The Study of a Predator-Prey Model with Fear Effect Based on State-Dependent Harvesting Strategy

In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- q ( q ≥ 1 ) periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- q periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB.

Asymptotic vectors of entire curves

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t<\infty $, $\left|z\left(t\right)\right|<\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|<\infty \right\}$. We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.

Optimization of Caving Technology in an Extrathick Seam with Longwall Top Coal Caving Mining

The optimization of top coal caving technology is an efficient method to improve the recovery ratio in longwall top coal caving (LTCC). In extrathick coal seams, the conventional single-opening sequential caving technology (SOSCT) shows the following problems: low recovery ratio, high rock mixed ratio, and poor drawing balance. For these problems, this research verifies the applicability of multiopening caving technology (MOCT) in extrathick coal seams theoretically. However, different drawing sequences have a great effect on the drawing mechanism. Based on the progressive drawing sequence of cluster-group-support, this paper firstly proposes a systematic naming method for the top coal caving technology. Furthermore, an independent cluster-group caving technology (ICGCT) is given, meaning that all supports are divided into several clusters, a cluster is divided into several groups, and clusters extract top coal in positive order while groups are in reverse order in the drawing process. By establishing an experimental model by the discrete element method PFC2D, the drawing mechanism is investigated under different caving technologies. The results show that ICGCT significantly improves the recovery ratio of the panel and mainly increases the drawing volume of top coal in the middle and upper end of the panel. The shape of the top coal boundary reflects the drawing efficiency. Due to the effect of drawing sequence in ICGCT, the generation and disappearance processes of coal ridge greatly decrease the residual top coal in the middle of the panel. The drawing body shape has a direct influence on the recovery ratio. Multiple complete drawing bodies exist in ICGCT, and the dispersion coefficient of drawing volume changes periodically in the range of 0.5–1.7, which is conducive to the management of drawing processes. In addition, discussing ICGCT and the dependent cluster-group caving technology (DCGCT), it is found that the recovery ratio of DCGCT has a slight increase, which enlarges the maximum drawing range of top coal at both panel ends, shortening the total drawing time of the panel. In summary, ICGCT provides a new approach for improving the recovery ratio and drawing balance in LTCC with an extrathick coal seam.

ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

In this paper, we shall prove Theorem 1 Let $f$ be nonconstant meromorphic in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\] holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives

In this paper we introduce some new subclasses of the p-valent analytic functions with higher-order derivatives that generalize some related subclasses of starlike and convex functions of a positive order. We found the order of (p,q)-valent starlikeness and convexity for the products of functions that belong to these classes. The order of (p,q)-valent starlikeness and convexity of certain integral operators for the product of functions of these classes were also obtained.

Latin America's Economic Chances in a Post-Covid-19 World

Received 01.12.2020. The effects of the COVID-19 pandemic have hit Latin America’s economy and social sphere more than any other crisis in the last hundred years. Coexistence with the virus caused a wave of restrictive measures, destroyed normal mechanisms of supply and demand, collapsed production and foreign trade, dramatically worsened the financial situation of the majority of Latin Americans, exacerbated domestic political problems. The article shows that the extreme severity of the corona crisis was caused not only by its specific features but also by the pairing with other social ills inherent in Latin America: fatal miscalculations of ruling populist regimes, deep-rooted corruption, a huge informal sector of the economy. At the same time, the author attempts to understand the reality of the countries of the region in the post-COVID&#8209;19 period, what can be done to mitigate the negative effects of the epidemic. And the main thing is whether Latin American states have the resources and opportunities to get out of the crisis with a more advanced technological structure of the regional economy. As a matter of fact, in international business and expert circles it is recognized that in recent years new and extremely important economic and social phenomena of the positive order have been born in the leading Latin American states: the modernization of the business community and the strengthening of the role of tecnolatinas, the improvement of the educational level of young people and, on this basis, the qualitative improvement of human capital, the deployment of modernization processes and technological renewal of the production apparatus. The consolidation and development of these trends create unique chances to overcome the extremely unfavourable situation in which the region found itself as a result of the epidemic of coronavirus.

On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0&lt;\alpha&lt;1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.