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.

ABOUT CHOOSING THE METHOD OF EXACT APPROXIMATIONS OF DISCRETE STATISTICS PROBABILITY DISTRIBUTIONS

In the paper, we consider the methods of exact approximations of statistics probabilities distribution. As the exact approximations, we consider ∆-exact distributions. The difference between the ∆-exact distributions and the exact approximations does not exceed a predefined arbitrary small value ∆ that defines the accuracy of the approximations. Besides, we consider the methods of the first and second multiplicity, which use statistic characteristics of samples. The first multiplicity method is based on the properties of the components of the first multiplicity vector, which are nonnegative integer solutions of a linear equation. The linear equation relates the alphabet sign frequency and the sample size. The second multiplicity method is based on the solution of a system of linear equations. The linear equations of the system relate the sample size and the alphabet cardinality with the number of the alphabet signs that have equal frequency in the sample. For the considered methods of exact approximations, we give expressions to estimate the computational complexity of exact approximations of distributions for any sample parameters. To provide the approximations accuracy of 10–5, and the computing resource with the performance of 1018 operations per second, we calculated the sample parameters. For these samples, we can calculate the exact approximations of distributions, using the considered methods, the available computing resource, and the declared accuracy. We formed the parameter regions for the samples, and the exact approximations of distributions can be calculated for these samples with the help of various methods. We compared the regions themselves and with the so-called region of uncertainty, which is limited from above not more than 5-fold excess of the sample size over the alphabet cardinality. On the base of the comparison of the parameter regions of the samples, which are suitable for calculation of the exact approximations of the distributions, we compared their calculation methods. It is shown that owing to the second multiplicity method, we can make calculations for all values of the alphabet cardinality from 2 to 256. In contrast to the second multiplicity method, the first multiplicity method does not allow calculations for the alphabet cardinality over 73. The parameter region of the samples, which are suitable for calculation of the limit approximations of the distributions by the second multiplicity method, contains the complete parameter region of the samples, suitable for calculation of the limit approximations of the distributions by the first multiplicity method, and exceeds it more than in 52 times. Owing to the comparison of the methods of exact approximations, it is proved that if we have the same computing resource, we can calculate the exact approximations with the help of the second multiplicity method for a greater number of samples with the increased parameters in comparison with the first multiplicity method. Hence, to calculate the exact approximations of statistics probability distributions, we choose the second multiplicity method. Practical significance of the research is possibility of calculation of the maximal values of the sample parameters. The current technological level of computer systems allows calculation of the exact approximations of the distributions for these values, which provide the minimal loss of criteria efficiency in comparison with the limit approximations used for the sample parameters. The scientific novelty of the research is the comparative analysis of the methods of exact approximations of distributions for calculation of distributions for the sample parameters, which do not allow calculation of the exact distributions due to their high computational complexity.

Analysis and Correction of the Attack against the LPN-Problem Based Authentication Protocols

This paper reconsiders a powerful man-in-the-middle attack against Random-HB# and HB# authentication protocols, two prominent representatives of the HB family of protocols, which are built based on the Learning Parity in Noise (LPN) problem. A recent empirical report pointed out that the attack does not meet the claimed precision and complexity. Performing a thorough theoretical and numerical re-evaluation of the attack, in this paper we identify the root cause of the detected problem, which lies in reasoning based on approximate probability distributions of the central attack events, that can not provide the required precision due to the inherent limitations in the use of the Central Limit Theorem for this particular application. We rectify the attack by employing adequate Bayesian reasoning, after establishing the exact distributions of these events, and overcome the mentioned limitations. We further experimentally confirm the correctness of the rectified attack and show that it satisfies the required, targeted accuracy and efficiency, unlike the original attack.

Testing the intrinsic scatter of the asteroseismic scaling relations with Kepler red giants

Asteroseismic scaling relations are often used to derive stellar masses and radii, particulaly for stellar, exoplanet, and Galactic studies. It is therefore important that their precisions are known. Here we measure the intrinsic scatter of the underlying seismic scaling relations for Δν and νmax, using two sharp features that are formed in the H–R diagram (or related diagrams) by the red giant populations. These features are the edge near the zero-age core-helium-burning phase, and the strong clustering of stars at the so-called red giant branch bump. The broadening of those features is determined by factors including the intrinsic scatter of the scaling relations themselves, and therefore it is capable of imposing constraints on them. We modelled Kepler stars with a Galaxia synthetic population, upon which we applied the intrinsic scatter of the scaling relations to match the degree of sharpness seen in the observation. We found that the random errors from measuring Δν and νmax provide the dominating scatter that blurs the features. As a consequence, we conclude that the scaling relations have intrinsic scatter of $\sim 0.5\%$ (Δν), $\sim 1.1\%$ (νmax), $\sim 1.7\%$ (M) and $\sim 0.4\%$ (R), for the SYD pipeline measured Δν and νmax. This confirms that the scaling relations are very powerful tools. In addition, we show that standard evolution models fail to predict some of the structures in the observed population of both the HeB and RGB stars. Further stellar model improvements are needed to reproduce the exact distributions.

Rate of convergence for traditional Pólya urns

Consider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.