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2 Effect of plant quality on the relative multiplication rate of the larch budmoth, k (calculations based on data from Benz 1974, table 8). Plant quality
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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 bet...
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... related to the fi nite rate of population increas � 'i.' (the prime is to remind us that this measure is not the true A. because it does ' not include egg and adult mortality ) � · Plotting). ' against needle length index reveals· a well-defined rela tionship between these two quantities, with a high coeffiCient of determina tion, r2 '.86 ( fig. 7.2). Interestingly, the alternative index�' raw fiber content, explains a somewhat lower percentage of variance in A. ' (r2\= .66; analysis based on' the same Benz data). Thus, the somewhat surprising conclusion is that n e edle' length appears to be a better index of food quality than raw fiber content. Cle a rly, food quality is a ...
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... now consider th e 'r e sults of the analysis of time series data on LBM density and needle lerigth during 1961-92 at Sils (Engadine Valley, Switzerland) (see figure 7.3a). Turchin et al. (2002) employed nonlinear regression to investigate � he cross-effects of LBM density and needle length on each other. ...
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... constructing the equation for LBM dynamics, we used the data depicted in figure 7.2. After trying several two-parameter relationships, we found that a negative exponential fun ction fits the data best (this is a purely phenomen ological approach, as we have no mechanistic basis for postulating a fu nc tional form). ...
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... predicted by the model match both the period and the amplitude of the observed LBM oscillations. Additionally, the model mimics the quantitative pattern of the quality index dynamics reasonably well, including the amplitude of variation and the timing of declines and increases (compare with figure 7.3a). However, the range of oscillations in Q, predicted by the model is somewhat lower than that observed. ...
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... if we examine the last documented LBM outbreak (peak in 1989), we notice that the plant quality index hardly declined at all, with needle lengths remaining at high levels through the whole period ( figure 7.3a). As discussed by Baltensweiler (1993b), a sequence of unusual weather in 1989-91 caused high egg mortality. ...
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... of nonlinear regression suggest that the parasitism rate is quite well resolved by model (7.4). Thus, the simple three-parameter equation (7.4b) resolves 71 % of the variance in the parasitism rate. ...
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... the model within these parameter ranges, we find that the model does very well for parameters at their median values (or very near to them). In particular, with slight modifi cations (specifically, Ro = 2.3, c = 0.9, and d = 100; note that with each of these modifications we are staying within I SE of the median estimates), the model output matches the data patterns very well ( figure 7.4). Quantitative measures of the observed time series pattern (periodicity, amplitude, and cross-correlations between LBM and parasitism or quality index) are also closely matched by the model-generated trajectories. ...
Citations
... is used to represent the plant's health. Previous work by Turchin and others [10,[13][14][15][16] has shown the necessity of including three species in their model to capture and sustain the budmoth population cycles: the larch trees represented by their PQI, the budmoths which infest them, and parasitoids which live off the budmoths. Turchin's tritrophic model [13,15] is able to reproduce the 9-year budmoth cycles seen in the Swiss Engadine valley but can reproduce no other feature mentioned above. ...
... Wasteful feeding by budmoth larvae on larch foliage leading to a scorched appearance of the entire landscape, has prompted several investigations of the population cycles of this insect pest (see for instance ref. 5). Previous studies on larch budmoth (LBM) population cycles have established the presence of a third trophic level -parasitoids which prey upon budmoth larvae [6][7][8][9] . The population densities of the budmoth, the parasitoids preying upon them and needle lengths of the larch 10 are all known to show periodic cycles which are mutually synchronized. ...
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.
... The larch budmoth populations in the European Alps are one system for which there is strong evidence of periodic travelling waves. The almost metronomic multi-year dynamics of larch budmoth populations has fascinated ecologists for decades (Turchin et al. 2002, ch. 9). ...
Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction-diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction-diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.
Cyclic types of ecological dynamics have been found in several biological species of forest Lepidoptera. There are several possible reasons for this, for example, interactions with consumers, predators, plant quality index, and interactions of density-dependent type. Furthermore, interaction with consumers and plant quality index is regarded as key ingredients to alter the dynamics for the parasitoid population. Consequently, the quality of food resources fluctuates due to the level of herbs. Such changes have been observed in different systems of forest pests. Lepidoptera (larch budmoth) is a destructive worm that affects high-altitude trees around the world and is rapidly declining and becoming extinct in large areas of the forest. Considering the interaction between the budmoth and plant quality index for larch trees in the mountain range in Switzerland (Swiss Alps), we discuss the dynamics of a discrete-time system. Ivlev type functional response regarding plant quality index is used for the formulation of discrete-time model concerning the interaction between the index of plant quality and budmoth. Moreover, the existence of steady-states, their local behaviors, and the boundedness of solutions are carried out for the discrete-time model under consideration. It is investigated that the model undergoes flip bifurcation about coexistence by applying the theory of normal forms and the center manifold theorem. Furthermore, the direction and the existence of Hopf bifurcation are explored for the model around its coexistence. Various methods of controlling chaos have been introduced to avoid system fluctuations and bifurcating attitudes. Validation of the analytical findings is illustrated through numerical simulations. Finally, the analytical results are validated by experimental and actual field data.
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.
This paper deals with the qualitative analysis of the travelling waves solutions of a reaction diffusion model that refers to the competition between the predator and prey with modified Leslie–Gower and Holling type II schemes. The well posedeness of the problem is proved. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of the forth degree exponential polynomial characteristic equation. We also prove the existence of a Hopf bifurcation which leads to periodic oscillating travelling waves by considering the diffusion coefficient as a parameter of bifurcation. Numerical simulations are given to illustrate the analytical study.
