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One of the most interesting observations from the time series presented in the section above is the following: when the magnitude of the Allee parameter (theta) is low, vegetation and prey densities are confined to low values. However, the predator densities deviate very significantly away from their mean. Now for very small (theta) the system is attracted to a periodic orbit, and so the large deviations are completely correlated with time and occur periodically. So they cannot be considered to be extreme events, as they are neither aperiodic, nor rare. But for larger (theta), both predator and prey densities can sometime shoot up over 7 standard deviations away from the mean value. This is evident clearly in Fig. 2c,e where one can see that both predator and prey populations exceed the (7sigma) threshold from time to time. The instants at which prey and predator populations exceed the (7sigma) threshold are now completely uncorrelated with time. This is consistent with the underlying chaotic dynamics that emerges under increasing Allee parameter (theta).In order to illustrate this, we mark the time instances at which a population exceeds the (7sigma) threshold, for different values of Allee parameter (theta). Figure 4 shows this for the vegetation, prey and predator populations. The density of points signifying the occurrence of extreme events is clearly the highest for the predator population. This indicates that the predator population has the greatest propensity for large deviations. It is also clear that vegetation has the least number of extreme events in the same time window. The uncorrelated nature of the extreme events is also evident in the scatter of these points, except in the small periodic windows that occur for certain special ranges of (theta). The increasing density of these points also illustrate the increasing probability of extreme events in the populations with increasing Allee parameter (theta).Figure 4Figure marking the time instances at which a population exceeds the (7sigma) threshold, for different values of Allee parameter (theta), for the case of (top to bottom) vegetation, prey and predator populations.Full size imageIn order to understand the phenomena quantitatively, we first estimate the maximum densities of vegetation, prey and predator populations (denoted by (u_{max}), (v_{max}) and (w_{max}) respectively) for varying the Allee parameter (theta). To estimate this, we find the global maximum of the populations sampled over a time interval (T=1000), averaged over a large set of random initial conditions.Figure 5 shows (u_{max}), (v_{max}) and (w_{max}), for Allee parameter (theta in [0,theta _{c})), scaled by their values at (theta = 0). These scaled maxima help us gauge the relative change in the maximum population densities arising due to the Allee effect. It is evident from our simulation results that the magnitude of the global maximum of vegetation does not change very significantly for increasing Allee parameter (theta), with its magnitude around (theta _c) being approximately 4 fold the value at (theta =0). However, the magnitude of maximum prey and predator populations change very significantly with respect to Allee parameter (theta) and exceeds over 10 fold the value obtained for (theta =0).Figure 5Global maximum of vegetation (u_{max}) (blue), prey (v_{max}) (red) and predator (black) populations, with respect to the Allee parameter (theta), scaled by their values obtained for (theta =0). Clearly, when Allee parameter (theta) is sufficiently large, the maximum prey and predator populations are an order of magnitude larger than that obtained in systems with no Allee effect.Full size imageWe then go on to numerically calculate the probability density of the vegetation, prey, and predator population densities, for increasing Allee effect parameter (theta). The tail of this probability density function reflects the influence of the Allee effect on the probability of obtaining extreme events. To illustrate this, we show the probability density function for the prey population in Fig. 6, for three different values of (theta). Extreme events are confined to the tail of the distribution that lie beyond the vertical red line, marking the (mu + 7 sigma) value in the figure. So it is clear from these probability distributions that the Allee effect in prey population promotes the occurrence of extreme events as the tail of the distribution is flatter and extends further with increasing Allee parameter (theta).Figure 6Probability Density Function (PDF) of the prey population v, for the system given by Eq. (1), with increasing magnitude of (theta) with (a) (theta =0), (b) (theta =0.015) and (c) (theta =0.02). The threshold for extreme event (mu + 7sigma) is denoted by vertical red dashed line.Full size imageIn order to ascertain that the extreme values are uncorrelated and aperiodic we examine the time intervals between successive extreme events in the population. Figure 7 (left panel) shows representative results for the return map of the intervals between extreme events in the prey population and it is clearly shows no regularity. The probability distribution of the intervals is also Poisson distributed and so the extreme population buildups are uncorrelated aperiodic events, as clearly evident from the right panel of the figure.Figure 7(Left) Return Map of (Delta t_{i+1}) versus (Delta t_i), and (right) Probability distribution of (Delta t_i) fitted with exponentially decaying function, where (Delta t_i) is the ith interval between successive extreme events, where an extreme event is defined at the instant when the prey population crosses the (mu +7sigma) line (cf. Fig. 2). Here (theta =0.024).Full size imageIn order to further quantify how Allee effect influences extreme events, we estimate the probability of obtaining large deviations, in a large sample of initial states tracked over a long period of time. We denote this probability by (P_{ext}), and we calculate it by following a large set of random initial conditions and recording the number of occurrences of the population crossing the threshold value in a prescribed period of time, with this time window being several orders of magnitude larger than the mean oscillation period. This time-averaged and ensemble-averaged quantity yields a good estimate of (P_{ext}). With no loss of generality, we choose the threshold for determining extreme events to be (mu + 7 sigma), i.e. when the variable crosses the (7 sigma) level, it is labelled as extreme.This probability, estimated for all three populations is shown in Fig. 8. First, it is clear from Fig. 8, that the probability of the occurrence of extreme events is the lowest for vegetation, and the highest for predator populations, for any value of the Allee parameter (theta in [0,theta _{c})). We also observe that, for values of the Allee parameter (theta) lower than a critical value denoted by (theta ^{u}_{c}) the probability of obtaining extreme events in the vegetation population tends to zero. Beyond the critical value (theta ^u_c), the vegetation population starts to exhibit extreme events. A similar trend emerges for the prey population. However, the critical value of the Allee parameter (theta) necessary for the emergence of a finite probability of extreme events, denoted by (theta ^{v}_{c}), is much smaller than (theta ^u_c). So for the prey population, a weaker Allee effect can induce extreme events.Figure 8Probability of obtaining extreme event in unit time ((P_{ext})), with respect to Allee parameter (theta), estimated by sampling a time series of length (T=5000), and averaging over 500 random initial states. Here we consider that an extreme event occurs when a population level crosses the threshold (mu + 7sigma). (P_{ext}) for vegetation, prey and predator are displayed in blue, red and black colors respectively. Note that there exists a narrow periodic window around (theta sim 0.02) (cf. Fig. 9), and so the large deviations in this window of Allee parameter are not associated with true extreme events, as they occur periodically.Full size imageNote that some mechanisms have been proposed for the generation of extreme events in deterministic dynamical systems, which typically have been excitable systems. These include interior crisis, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion. However the extreme events generated by these mechanisms occur typically at very specific critical points in parameter space, or narrow windows around it. The first important difference in our system here is that the extreme events do not emerge only at some special values alone. Rather, there is a broad range in Allee parameter space where extreme events have a very significant presence. This makes our extreme event phenomenon more robust, and thus increases its potential observability. This also rules out the intermittency-induced mechanisms that have been proposed, as is evident through the lack of sudden expansion in attractor size in our bifurcation diagram (Fig. 3) in general.However, interestingly, the system does have one parameter window where there is attractor widening and this gives rise to a markedly enhanced extreme event count. The peak observed in Fig. 8 can be directly correlated with a sudden attractor widening leading to a marked increase of extreme event in a narrow window of parameter space located near the crisis (see Fig. 9). Additionally, for a narrow window around (theta sim 0.02), the emergent dynamics is periodic. So the large deviations are no longer uncorrelated, and so they are not extreme events in the true sense.Figure 9Bifurcation diagram of prey populations with respect to Allee parameter, in the range (theta in [0.0189 : 0.0191]). Here we display the local maxima of the prey population. The parameter values in Eq. (1) are as mentioned in the text.Full size imageLastly we notice that the predator population shows extreme events for all values of (theta in [0,theta _{c})). So the predator population is most prone to experiencing unusually large deviations from the mean. We also observe that the probability of occurrence of extreme events in the predator population is not affected significantly by the Allee effect. This is in marked contrast to the case of vegetation and prey, where the Allee effect crucially influences the advent of extreme events. Also, for the predator population there is no marked transition from zero to finite (P_{ext}) under increasing Allee parameter (theta), as evident for vegetation and prey populations. More

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