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- Modelling and Analysis of an One-predator Two-prey Ecological System with Fear Effect

pp. 585-604 | DOI: 10.5890/DNC.2021.12.001

Anindita Bhattacharyya, Sanghita Bose, Ashok Mondal, A. K. Pal

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The present study deals with the dynamical response of a two-prey onepredator model inculcating the anti-predator fear effect. The proposed model considers a Holling type II response function and it is intended to investigate the effect of the presence of fear among preys due to a predator. It is first shown that the system is bounded and the conditions of existence and stability of the equilibria of the proposed model have been furnished. Next the presence of Hopf bifurcation and limit cycles have been shown to explain the transition of the model from a stable to an unstable one. The study reveals that along with fear the interaction between the preys and predator can also be effectively stated as a control factor in determining dynamics of the model. The effect of anti-predator fear and mutual interaction between the preys and predator has been numerically simulated in order to potray the dynamics of the model and the occurrence of limit cycles.

- Group classification and solutions of a mathematical model from tumour biology

pp. 605-615 | DOI: 10.5890/DNC.2021.12.002

N.H. Ibragimov, R. Tracina, E.D. Avdonina

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We are interested in symmetries of a mathematical model of a malignant tumour dynamics due to haptotaxis. The model is formulated as a system of two nonlinear partial differential equations with two independent variables and contains two unknown functions of the dependent variables. When the unknown functions are arbitrary, themodel has only two symmetries. These symmetries allow to investigate only travelling wave solutions. The aim of the present paper is to make the group classification of the mathematical model under consideration and find the cases when themodel has additional symmetries and hence additional group invariant solutions.

- New relationship between Energy and Estrada index

pp. 617-623 | DOI: 10.5890/DNC.2021.12.003

Yalan Li, Bo Deng, Chengfu Ye

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Let $G$ be a graph on $n$ vertices, and let $\lambda_{1}, \cdots,\lambda_{n}$ be its eigenvalues. The energy $E(G)$ of a graph $G$ is defined as the sum of absolute values of the eigenvalues of $G$. The Estrada index of the graph $G$ is defined as $EE(G)=\sum ^{n}_{i=1}e^{\lambda_{i}}$. We get some new bounds for $EE(G)$. Some special inequalities are used to obtain the relationship between $E(G)$ and $EE(G)$.

- New stability estimates of solutions to strong damped wave equation with logarithmic external forces

pp. 625-634 | DOI: 10.5890/DNC.2021.12.004

Nabil Houma, Khaled Zennir, Abderrahmane Beniani, Abdelhak Djebabela

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In ths paper, we consider a new stability results of solutions to class of wave equations with weak, strong damping terms and logarithmic source in $\mathbb{R}^n$. We prove general stability estimates by introducing suitable Lyapunov functional.

- The effect of damping terms on decay rate for system of three nonlinear wave equations with weak-memories

pp. 635-647 | DOI: 10.5890/DNC.2021.12.005

Derradji Guidad, Khaled Zennir, Abdelhak Berkane, Mohamed Berbiche

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In this paper, we consider a very important problem from the point of view of application in sciences and engineering. A system of three wave equations having a different damping effects in an unbounded domain with strong external forces. Using the Faedo-Galerkin method and some energy estimates, we will prove the existence of global solution in $\mathbb{R}^n$ owing to to the weighted function. By imposing a new appropriate conditions, which are not used in the literature, with the help of some special estimates and generalized Poincar´e’s inequality, we obtain an unusual decay rate for the energy function.

- A study of approximation properties of Beta type summation-integral operator

pp. 649-662 | DOI: 10.5890/DNC.2021.12.006

Dhawal J. Bhatt, Vishnu Narayan Mishra, Ranjan Kumar Jana

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In the present paper we introduce Durrmeyer-type operator involving the beta function and Baskakov basis function and study its approximation properties. We obtain the rate of convergence in different terms. The uniform convergence of sequence of these operators is achieved using Korovkin’s theorem. Order of approximation for functions of some special class is also obtained. We establish the Voronovskaja type asymptotic result for this operator and a direct estimate of approximation for sequence of these operator.

- Period-3 motions to chaos in an inverted pendulum with a periodic base movement

pp. 663-680 | DOI: 10.5890/DNC.2021.12.007

Chuan Guo, Albert C.J. Luo

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A bifurcation tree of period-3 motions to chaos in an inverted pendulum with a periodic base movement is presented through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are also carried out. Frequencyamplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Numerical simulations of periodic motions on the bifurcation are completed. The numerical and analytical results are presented for comparison. Except for period-1 motion to chaos studied before, this study focuses on other periodic motions to chaos existing in the inverted pendulum with periodic base movement. In the earthquake testing, one tests structures from 1hz to 33 hz. However, during such a frequency range, one not only can observe the period-1 motion to chaos, but one can observe period-3 motion to chaos in such an inverted pendulum with a periodic base movement. Thus, in the building design, period-3 motions to chaos should be considered, which have different dynamical behaviors from the period-1 motions to chaos.

- Parameters estimation of a constrained predator prey dynamical model with incomplete data

pp. 681-691 | DOI: 10.5890/DNC.2021.12.008

C. Muzvondiwa, A.A. Adeniji, I. Fedotov, M.Y. Shatalov, A.C. Mkolesia

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Substantial frameworks for a predator prey dynamical model have been developed in the past. A Method for estimating parameters of a constrained dynamical model with incomplete data assuming information about predator (known) and prey (unknown) was investigated. Unknown parameters for the dynamical model was estimated using undetermined Lagrangian multiplier method. The method for estimating the parameters are based on the construction of a quadratic goal function minimization from a set of ordinary differential equations. A non-homogeneous system was generated using ordinary least square method applied to the goal function in order to estimate the parameters. The derivation of the parameters process was one of the inverse problem and the model with estimated parameters gave some satisfactory fit.

- Well-posedness and stability for a Moore-Gibson-Thompson equation with internal distributed delay

pp. 693-703 | DOI: 10.5890/DNC.2021.12.009

Abdelkader Braik, Abderrahmane Beniani, Khaled Zennir

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In this work, we consider the Moore-Gibson-Thompson equation with distributed delay. We prove, under an appropriate assumptions and a smallness conditions on the parameters $\alpha$, $\beta$, $\gamma$ and $\mu$, that this problem is well-posed and then by introducing suitable energy and Lyapunov functionals, the solution of (1) and (2) decays to zero as $t$ tends to infinity.

- Dynamical complexity in a tritrophic food chain model with prey harvesting

pp. 705-722 | DOI: 10.5890/DNC.2021.12.010

Krishnendu Sarkar, Nijamuddin Ali, Lakshmi Narayan Guin

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The present investigation deals with a tritrophic food web model with Holling-Tanner type II functional response to clarify the dynamical complexity of the eco-systems in the natural environment. The objective of this study is to explore the harvesting mechanism scenario in a threedimensional interacting species system such as one prey and two specialist predators. Attention has been given to demonstrate the system characteristics near the biologically feasible equilibria. Specifically, stability, Hopf-Andronov bifurcation for the respective system parameters and dissipativeness has been performed in order to scrutinize the system behaviour. Lyapunov exponents are worked out numerically and an unstable scenario for significant parameters of the model system has been executed to characterize the complex dynamics. In addition to, we put forward a detailed numerical simulation to justify the chaotic dynamics of the present system. We conclude that chaotic dynamics can be executed by the prey harvesting parameters.

- Domination Polynomials of certain hexagon lattice graphs

pp. 723-731 | DOI: 10.5890/DNC.2021.12.011

Caibing Chang, Haizhen Ren, Zijian Deng, Bo Deng

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Let $G$ be a simple graph with order $n$. The domination polynomial of graph $G$ is defined by $D(G,x)=\sum_{i=|\gamma(G)|}^{n}d(G,i)x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $\gamma(G)$ is the domination number of $G$. Calculating the domination polynomial of $G$ is difficult in general, as determining whether $\gamma(G)\leq k$ is known to be $NP$-complete. This has led to an emphasis on studying this problem in particular classes of graphs. In this paper, we consider the following two kinds of graphs. One is the benzene graph $F_{6,n}$ which constructed by selecting one vertex in each of $n$ benzenes$(i.e \ C_{6})$ and identifying them. The other is the $n-book$ hexagon lattice graph $B_{n,6}$ which identifying $n$-copies of the $C_{6}$ with three common edges. Their closed form expressions for domination polynomial are all given.

- Blow-up of result in a nonlinear wave equation with delay and source term

pp. 733-741 | DOI: 10.5890/DNC.2021.12.012

Tayeb Lakroumbe, Mama Abdelli, Abderrahmane Beniani

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In this paper we consider the initial boundary value problem for a nonlinear damping and a delay term of the form: $$ |u_t|^{l}u_{tt}-\Delta u (x,t) -\Delta u_{tt}+\mu_1|u_t|^{m-2}u_t\\+\mu_2|u_t(t-\tau)|^{m-2}u_t(t-\tau)=b|u|^{p-2}u, $$ with initial conditions and Dirichlet boundary conditions. Under appropriate conditions on $\mu_1$, $\mu_2$, we prove that there are solutions with negative initial energy that blow-up finite time if $p \geq \max\{l+2,m\}$.

- Darcy-Forchheimer flow of MHD Powell-Eyring nanoliquid over a nonlinear radially stretching disk with the impact of activation energy

pp. 743-753 | DOI: 10.5890/DNC.2021.12.013

Madhu Macha, Besthapu Prabhakar

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This study proclaims the Darcy-Forchheimer flow of Powell- Eyring nanoliquid subjected to nonlinear radially stretching disk. Further the impact of activation energy retained in concentration expression. In addition to this, convective boundary condition is adopted together with a modified version of mass flux condition is used. The modeled partial differential equations have been remodeled into system of ordinary differential equations via appropriate similarity variables. These ODEs are solved by Runge-Kutta fourth order scheme along with shooting technique. Graphs have been prepared to analyze the features of various influential parameters on velocity, temperature and concentration fields. Significant effects are found for various estimations of the fluid parameter on velocity, temperature and concentration profiles. Velocity field is reduced for growing values of porosity as well as inertia coefficient. Concentration rises for larger values of energy parameter but it is depreciated for higher values of chemical reaction rate.

- Some new results in the $q$-calculus

pp. 755-763 | DOI: 10.5890/DNC.2021.12.014

Jorge A. Castillo Medina, Salvador Cruz Garcıa, Juan E. Napoles Valdes, Thelma Galeana Moyaho

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In this paper, we present some results for a fractional derivative of type $q$ uniform defined by the authors in a previous work, and which are generalizations of known classical results of ordinary calculus.

- Demand Behaviour for Weather Index Insurance Products in Regions Prone to Agricultural Droughts

pp. 765-780 | DOI: 10.5890/DNC.2021.12.015

Dmitry V. Kovalevsky, Marıa Manez Costa

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Weather index insurance (WII) is a promising insurance scheme relevant for the agricultural sector, particularly in many low-income countries, including regions prone to agricultural droughts. WII policies might be more affordable to farmers, as the payouts in these insurance schemes are based on weather indices objectively determined for the specific agricultural regions, and therefore a costly individual loss assessment is not necessary. To successfully implement and scale up WII schemes, the development of transdisciplinary models properly addressing the complexity of relevant socio-natural processes and their interplay is necessary. We develop a stochasticmodel to simulate the demand for insurance policies in a droughtprone region under an assumption that weather and climate services, as tailored products for regions vulnerable to droughts, provide forecasts of the weather index on which the insurance scheme is based. Therefore, the simulated demand for the insurance policy might depend on the quality of the available weather index forecast. Presented modelling results suggest that both the income of individual producers and the dynamics of aggregate demand for insurance policiesmight be sensitive to the quality of the available weather index forecast. The developed modelling approaches can inform the design and implementation of WII schemes in drought-prone regions.