food-chain model, with generalist top predator. The model based on a modified version of the Leslie-Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. We show that the model exhibits finite time blowup in certain parameter range and for large enough initial data. This result implies that finite time blowup is possible in a large class of such three-species food-chain models. We propose a modification to the model and prove that the modified model has globally existing classical solutions, as well as a global attractor. We reconstruct the attractor using nonlinear time series analysis and show that it pssesses rich dynamics, including chaos in certain parameter regime, whilst avoiding blowup in any parameter regime. We also provide estimates on its fractal dimension as well as provide numerical simulations to visualise the spatiotemporal chaos.

A remark on “Study of a Leslie–Gower-type tritrophic population model” [Chaos, Solitons and Fractals 14 (2002) 1275–1293]

In Aziz-Alaoui (2002) a three species ODE model, based on a modified Leslie–Gower scheme is investigated. It is shown that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, which eventually enter an invariant attracting set. We show that this is not true. To the contrary, solutions to the model can blow up in finite time, even under the restrictions derived in Aziz-Alaoui (2002), if the initial data is large enough. We also prove similar results for the spatially extended system. We validate all of our results via numerical simulations.

Analysis of the Trojan Y chromosome model for eradication of invasive species in a dendritic riverine system

The use of Trojan Y chromosomes has been proposed as a genetic strategy for the eradication of invasive species. The strategy is particularly relevant to invasive fish species that have XY sex determination system and are amenable to sex-reversal. In this paper we study the dynamics of an invasive fish population occupying a dendritic domain in which Trojan individuals bearing multiple Y chromosomes have been released as a means of eradication. We demonstrate the existence of a bounded absorbing set that represents extinction of the invasive species irrespective of the dendritic configuration. The method of analysis used to obtain global estimates could be applied to other population problems and other geometries.

A comparison of the Trojan Y Chromosome and daughterless carp eradication strategies

Two autocidal genetic biocontrol methods have been proposed as a means to eliminate invasive fish by changing the sex ratio of the population: the Trojan Y Chromosome (TYC) strategy and the Daughterless Carp (DC) strategy. Both strategies were modeled using ordinary differential equations that allow the kinetics of female decline to be assessed under identical modeling conditions. When compared directly in an ordinary differential equation (ODE) model, the TYC strategy was found to result in female extinction more rapidly than a DC strategy (in each of three models tested in which the Daughterless autocidal fish contained an aromatase inhibitor gene in either two or eight copies). The TYC strategy additionally required the introduction of fewer autocidal fish to the target population to achieve local extinction of females as compared to the DC approach. The results suggest that the relatively lower efficiency of female reduction associated with the DC approach is a consequence of a greater capacity to produce females and also a reduced capacity to produce males as compared to the TYC system.

A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners.

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. 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 T² 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.

Prey cannibalism alters the dynamics of Holling–Tanner-type predator–prey models

Cannibalism, which is the act of killing and consumption of conspecifics, has been considered primarily in the predator, despite strong ecological evidence that it exists among prey. In the current manuscript, we investigate both the ODE and spatially explicit forms of a Holling–Tanner model, with ratio-dependent functional response, and show that cannibalism in the predator provides a stabilizing influence as expected. However, when cannibalism in the prey is considered, we show that it cannot stabilize the unstable interior equilibrium in the ODE case, in certain parameter regime, but can destabilize the stable interior equilibrium, leading to a stable limit cycle or “life boat” mechanism, for prey. We also show that prey cannibalism can lead to pattern forming Turing dynamics, which is an impossibility without it. The effects of a stochastic prey cannibalism rate are also considered.

Stochastic models for the Trojan Y-Chromosome eradication strategy of an invasive species

ABSTRACT The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we develop a Markov jump process model for this strategy, and we verify that there is a positive probability for wild-type females going extinct within a finite time. Moreover, when sex-reversed Trojan females are introduced at a constant population size, we formulate a stochastic differential equation (SDE) model as an approximation to the proposed Markov jump process model. Using the SDE model, we investigate the probability distribution and expectation of the extinction time of wild-type females by solving Kolmogorov equations associated with these statistics. The results indicate how the probability distribution and expectation of the extinction time are shaped by the initial conditions and the model parameters.

Long time dynamics of a three-species food chain model with Allee effect in the top predator

The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the model, that is upper semicontinuous in the Allee threshold parameter m . Next, we numerically investigate the decay rate to a target attractor, that is when m = 0 , in terms of m . We find decay estimates that are O ( m γ ) , where γ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m . Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.