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... Iyengar et al. [15] investigated impact of climate change on the interaction of larch budmoth system. In order to give an explanation for the irregular larch budmoth cyclic outbreaks observed in the French Alps, Balakrishnan et al. [16] proposed important modifications in models related to larch budmoth interaction by implementing a slow time dependence in one of the species-specific parameters. Taking into account the interaction between plant quality and larch budmoth in Swiss Apls, recently Ali et al. [17] reported period-doubling bifurcation, Neimark-Sacker bifurcation and chaos control for a class of discrete-time system with Ricker equation. ...
... According to [10,15,16], some seasonal effects may result migration of budmoth population, and a change in plant quality index. Keeping in view these studies, the biological interpretation of control system (5.4) is given as follows: ...
... Iyengar et al. [16] investigated the impact of climate change on the interaction of larch budmoth system. In order to give an explanation for the irregular larch budmoth cyclic outbreaks observed in the French Alps, Balakrishnan et al. [17] proposed important modifications in models related to larch budmoth interaction by implementing a slow time dependence in one of the species-specific parameters. Taking into account the interaction between plant quality and larch budmoth in Swiss Apls, recently Ali et al. [18] reported period-doubling bifurcation, Neimark-Sacker bifurcation and chaos control for a class of discrete-time system with Ricker equation. ...
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Keeping in mind the interactions between budmoths and the quality of larch trees located in the Swiss Alps (a mountain range in Switzerland), a discrete-time model is proposed and studied. The novel model is proposed with implementation of a nonlinear functional response that incorporates plant quality. The proposed functional response is validated with real observed data of larch budmoth interactions. Furthermore, we investigate the qualitative behavior of the proposed discrete-time system with interactions between budmoths and the quality of larch trees. Proofs of the boundedness of solutions, and the existence of fixed points and their topological classification are carried out. It is proved that the system experiences period-doubling bifurcation at its positive fixed point using the center manifold theorem and normal forms theory. Moreover, existence and direction for the torus bifurcation are also investigated for larch budmoth interactions. Bifurcating and fluctuating behaviors of the system are controlled through utilization of chaos control strategies. Numerical simulations are presented to demonstrate the theoretical findings. At the end, theoretical investigations are validated with field and experimental data.
... Regarding [n] (q−1) as a deformed number, a scheme of q-deformation of nonlinear maps was proposed in [70] and this proposal has been found to have many applications (see, e.g., [71][72][73][74][75][76]). Equation (108) shows that we can write e q (x) as a φ-exponential function: ...
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The Tsallis q-exponential function eq(x)=(1+(1q)x)11qe_{q}(x) = (1+(1-q)x)^{\frac {1}{1-q}} is found to be associated with the deformed oscillator defined by the relations [N,a]=a\left [N,a^{\dagger }\right ] = a^{\dagger }, [N,a] = −a, and [a,a]=ϕT(N+1)ϕT(N)\left [a,a^{\dagger }\right ] = \phi _{T}(N+1)-\phi _{T}(N), with ϕT(N) = N/(1 + (q − 1)(N − 1)). In a Bargmann-like representation of this deformed oscillator the annihilation operator a corresponds to a deformed derivative with the Tsallis q-exponential functions as its eigenfunctions, and the Tsallis q-exponential functions become the coherent states of the deformed oscillator. When q = 2 these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when q = 1 this deformed oscillator is a canonical boson oscillator, when 1 < q < 2 its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when q→2 it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state. https://doi.org/10.1007/s10773-020-04534-w
Article
In this paper, we study some dynamics concerning the interaction of budmoth and plant quality index of larch situated in the Swiss Alps. Taking into account this interaction, a two-dimensional discrete-time system is formulated and discussed. The new model is formulated with an application of Holling type III functional response for the plant quality index. Furthermore, the proposed functional response is supported by actual observed data related to larch budmoth interaction. In addition to proving that solutions are uniformly bounded, the existence of biologically feasible fixed points and the local dynamics of the proposed model regarding its fixed points are also studied. It is shown that the proposed model undergoes period-doubling bifurcation around its coexistence with the implementation of the center manifold and normal form theories. Furthermore, the existence and direction of Neimark–Sacker bifurcation about positive fixed point are discussed. Taking into account the biological relevance of chaos control strategies, different methods of controlling chaos are implemented with their biological relevance. Numerical simulation is demonstrated to illustrate the theoretical discussion. Finally, theoretical discussion is validated with experimental and field data.
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1 Outbreaks of the larch budmoth (LBM) in the European Alps are among the most documented population cycles and their historical occurrence has been reconstructed over 1200 years. 2 Causes and consequences of cyclic LBM outbreaks are poorly understood and little is known about populations near the margin of the host’s distribution range. 3 In the present study, we quantify historical LBM outbreaks and associated growth reductions in host trees (European larch). Tree-ring data collected from 18 sites between approximately 500 and 1700m a.s.l. in the Northern pre-Alps are compared with data from the Western Alps and Tatra Mountains, as well as with nonhost Norway spruce. 4 Highly synchronized host and nonhost growth in the Northern pre-Alps shows that periodic LBM outbreaks are largely absent near the distributional limit of larch. By contrast, growth patterns in the Western Alps LBM core region are indicative of LBM events. Although climatic conditions in the Northern pre-Alps and Tatra Mountains would allow LBM outbreaks, low host plant abundance is likely the key driver for the absence of cyclic outbreaks in these regions. 5 The results obtained in the present study suggest that, in addition to the climatic conditions, host-species abundance is critically important for the occurrence of periodic LBM outbreaks and the determination of the respective outbreak range.
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ABSTRACT: The intensity of cyclic larch budmoth ( Zeiraphera diniana GuenÈe; LBM) outbreaks across the European Alps has been reported to have weakened since the early 1980s. In addition to a warmer climate, changes in land-use cover over modern and historical times may have affected the LBM system. Here, we present tree-ring-based reconstructions of LBM outbreaks from a mixed subalpine larch-pine forest in the French Alps for the period 1700-2010. Temporal variation in LBM outbreak severity was mainly driven by land-use changes, including varying forest structure and species composition. Human population pressure and associated resource demands for fuel wood and construction timber not only resulted in a reduction of larch and subsequent suppression of pine, but also supported an overall grassland expansion for livestock. Superimposed on modern land abandonment and pine re-colonization is a strong warming trend, which may also contribute to the observed late 20th-century weakening of Alpine-wide cyclic LBM outbreaks. Our results suggest that a complex interplay of different factors triggered less synchronized LBM outbreaks at broader scales, with overall significantly lower intensities at local scales.
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During the late 1940s, immediately after World War II, the lush green forests of the Engadine Valley, high in the Swiss Alps, turned an ugly red-brown in the midst of the tourist season. This was due to a spectacular outbreak of the larch budmoth, Zeiraphera diniana Guenée (Lepidoptera: Tortricidae). Preparing for a revival of the tourist industry, and having the new insecticide DDT at hand, it seemed only appropriate that the tourist office urge the forest service to control the pest. This was the beginning of what was to become a 34-year study of the population dynamics of the larch budmoth (Fig. 1).
Chapter
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The population dynamics of the larch budmoth (LBM), Zeiraphera diniana, in the Swiss Alps are perhaps the best example of periodic oscillations in ecology (figure 7.1). These oscillations are characterized by a remarkably regular periodicity, and by an enormous range of densities experienced during a typical cycle (about 100,000-fold difference between peak and trough numbers). Furthermore, nonlinear time series analysis of LBM data (e.g., Turchin 1990, Turchin and Taylor 1992) indicates that LBM oscillations are definitely generated by a second-order dynamical process (in other words, there is a strong delayed density dependence—see also chapter 1). Analysis of time series data on LBM dynamics from five valleys in the Alps suggests that around 90% of variance in Rt is explained by the phenomenological time series model employing lagged LBM densities, R, =f(Ni-1,Ni-2,) (Turchin 2002). As discussed in the influential review by Baltensweiler and Fischlin (1988) about a decade ago, ecological theory suggests a number of candidate mechanisms that can produce the type of dynamics observed in the LBM (see also chapter 1). Baltensweiler and Fischlin concluded that changes in food quality induced by previous budmoth feeding was the most plausible explanation for the population cycles. During the last decade, the issue of larch budmoth oscillations was periodically revisited by various population ecologists looking for general insights about insect population cycles (e.g., Royama 1977, Bowers et al. 1993, Ginzburg and Taneyhill 1994, Den Boer and Reddingius 1996, Hunter and Dwyer 1998, Berryman 1999). These authors generally concurred with the view that budmoth cycles are driven by the interaction with food quality. A recent reanalysis of the rich data set on budmoth population ecology collected by Swiss researchers over a period of several decades, however, suggested that the role of parasitism is underappreciated (Turchin et al. 2002). Before focusing on the roles of food quality and parasitism in LBM dynamics, we briefly review the status of other hypotheses that were discussed in the literature on LBM cycles. First, the natural history of the LBM-larch system is such that food quantity is an unlikely factor to explain LBM oscillations.
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In an earlier study (Holling, 1959) the basic and subsidiary components of predation were demonstrated in a predator-prey situation involving the predation of sawfly cocoons by small mammals. One of the basic components, termed the functional response, was a response of the consumption of prey by individual predators to changes of prey density, and it appeared to be at least theoretically important in population regulation: Because of this importance the functional response has been further examined in an attempt to explain its characteristics.
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The generalized Tsallis statistics produces a distribution function appropriate to describe the interior solar plasma, thought as a stellar polytrope, showing a tail depleted with respect to the Maxwell-Boltzmann distribution and reduces to zero at energies greater than about 20kBT. The Tsallis statistics can theoretically support the distribution suggested in the past by Clayton and collaborators, which shows also a depleted tail, to explain the solar neutrino counting rate.
Book
Why do organisms become extremely abundant one year and then seem to disappear a few years later? Why do population outbreaks in particular species happen more or less regularly in certain locations, but only irregularly (or never at all) in other locations? Complex population dynamics have fascinated biologists for decades. By bringing together mathematical models, statistical analyses, and field experiments, this book offers a comprehensive new synthesis of the theory of population oscillations. Peter Turchin first reviews the conceptual tools that ecologists use to investigate population oscillations, introducing population modeling and the statistical analysis of time series data. He then provides an in-depth discussion of several case studies--including the larch budmoth, southern pine beetle, red grouse, voles and lemmings, snowshoe hare, and ungulates--to develop a new analysis of the mechanisms that drive population oscillations in nature. Through such work, the author argues, ecologists can develop general laws of population dynamics that will help turn ecology into a truly quantitative and predictive science. Complex Population Dynamics integrates theoretical and empirical studies into a major new synthesis of current knowledge about population dynamics. It is also a pioneering work that sets the course for ecology's future as a predictive science.
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Hypotheses for the causes of regular cycles in populations of forest Lepidoptera have invoked pathogen-insect or foliage-insect interactions. However, the available data suggest that forest caterpillar cycles are more likely to be the result of interactions with insect parasitoids, an old argument that seems to have been neglected in recent years.
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