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q -deformations and the dynamics of the larch bud-moth population cycles
Abstract
The concept of q-deformation of numbers is applied here to improve and modify a tritrophic population dynamics model to understand defoliation of the coniferous larch trees due to outbreaks of the larch bud-moth insect population. The results are in qualitative agreement with observed behavior, with the larch needle lengths, bud-moth population and parasitoid populations all showing 9-period cycles which are mutually synchronized. © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
... Quantum deformation (q-deformation) is a relatively recent branch of mathematical analysis yet it has been shown to have many applications. Applications in nuclear physics are given in [43], while application in population dynamics is given in [19]. Other applications in Yang-Baxter models are presented in [8]. ...
... The Neimark-Sacker bifurcation of system (3) occurs at the equilibrium point P 3 (x * , y * ) if the two conditions (19) and (23) ...
... Quantum deformation (q-deformation) is a relatively recent branch of mathematical analysis yet it has been shown to have many applications. Applications in nuclear physics are given in [43], while application in population dynamics is given in [19]. Other applications in Yang-Baxter models are presented in [8]. ...
... The Neimark-Sacker bifurcation of system (3) occurs at the equilibrium point P 3 (x * , y * ) if the two conditions (19) and (23) ...
The aim of this work is to analytically investigate the nonlinear dynamic behaviors of a proposed reduced Lorenz system based on q-deformations. The effects of varying the new q-deformation parameter on the dynamical behaviors of the system along with the induced bifurcations of fixed points are explored. In particular, the codimension-one bifurcation analysis is carried out at interior fixed point of the q-deformed system. Explicit conditions for the existence of pitchfork and Neimark–Sacker bifurcations are obtained. Numerical simulations are performed to confirm stability and bifurcation analysis in addition to investigate the effects of variations in system parameters. The changes in system dynamics are explored via the bifurcation diagrams, phase portraits and time series diagrams. Moreover, the quantification of system complex behaviors is depicted through the maximal Lyapunov exponent plots. A cascaded version of the model is proposed to boost its complex dynamics. Then, a chaos-based image encryption scheme, relying on a proposed key-distribution algorithm, is introduced as an application. Finally, several aspects of security analysis are examined for the encryption system to prove its efficiency and reliability against possible attacks.
... The q deformation also plays a vital role in modelling various biological systems for understanding their dynamics. In Ref. [16], the concept of q deformation is applied to improve and modify a tritrophic population model to understand the defoliation of coniferous Larch trees. The q deformation technique has been used to examine the dynamics of different models, such as logistic maps [10,17], Gaussian maps [9,23], etc. ...
We apply the deformation scheme to the classical Ricker map and obtain aq-deformed Ricker map, namely, q-Ricker map. The aim of the paper is to investigate the nonlinear dynamics, bifurcation structure, and topological entropy of q-Ricker map. In particular, we show that q-Ricker map proclaims many exciting phenomena that are remarkable in one-dimensional dynamical systems, such as the presence of coexisting attractors, physically non-observable chaos, hydra paradox, bubbling effect, and extinction. We discuss fold and flip bifurcations and further the presence of stochastically stable chaos. Finally, we show that a certain amount of deformation in the system can keep it in equilibrium; however, excessive deformation causes extinction.
... In ref. 11 we considered this tritrophic system using the dimensionless scaling introduced in ref. 12; however the decay of foliage was no longer represented by a constant, but rather by a density-dependent function of the PQI. ...
Periodic outbreaks of the larch budmoth Zeiraphera diniana population (and the massive forest defoliation they engender) have been recorded in the Alps over the centuries and are known for their remarkable regularity. But these have been conspicuously absent since 1981. On the other hand, budmoth outbreaks have been historically unknown in the larches of the Carpathian Tatra mountains. To resolve this puzzle, we propose here a model which includes the influence of climate and explains both the 8-9 year periodicity in the budmoth cycle and the variations from this, as well as the absence of cycles. We successfully capture the observed trend of relative frequencies of outbreaks, reproducing the dominant periodicities seen. We contend that the apparent collapse of the cycle in 1981 is due to changing climatic conditions following a tipping point and propose the recurrence of the cycle with a changed periodicity of 40 years-the next outbreak could occur in 2021. Our model also predicts longer cycles.
This paper concerns the dynamical study of the q-deformed homographic map, namely, the q-homographic map, where q-deformation is introduced by Jagannathan and Sinha with the inspiration from Tsalli’s q-exponential function. We analyze the q-homographic map by computing its basic nonlinear dynamics, bifurcation analysis, and topological entropy. We use the notion of a false derivative and the generalized Lambert W function of the rational type to estimate the upper bound on the number of fixed points of the q-homographic map. Furthermore, we discuss chaotification of the q-deformed map to enhance its complexity, which consists of adding the remainder of multiple scaling of the map’s value for the next generation using the multiple remainder operator. The chaotified q-homographic map shows high complexity and the presence of robust chaos, which have been theoretically and graphically analyzed using various dynamical techniques. Moreover, to control the period-doubling bifurcations and chaos in the q-homographic map, we use the feedback control technique. The theoretical discussion of chaos control is illustrated by numerical simulations.
The concept of q-deformation has been expanded in a variety of situations including q-deformed nonlinear maps. Coupled q-deformed logistic maps with positive and negative q-diffusive coupling are investigated. The transition to state of synchronized fixed point in the thermodynamic limit for random initial conditions is studied as dynamic phase transitions. Though rare for one-dimensional maps, q-deformed maps show multistability. For weak coupling, the synchronized state may not be observed in the thermodynamic limit with random initial conditions even if it is linearly stable. Thus multistability persists. However, for large lattices with random initial conditions, five well-defined critical points are observed where the transition to a synchronized state occurs. Two of these show continuous phase transition. One of them is in the directed percolation universality class. In another case, the order parameter shows power-law decay superposed with oscillations on a logarithmic scale at the critical point and does not match with known universality classes.
We apply Tsalli's deformation scheme to the classical Ricker map and obtain a q-deformed Ricker map, namely, q-Ricker map. The aim of the paper is to investigate the nonlinear dynamics, bifurcation structure, and topological entropy of q-Ricker map. In particular, we show that q-Ricker map proclaims many exciting phenomena that are remarkable in one-dimensional dynamical systems, such as the presence of coexisting attractors, physically non-observable chaos, hydra paradox, bubbling effect, and extinction. We discuss fold and flip bifurcations and further the presence of stochastically stable chaos. Finally, we show that a certain amount of deformation in the system can keep it in equilibrium; however, excessive deformation causes extinction.
In 1991, T. Puu proposed a cubic investment map in order to analyze chaos in business cycles [34]. In this study, we propose a q-deformation of the aggregate variable income difference in order to analyze the dynamics of the q-deformed version considering that the original model is a particular case of the q-deformation case with q=1.
We analyze the q-deformed logistic map, where the q-deformation follows the scheme inspired in the Tsallis q-exponential function. We compute the topological entropy of the dynamical system, obtaining the parametric region in which the topological entropy is positive and hence the region in which chaos in the sense of Li and Yorke exists. In addition, it is shown the existence of the so-called Parrondo's paradox where two simple maps are combined to give a complicated dynamical behavior.
We study the dynamics of a discrete-time tritrophic model which mimics the observed periodicity in the population cycles of the larch budmoth insect which causes widespread defoliation of larch forests at high altitudes periodically. Our model employs q-deformation of numbers to model the system comprising the budmoth, one or more parasitoid species, and larch trees. Incorporating climate parameters, we introduce additional parasitoid species and show that their introduction increases the periodicity of the budmoth cycles as observed experimentally. The presence of these additional species also produces other interesting dynamical effects such as periodic bursting and oscillation quenching via oscillation death, amplitude death, and partial oscillation death which are also seen in nature. We suggest that introducing additional parasitoid species provides an alternative explanation for the collapse of the nine year budmoth outbreak cycles observed in the Swiss Alps after 1981. A detailed exploration of the parameter space of the system is performed with movies of bifurcation diagrams which enable variation of two parameters at a time. Limit cycles emerge through a Neimark-Sacker bifurcation with respect to all parameters in all the five and higher dimensional models we have studied.
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