scholarly journals Stability and Bifurcation Analysis of a Singular Delayed Predator-Prey Bioeconomic Model with Stochastic Fluctuations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yue Zhang ◽  
Qingling Zhang
Keyword(s):  
Deterministic Model ◽  
Gaussian White Noise ◽  
Differential Algebraic Equations ◽  
Bioeconomic Model ◽  
Algebraic Equations ◽  
Stability Switches ◽  
Transform Method ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Stochastic Fluctuations

This study investigates a singular delayed predator-prey bioeconomic model with stochastic fluctuations, which is described by differential-algebraic equations because of economic factors. The interior equilibrium of the singular delayed predator-prey bioeconomic model switches from being stable to unstable and then back to being stable, with the increase in time delay. The critical values for stability switches and Hopf bifurcations can be analytically determined. Subsequently, the effect of a fluctuating environment on the singular stochastic delayed predator-prey bioeconomic model obtained by introducing Gaussian white noise terms to the aforementioned deterministic model system is discussed. The fluctuation intensity of the population and harvest effort are calculated by Fourier transform method. Numerical simulation results are presented to verify the effectiveness of the conclusions.

Stability and Hopf Bifurcation of a Predator–Prey Biological Economic System with Nonlinear Harvesting Rate

2015 ◽  
Vol 16 (6) ◽  
pp. 249-258 ◽  
Author(s):  
Weiyi Liu ◽  
Chaojin Fu ◽  
Boshan Chen
Keyword(s):  
Hopf Bifurcation ◽  
Economic System ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Matlab Simulation ◽  
Predator Prey ◽  
Complex Models ◽  
Nonlinear Harvesting ◽  
Nonlinear Harvesting Rate ◽  
The Stability

AbstractIn this paper, we analyze the stability and Hopf bifurcation of a biological economic system with harvesting effort on prey. The model we consider is described by differential-algebraic equations because of economic revenue. We choose economic revenue as a positive bifurcation parameter here. Different from previous researchers’ models, this model with nonlinear harvesting rate is more general. Furthermore, the improved calculation process of parameterization is much simpler and it can handle more complex models which could not be dealt with by their algorithms because of enormous calculation. Finally, by MATLAB simulation, the validity and feasibility of the obtained results are illustrated.

Stochastic stability and Hopf bifurcation analysis of a singular bio-economic model with stochastic fluctuations

2019 ◽  
Vol 12 (08) ◽  
pp. 1950083
Author(s):  
Yi Zhang ◽  
Na Li ◽  
Jianyu Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Diffusion Process ◽  
Stochastic Stability ◽  
Measure Theory ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Differential Algebraic Equations ◽  
Singular Boundary ◽  
Algebraic Equations ◽  
Stochastic Fluctuations

In this paper, we study a class of singular stochastic bio-economic models described by differential-algebraic equations due to the influence of economic factors. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. Stochastic stability and Hopf bifurcations can be analytically determined based on the singular boundary theory of diffusion process, the Maximal Lyapunov exponent and the invariant measure theory. The critical value of the stochastic Hopf bifurcation parameter is obtained and the position of Hopf bifurcation drifting with the parameter increase is presented as a result. Practical example is presented to verify the effectiveness of the results.

scholarly journals Analytical Solution of a Nonlinear Index-Three DAEs System Modelling a Slider-Crank Mechanism

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal
Keyword(s):  
Analytical Solution ◽  
Series Solution ◽  
Perturbation Parameter ◽  
Differential Algebraic Equations ◽  
Modern Technology ◽  
Power Series Solution ◽  
System Modelling ◽  
Algebraic Equations ◽  
Transform Method ◽  
Crank Mechanism

The slider-crank mechanism (SCM) is one of the most important mechanisms in modern technology. It appears in most combustion engines including those of automobiles, trucks, and other small engines. The SCM model considered here is an index-three nonlinear system of differential-algebraic equations (DAEs), and therefore difficult to integrate numerically. In this work, we present the application of the differential transform method (DTM) to obtain an approximate analytical solution of the SCM model in convergent series form. In addition, we propose a posttreatment of the power series solution with the Padé resummation method to extend the domain of convergence of the approximate series solution. The main advantage of the proposed technique is that it does not require an index reduction and does not generate secular terms or depend on a perturbation parameter.

scholarly journals Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes
Keyword(s):  
Perturbation Parameter ◽  
Convergent Series ◽  
Differential Algebraic Equations ◽  
Differential Transform Method ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Partial Differential Algebraic Equations ◽  
Differential Transform ◽  
Reduced Differential Transform Method

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter.

scholarly journals Analytical and approximate solutions of Fractional Partial Differential-Algebraic Equations

2020 ◽  
Vol 5 (1) ◽  
pp. 109-120 ◽  
Author(s):  
Hatıra Günerhan ◽  
Ercan Çelik
Keyword(s):  
Numerical Solution ◽  
Approximate Solutions ◽  
Differential Algebraic Equations ◽  
Differential Transform Method ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Partial Differential Algebraic Equations ◽  
Differential Transform ◽  
Fractional Differential

AbstractIn this paper, we have extended the Fractional Differential Transform method for the numerical solution of the system of fractional partial differential-algebraic equations. The system of partial differential-algebraic equations of fractional order is solved by the Fractional Differential Transform method. The results exhibit that the proposed method is very effective.

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