Spiral periodic structures in a parameter plane of an ecological model

doi:10.1016/j.amc.2014.12.115

Applied Mathematics and Computation

Volume 254, 1 March 2015, Pages 9-13

https://doi.org/10.1016/j.amc.2014.12.115 Get rights and content

Abstract

We investigate a parameter plane of a set of three autonomous, ten-parameter, first-order nonlinear ordinary differential equations, which models a tri-trophic food web system. By using Lyapunov exponents, bifurcation diagrams, and trajectories in the phase-space, to numerically characterize the dynamics of the model in a parameter plane, we show that it presents typical periodic structures embedded in a chaotic region, forming a spiral structure that coils up around a focal point while period-adding bifurcations take place.

Introduction

Investigations in parameter-planes of continuous-time nonlinear dynamical systems, using Lyapunov exponents to numerically characterize the dynamics by discriminating periodic, quasiperiodic, and chaotic behaviors, have grown substantially in recent years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47]. Note that we have cited above only papers published since 2010. Typical self-organized periodic structures have been observed embedded in chaotic regions of these parameter planes. Some of these are the shrimp-shaped periodic structures, first described by Gallas [48] in a discrete-time nonlinear dynamical system, and after observed also in a wide range of both discrete- and continuous-time nonlinear dynamical systems, modeled respectively by maps and sets of first-order ordinary differential equations. Many properties of these periodic structures have been described since then, showing that they may appear highly organized in different ways in a parameter plane of a particular system. This is true for a large number of different systems, in different fields, including chemical reactions, population dynamics, electronic circuits, lasers, forced oscillators, neural networks, among others. One such type of organization consists of a set of shrimp-shaped periodic structures forming a spiral that coils up around a focal point while period-adding bifurcations take place. To our knowledge these spiral bifurcations have been observed in parameter planes of electronic circuits [1], [29], [49], a Rössler model [50], a chemical oscillator [1], a Hopfield neural network [30], modified optical injection semiconductor lasers [31], and a tumor growth mathematical model [43]. The spiral periodic structures were experimentally detected in electronic circuits [51], and the global mechanism responsible for its origin and organization was reported simultaneously by Vitolo et al. [14] and Barrio et al. [15].

Here we report the period-adding spiral bifurcation of the shrimp-shaped periodic structures in a parameter plane of a tri-trophic food web system [52], which is modeled by a set of three autonomous, ten parameter, first-order ordinary differential equations. The Letter is organized as follows. In Section 2 the mathematical model of the considered tri-trophic food web system is established. In Section 3 parameter planes and other numerical results are presented and interpreted. The Letter is finalized in Section 3.

Section snippets

The model

The equations of motion that theoretically describe the dynamical behavior of the tri-trophic food web system here considered, were recently proposed by Priyadarshi and Gakkhar [52], and are given by $\begin{matrix} \dot{X} = a_{0} X (1 - \frac{X}{K}) - \frac{a_{1} XY}{1 + b_{1} X} - \frac{a_{2} XZ}{1 + b_{2} X + b_{3} Y}, \\ \dot{Y} = Y (- r + \frac{a_{1} cX}{1 + b_{1} X}) - \frac{a_{3} YZ}{1 + b_{2} X + b_{3} Y}, \\ \dot{Z} = S_{0} (Z^{2} - \frac{Z^{2}}{S_{3} + S_{1} X + S_{2} Y}), \end{matrix}$ where X is the bottom prey population density, Y is the specialist predator population density, and Z is the generalist predator population density. The parameter $a_{0}$ is the intrinsic growth rate of the bottom

Numerical results

Fig. 1 shows a $(w_{3}, w_{2})$ parameter plane for the tri-trophic food web nondimensionalized model (2), which is a 2-dimensional cross-section of their 10-dimensional parameter-space, obtained by plotting the largest Lyapunov exponent (LLE) on a $10^{3} \times 10^{3}$ mesh of points. Following Ref. [52], the remaining parameters in Eqs. (2) were kept fixed as $w_{1} = 1.4$ , $w_{4} = 1.0, w_{5} = 0.16, w_{6} = 0.1, w_{7} = 0.1, w_{8} = 0.5$ , $w_{9} = 8.0$ , and $w_{10} = 8.0$ . System (2) was integrated using a fourth order Runge–Kutta algorithm with a fixed time step

Summary

We carried out numerical simulations on a set of dimensionless three autonomous, ten-parameter, first-order nonlinear ordinary differential equations that models a tri-trophic food web system. By using the largest Lyapunov exponent as a measure of chaotic or periodic behavior, we have constructed a parameter plane varying two of these ten parameters that control the dynamics of the system. We have shown that this parameter plane presents organized periodic structures embedded in a chaos region,

Acknowledgments

The authors thank Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, and Fundação de Amparo à Pesquisa e Inovação do Estado de Santa catarina-FAPESC, Brazilian Agencies, for financial support.

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Spiral periodic structures in a parameter plane of an ecological model

Abstract

Introduction

Section snippets

The model

Numerical results

Summary

Acknowledgments

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Comput. Math. Appl.

Neurocomputing

Phys. Lett. A

Commun. Nonlinear Sci. Numer. Simul.

Phys. Lett. A

Opt. Commun.

Neural Networks

Appl. Math. Comput.

Appl. Math. Comput.

Chaos Solitons Fract.

Commun. Nonlinear Sci. Numer. Simul.

Chaos Solitons Fract.

Physica D

Commun. Nonlinear Sci. Numer. Simul.

The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows

Int. J. Bifurcation Chaos

Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity

Chaos

Regular and chaotic dynamics of the Lorenz–Stenflo system

Int. J. Bifurcation Chaos

Dynamics of a particular Lorenz type system

Int. J. Mod. Phys. C

Abundance of stable periodic behavior in a Red Grouse population model with delay: a consequence of homoclinicity

Chaos

Labyrinth bifurcations in optically injected diode lasers

Eur. Phys. J. D

Self-similarities and period-adding in the parameter-space of a nonlinear resonant coupling process

Int. J. Nonlinear Sci.

Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback

Phys. Rev. E

Lyapunov exponent diagrams of a 4-dimensional Chua system

Chaos