Complex Dynamical Behavior of Holling–Tanner Predator-Prey Model with Cross-Diffusion

Spatial predator-prey models have been studied by researchers for many years, because the exact distributions of the population can be well illustrated via pattern formation. In this paper, amplitude equations of a spatial Holling–Tanner predator-prey model are studied via multiple scale analysis. First, by amplitude equations, we obtain the corresponding intervals in which different kinds of patterns will be onset. Additionally, we get the conclusion that pattern transitions of the predator are induced by the increasing rate of conversion into predator biomass. Specifically, pattern transitions of the predator between distinct Turing pattern structures vary in an orderly manner: from spotted patterns to stripe patterns, and finally to black-eye patterns. Moreover, it is discovered that pattern transitions of prey can be induced by cross-diffusion; that is, patterns of prey transmit from spotted patterns to stripe patterns and finally to a mixture of spot and stripe patterns. Meanwhile, it is found that both effects of cross-diffusion and interaction between the prey and predator can lead to the complicated phenomenon of dynamics in the system of biology.

Vibration control through the robust nonlinear absorber with negative stiffness and internal resonance creation

In this study, a nonlinear absorber that works with a negative stiffness mechanism is suggested to mitigate vibration, and its effect on the reduction of vibration is investigated. The negative stiffness, which is inherently nonlinear, creates internal resonance; therefore, the vibration energy can be transmitted from low-frequency to high-frequency vibrating modes, causing vibration suppression. The nonlinear absorber is added to the primary nonlinear system, and when the main system is subjected to external resonance due to harmonic excitation, the negative stiffness parameter of absorber is so adjusted that autoparametric resonance occurs and vibration is reduced. First, the mathematical model of the system is presented and the governing differential equations of the motion are derived, and then, using the multiple scale method, the equations are solved for the case without, and with the 1:3 internal resonance. The responses and their stability are inspected, discussed, and compared. After that, the effect of negative stiffness and damping parameters on vibration amplitude reduction is investigated and the adequacy of the proposed absorber will be demonstrated by numerical analysis. Finally, the energy exchange between the primary system and the absorber will be demonstrated by plotting the responses in the state space and the displacement response Fourier spectrum.

Scale-Covariant and Scale-Invariant Gaussian Derivative Networks

This paper presents a hybrid approach between scale-space theory and deep learning, where a deep learning architecture is constructed by coupling parameterized scale-space operations in cascade. By sharing the learnt parameters between multiple scale channels, and by using the transformation properties of the scale-space primitives under scaling transformations, the resulting network becomes provably scale covariant. By in addition performing max pooling over the multiple scale channels, or other permutation-invariant pooling over scales, a resulting network architecture for image classification also becomes provably scale invariant. We investigate the performance of such networks on the MNIST Large Scale dataset, which contains rescaled images from the original MNIST dataset over a factor of 4 concerning training data and over a factor of 16 concerning testing data. It is demonstrated that the resulting approach allows for scale generalization, enabling good performance for classifying patterns at scales not spanned by the training data.

Modulation Instability of Hydro-Elastic Waves Blown by a Wind with a Uniform Vertical Profile

An interesting physical phenomenon was recently observed when a fresh-water basin is covered by a thin ice film that has properties similar to the property of a rubber membrane. Surface waves can be generated under the action of wind on the air–water interface that contains an ice film. The modulation property of hydro-elastic waves (HEWs) in deep water covered by thin ice film blown by the wind with a uniform vertical profile is studied here in terms of the airflow velocity versus wavenumber. The modulation instability of HEWs is studied through the analysis of coefficients of the nonlinear Schrödinger (NLS) equation with the help of the Lighthill criterion. The NLS equation is derived using the multiple scale method in the presence of airflow. It is demonstrated that the potentially unstable hydro-elastic waves with negative energy appear for relatively small wind speeds, whereas the Kelvin–Helmholtz instability arises when the wind speed becomes fairly strong. Estimates of parameters of modulated waves for the typical conditions are given.

Spatial Representativeness of Gross Primary Productivity from Carbon Flux Sites in the Heihe River Basin, China

Studying the spatial representativeness of carbon flux measurement data for typical land cover types can provide important information for benchmarking Earth system models and validating multiple-scale remote sensing products. In our study, daily gross primary productivity (GPP) was firstly derived from eddy covariance observation systems and seasonal variations in field GPP were analyzed at nine flux tower sites for typical land cover types in the Heihe River Basin, China. Then, the real-time footprint distance and climate footprint distance of the field GPP were obtained by using a footprint source area model. Lastly, multiple-scale GPP products were validated at footprint scale, and the impacts (measurement height, surface roughness and turbulent state of the atmosphere) on the footprint distance of field GPP were analyzed. The results of this paper demonstrated that climate footprint distances ranged from about 500 m to 1500 m for different land cover types in the Heihe River Basin. The accuracy was higher when validating MODIS GPP products at footprint scale (R2 = 0.56, RMSE = 3.07 g C m−2 d−1) than at field scale (R2 = 0.51, RMSE = 3.34 g C m−2 d−1), and the same situation occurred in the validation of high-resolution downscaled GPP (R2 = 0.85, RMSE = 1.34 g C m−2 d−1 when validated at footprint scale; R2 = 0.82, RMSE = 1.47 g C m−2 d−1 when validated at field scale). The results of this study provide information about the footprints of field GPP for typical land cover types in arid and semi-arid areas in Northwestern China, and reveal that precision may be higher when validating multiple-scale remote sensing GPP products at the footprint scale than at the field scale.

Instability of a dusty Kolmogorov flow

Suspended particles can significantly alter the fluid properties and, in particular, can modify the transition from laminar to turbulent flow. We investigate the effect of heavy particle suspensions on the linear stability of the Kolmogorov flow by means of a multiple-scale expansion of the Eulerian model originally proposed by Saffman (J. Fluid Mech., vol. 13, issue 1, 1962, pp. 120–128). We find that, while at small Stokes numbers particles always destabilize the flow (as already predicted by Saffman in the limit of very thin particles), at sufficiently large Stokes numbers the effect is non-monotonic in the particle mass fraction and particles can both stabilize and destabilize the flow. Numerical analysis is used to validate the analytical predictions. We find that in a region of the parameter space the multiple-scale expansion overestimates the stability of the flow and that this is a consequence of the breakdown of the scale separation assumptions.