Annals of Applied Mathematics AAM Volume 34, Number 2

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Volume 34, Number 2

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Nonlinear Ocean Internal Waves

Boling Guo & Yitong Pei

Ann. Appl. Math., 34 (2018), pp. 111-125.

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Abstract

In this paper, several important mathematical models of nonlinear ocean internal waves are introduced, and the related results are given.

Estimation of Parameters in a Closed Pygmy Population in Cameroon

Yannick Tchaptchie Kouakep & David Bekolle

Ann. Appl. Math., 34 (2018), pp. 126-137.

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Abstract

Inspired by Kouakep [16], we consider in this note a well-posed model with differential susceptibility and infectivity adding continuous age structure to an ODE model for a “Baka” pygmy group in the East of Cameroon (Africa). Assuming a very low contribution of carriers to infection compared to acute infection, we estimate a probability $p(a)$ (to develop symptomatic Hepatitis B state at age $a$) and acute carriers’ transmission rate. The value $R_0 = 2:67 > 1$ of the basic reproduction number estimated from data in the east of Cameroon confirms that HBV is endemic in the Baka pygmy group.

A Block-Coordinate Descent Method for Linearly Constrained Minimization Problem

Xuefang Liu & Zheng Peng

Ann. Appl. Math., 34 (2018), pp. 138-152.

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Abstract

In this paper, a block coordinate descent method is developed to solve a linearly constrained separable convex optimization problem. The proposed method divides the decision variable into a few blocks based on certain rules. Then the candidate solution is iteratively obtained by updating one block at each iteration. The problem, whether or not there are overlapping regions between two immediately adjacent blocks, is investigated. The global convergence of the proposed method is established under some suitable assumptions. Numerical results show that the proposed method is effective compared with some “full-type” methods.

A Commensal Symbiosis Model with Holling Type Functional Response and Non-Selective Harvesting in a Partial Closure

Yu Liu & Xinyu Guan

Ann. Appl. Math., 34 (2018), pp. 153-164.

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Abstract

A two species commensal symbiosis model with Holling type functional response and non-selective harvesting in a partial closure is considered. Local and global stability property of the equilibria are investigated. Depending on the the area available for capture, we show that the system maybe extinct or one of the species will be driven to extinction, while the rest one is permanent, or both of the species coexist in a stable state. The dynamic behaviors of the system is complicated and sensitive to the fraction of the harvesting area.

New Dynamic Inequalities for Decreasing Functions and Theorems of Higher Integrability

S.H. Saker, D. O’Regan, M.M. Osman & R.P. Agarwal

Ann. Appl. Math., 34 (2018), pp. 165-177.

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Abstract

In this paper we establish some new dynamic inequalities on time scales which contain in particular generalizations of integral and discrete inequalities due to Hardy, Littlewood, Pόlya, D’Apuzzo, Sbordone and Popoli. We also apply these inequalities to prove a higher integrability theorem for decreasing functions on time scales.

The Bounds about the Wheel-Wheel Ramsey Numbers

Lili Shen & Xianzhang Wu

Ann. Appl. Math., 34 (2018), pp. 178-182.

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Abstract

In this paper, we determine the bounds about Ramsey number $R(W_m, W_n),$ where $W_i$ is a graph obtained from a cycle $C_i$ and an additional vertex by joining it to every vertex of the cycle $C_i.$ We prove that $3m+1 ≤ R(W_m, W_n) ≤ 8m − 3$ for odd $n,$ $m ≥ n ≥ 3,$ $m ≥ 5,$ and $2m + 1 ≤ R(W_m, W_n) ≤ 7m − 2$ for even $n$ and $m ≥ n + 502.$ Especially, if $m$ is sufficiently large and $n = 3,$ we have $R(W_m, W_3) = 3m + 1.$

Dynamical Analysis of a Single-Species Population Model with Migrations and Harvest Between Patches

Chenjia Wang & Fengying Wei

Ann. Appl. Math., 34 (2018), pp. 183-198.

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Abstract

A single-species population model with migrations and harvest between the protected patch and the unprotected patch is formulated and investigated in this paper. We study the local stability and the global stability of the equilibria. The research points out, under some suitable conditions, the single-species population model admits a unique positive equilibrium, which is globally asymptotically stable. We also derive that the trivial solution is globally asymptotically stable when the harvesting rate exceeds the threshold. Further, we discuss the practical effects of the protection zones and the harvest. The main results indicate that the protective zones indeed eliminate the extinction of the species under some cases, and the theoretical threshold of harvest to the practical management of the endangered species is provided as well. To end this contribution and to check the validity of the main results, numerical simulations are separately carried out to illustrate these results.

Dynamics of a Predator-Prey Reaction-Diffusion System with Non-Monotonic Functional Response Function

Huan Wang & Cunhua Zhang

Ann. Appl. Math., 34 (2018), pp. 199-220.

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Abstract

In this article, a two-species predator-prey reaction-diffusion system with Holling type-IV functional response and subject to the homogeneous Neumann boundary condition is regarded. In the absence of the spatial diffusion, the local asymptotic stability, the instability and the existence of Hopf bifurcation of the positive equilibria of the corresponding local system are analyzed in detail by means of the basic theory for dynamical systems. As well, the effect of the spatial diffusion on the stability of the positive equilibria is considered by using the linearized method and analyzing in detail the distribution of roots in the complex plane of the associated eigenvalue problem. In order to verify the obtained theoretical predictions, some examples and numerical simulations are also included by applying the numerical methods to solve the ordinary and partial differential equations.

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