scholarly journals Dynamical Systems Analysis of a Five-Dimensional Trophic Food Web Model in the Southern Oceans

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Scott A. Hadley ◽  
Lawrence K. Forbes
Keyword(s):  
Hopf Bifurcation ◽  
Theoretical Model ◽  
Systems Analysis ◽  
Floquet Theory ◽  
The Other ◽  
Solution Branch ◽  
Predator Prey ◽  
Dynamical Systems Analysis ◽  
Food Web Model ◽  
Population Numbers

A theoretical model developed by Stone describing a three-level trophic system in the Ocean is analysed. The system consists of two distinct predator-prey networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys, in the sense that the presence of one is advantageous to the other when nutrients are low. It is shown that spontaneous oscillations in population numbers are possible, and that they result from a Hopf bifurcation. The limit cycles are analysed using Floquet theory and are found to change from stable to unstable as a solution branch is traversed.

SIZE MEASURE RELATIONSHIP METHOD FOR FRACTAL ANALYSIS OF SIGNALS

Fractals ◽  
2008 ◽  
Vol 16 (03) ◽  
pp. 235-241 ◽  
Author(s):  
P. PARAMANATHAN ◽  
R. UTHAYAKUMAR
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Systems Analysis ◽  
Computation Time ◽  
The Other ◽  
Dynamical Systems Analysis ◽  
Specific Complex ◽  
Size Measure ◽  
Electroencephalographic Signals ◽  
Irregular Behavior

The fractal dimension of signals represents a powerful tool for analyzing the irregular behavior and state of the dynamical systems. Analysis of waveforms has been used to identify and distinguish specific complex patterns. A variety of algorithms are available for the computation of fractal dimension of waveforms. In this paper we evaluate the performance of our algorithm based on size measure relationship method, quantifying the synthetic waveforms and electroencephalographic signals. Compared to Katz's, Higuchi's and Petrosian's algorithm advantages of this method include greater speed and not affected by noise. The computation time for the algorithm suggested in this paper is much less than the other methods.

Eurasian "Theory of Economy Management" as an Organic Part of the Russian Economic School

2003 ◽  
pp. 95-110
Author(s):  
M. Voeykov
Keyword(s):  
Economic Development ◽  
Theoretical Model ◽  
Initial Step ◽  
State Regulation ◽  
Soviet Economy ◽  
The Other ◽  
Large Share ◽  
Organic Part ◽  
Two Sides ◽  
Enterprise Activity

The original version of "the theory of economy management", developed in the 1920s by Russian economists-emigrants who called themselves "Eurasians" (N. Trubetskoy, P. Savitskiy, etc.) is analyzed in the article. They considered this theory to be the basis of the original Russia's way of economic development. The Eurasian theory of economy management focuses on two sides of enterprise activity: managerial as well as social and moral. The Eurasians accepted the Soviet economy with the large share of state regulation as the initial step of development. On the other hand they paid much attention to the private sector activity. Eurasians developed a theoretical model of the mixed economy which can be attributed as the Russian economic school.

scholarly journals Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 785
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Equilibrium Point ◽  
Power Law ◽  
Equilibrium Points ◽  
Essential Difference ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals Triggering avalanches by transverse perturbations in a rotating drum

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Vicente Salinas ◽  
Cristóbal Quiñinao ◽  
Sebastián González ◽  
Gustavo Castillo
Keyword(s):  
Hopf Bifurcation ◽  
Theoretical Model ◽  
Order Parameter ◽  
Small Scale ◽  
Rotating Drum ◽  
Governing Parameter ◽  
Stick Slip ◽  
Slip Regime ◽  
Axis Of Rotation

AbstractWe study the role of small-scale perturbations in the onset of avalanches in a rotating drum in the stick-slip regime. By vibrating the system along the axis of rotation with an amplitude orders of magnitude smaller than the particles’ diameter, we found that the order parameter that properly describes the system is the kinetic energy. We also show that, for high enough frequencies, the onset of the avalanche is determined by the amplitude of the oscillation, contrary to previous studies that showed that either acceleration or velocity was the governing parameter. Finally, we present a theoretical model that explains the transition between the continuous and discrete avalanche regimes as a supercritical Hopf bifurcation.

scholarly journals Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

2018 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
Jai Prakash Tripathi ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Interaction Model ◽  
Dynamical Analysis ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Predator Species ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model

AbstractIn this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

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