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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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As stated, the primary goal of this study is to promote an objective decision support framework for water resource planning and management purposes within the context of the WEF security nexus, which takes into account the uncertainties imposed by the climate change phenomenon. Such a framework is “robust” since it takes the multi-dimensionality of water-related problems into account while addressing the uncertainties imposed by climate change projections. The basic components of this decision-making paradigm are depicted in Fig. 1. In principle, while this framework is sensitive to the uncertainties associated with the climate change projections, it can provide a dynamic water resources planning and management scheme promoted within the WEF security network. Thus, in addition to the status quo, a series of climate change projections (i.e., RCP 2.6, RCP 4.5, and RCP 8.5) are also integrated into the proposed decision support framework. In essence, the main components of the proposed framework are simulation and operation of the water resources system based on the standard operation policy (SOP), evaluating the system’s efficiency through a series of quantitative performance criteria, and finally, applying the MADM-based framework to opt for a robust system renovation setting.Figure 1Basic components of the robust decision-making paradigm for water resources planning and management.Full size imageSimulating the water resources systemSOP is a primitive, and perhaps the most-well-known real-time operation policy in water resources planning and management14. The core principle here is to minimize the prioritized water shortage at the current time step with no conservation policy (e.g., hedging rules) in place. SOP, as a standard rule curve (RC), determines how the operator should behave at any given state of a reservoir15,16. This rule curve is established as an attempt to balance various water demands including but not limited to flood control, hydropower, water supply, and recreation17. A SOP operating system attempts to release water to meet a water demand at the current time, with no regard to the future.In general, SOP can be mathematically expressed as18:$$R_{t} = left{ {begin{array}{*{20}c} {D_{t} } \ {AW_{t} } \ 0 \ end{array} – S_{min } } right.begin{array}{*{20}c} {} & {if} & {AW_{t} > S_{min } } \ {} & {if} & {AW_{t} > S_{min } } \ {} & {if} & {AW_{t} le S_{min } } \ end{array} begin{array}{*{20}c} {} & {and} & {AW_{t} – S_{min } ge D_{t} } \ {} & {and} & {AW_{t} – S_{min } < D_{t} } \ {} & {} & {} \ end{array} quad t = { 1},{ 2},{ 3}, , ... , ,T$$ (1a) where$$AW_{t} = S_{t} + Q_{t} - Loss_{t}$$ (1b) in which Rt = amount of water supplied during the tth time step; Dt = consumers’ water demand during the tth time step; AWt = amount of available water during the tth time step; St = amount of stored water during the tth time step; Smin = dead storage of the reservoir; Qt = inflow during the tth time step; Losst = net water loss (i.e., precipitation minus evaporation) of the reservoir during the tth time step; and T = total number of time steps in the operational horizon.In practice, however, a different type of water demand leads to a different interpretation of water shortage. There are cases in which the stakeholders’ needs are represented by a set of volumetric demand targets, and the decision-makers’ objective would be to minimize the water deficit based on a set of priorities for these demands. This is a typical case for agricultural, domestic, industrial, and environmental demands. For hydropower generation, however, a conventional interpretation of SOP would be to generate maximum electricity permitted by the power plant capacity (PPC) at each given time step19. For a hydropower system, the amount of water needed to reach a power plant capacity is given by19:$$R_{t} = frac{{86400 times PF times Countday_{t} times PPC}}{{gamma_{w} times g times eta times Delta H_{t} }}$$ (2) in which, γw = water specific weight; g = gravitational acceleration; η = efficiency of the hydropower system; ΔHt = height difference between the reservoir water level and the tailwater level at time step t; Countdayt = number of days within time step t; and PF = plant factor of the hydropower system.As stated earlier, applying an SOP-based plan requires a set of pre-defined priorities to advise decision-makers concerning the order, in which each of these demands is to be met. The major water demands include drinking, industry, environment, agriculture, and hydropower. Thus, according to the SOP’s principle, the decision-makers, first, allocate the available water to meet the demand of the stakeholder with the highest priority (i.e., the domestic and industrial demand). After this first water demand is fully satisfied, the available water can be used for the next demand. Such an allocation process continues until no water is available. It should be noted, however, that if the released water in each stage passes through the penstock equipped with the turbines, electricity can be generated. The amount of energy generated in previous stages must be accounted for before computing the amount of water released for hydropower purposes.Performance criteriaPerformance criteria are, in essence, quantitative measures that can provide a practical insight for the decision-makers regarding the status of a system. This definition covers a broad spectrum of mathematical representations, which can range from simple mathematical formulas such as the average of a specific output to more complex and probability-based entities20,21. The most fundamental and universal probability-based performance criteria are reliability, resiliency, and vulnerability22,23,24. In essence, reliability is the probability of successful function of a system; resiliency measures the probability of successful functioning following a system failure; and vulnerability quantifies the severity of failure during an operation horizon25. It should be noted that these three criteria assess different aspects of a water resources system, and as such, they complement one another26. For more information regarding these probabilistic performance criteria, the readers can refer to Sandoval-Solis et al.27 and Zolghadr-Asli et al.20.In this study, the concept of levelized cost of energy (LCOE) is utilized for economic evaluation. The LCOE of a given hydropower system is the ratio of lifetime costs to lifetime electricity generation, both of which are discounted back to a common year using a discount rate that reflects the average cost of capital28. The LCOE of renewable energy systems depends on the technology, geographic criteria, capital and operating costs, and the efficiency of the system. The LCOE can be mathematically expressed as follows29:$$LCOE = frac{{sumnolimits_{t = 1}^{n} {frac{{I_{t} + M_{t} + F_{t} }}{{left( {1 + r} right)^{t} }}} }}{{sumnolimits_{t = 1}^{n} {frac{{E_{t} }}{{left( {1 + r} right)^{t} }}} }}$$ (3) in which It = investment expenditures in year t; Mt = operation and maintenance expenditures in year t; Ft = fuel expenditures in year t; Et = electricity generation in year t; r = discount rate; and n = economic life expectancy of the system.MADMMADM is an umbrella term to describe a series of frameworks, which aim to help individuals or a group of individuals to prioritize a series of discretely defined alternatives with regard to a set of evaluation attributes30,31. MADM can provide the necessary means to conduct planning and management under changing circumstances such as those under climate change conditions10,32. According to one of the basic principles of MADM, the decision-maker can use the similarity of the feasible alternatives and the preferential result and/or incongruity of the undesirable alternatives. The notion mentioned above is, chiefly, the core principle of the reference-dependent theory33. Accordingly, the reference-based branch of the MADM methods can, itself, be classified into two major groups: screening methods and ranking methods. Screening methods eliminate alternatives that cannot satisfy the pre-determined conditions for the desirable solution, while ranking methods order all the alternatives from the best to the worst34.Pioneered by Hwang and Yoon35, the technique for order references by similarity to an ideal solution (TOPSIS) is a compensatory, objective MADM solving method rooted from the basic principles of the reference-dependent theory. The core idea is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution36. The basic computation algorithm of TOPSIS can be summarized as follows37,38:Step I: Construct the original decision matrix (X), where m feasible alternatives are to be evaluated based on n evaluation criteria:$$X = left[ {begin{array}{*{20}c} {x_{11} } & {x_{12} } & cdots & {x_{1n} } \ {x_{21} } & {x_{22} } & cdots & {x_{2n} } \ vdots & vdots & ddots & vdots \ {x_{m1} } & {x_{m2} } & cdots & {x_{mn} } \ end{array} } right]$$ (4) in which xij = the element of the ith alternative concerning the jth criterion.Step II: Defining the reference alternatives [i.e., the ideal solution (s+) and the negative-ideal solution (s−)]. To do so, first, the elements of the decision matrix that are associated with negative criteria must be redefined by using the following equation:$$x_{ij}^{ * } = frac{1}{{x_{ij} }}$$ (5a) The elements of the decision matrix that are associated with positive criteria would remain the same:$$x_{ij}^{ * } = x_{ij}$$ (5b) The ideal alternative is an arbitrarily defined vector, which describes the aspired solution to the given problem, while the inferior alternative is an arbitrarily defined solution that represents the most undesirable option for the given MADM problem. Here, the ideal and negative-ideal solutions would be represented with two separate vectors where each pair of the corresponding elements in these vectors is, respectively, the maximum and minimum values of (x_{ij}^{ * }) with regard to each of the evaluation criteria.Step III: Each element of the decision matrix should be normalized by using the following equation:$$p_{ij} = frac{{x_{ij}^{ * } }}{{sqrt {sumnolimits_{i = 1}^{m} {x_{ij}^{ * 2} } } }}$$ (6) in which pij = the normalized performance value for the ith alternative with respect to the jth criterion.Step IV: The weighted normalized preference value (zij) can be computed as follows:$$z_{ij} = p_{ij} times w_{j} quad forall i,j$$ (7) in which wj = the weight (i.e., the importance value) of the jth criterion. The weights assigned to the evolution criteria reflect their relative importance to the decision-makers. The higher the weights are, the more crucial their roles would be in the selection process. Chiefly, these weighting mechanisms are either subjective in nature or follow an objective procedure. In the subjective approaches, the weights of the attributes are assigned based on the performance information given by the decision-maker, whereas in the objective approaches, the weights of the evaluation attributes would be obtained by using the objective information extracted from the decision matrix39. Shannon’s Entropy method, used in this study as the weight assignment mechanism, is a well-known objective weighting technique40. This method tends to assign the highest weight to an evaluation attribute with the highest dispersity in its values. For more information on the computational framework of this method, the readers can refer to Lotfi and Fallahnejad41.Step V: In this step, every given alternative is compared to the reference points, namely, the ideal and inferior alternatives. The described procedure, which is known as the separation measurement in TOPSIS, can be mathematically expressed as follows35:$$D_{i}^{ + } = sqrt {sumlimits_{j = 1}^{n} {left( {z_{ij} - z_{j}^{ + } } right)^{2} } }$$ (8) And$$D_{i}^{ - } = sqrt {sumlimits_{j = 1}^{n} {left( {z_{ij} - z_{j}^{ - } } right)^{2} } }$$ (9) in which (D_{j}^{ + }) and (D_{j}^{ - }) = separation measurements of the jth criterion with respect to the ideal and inferior alternatives, respectively.Step VI: The relative closeness to the ideal solution (χi), which can be used to rank the desirability of the feasible alternative, can be computed as follows35:$$chi_{i} = frac{{D_{i}^{ - } }}{{D_{i}^{ + } + D_{i}^{ - } }}quad forall i$$ (10) The further this distance (i.e., larger values of χi), the more desirable the alternative would be.Robust multi-attribute frameworkAs stated, each climate change scenario depicts a unique future with regard to the changing climate, which in turn introduces an element of uncertainty to the projected performance of water resources systems during their operation horizon. Furthermore, downscaling methods, which link these projected changes in the global climatic pattern to a local or regional scale, can be another source of uncertainty. Naturally, for long-lasting water infrastructure such as a hydropower system, addressing these uncertainties in a proper and timely manner can be one of the key components of a robust project. Thus, this study aims to not only evaluate the system’s performance under the status quo but also assess the credibility of the system under the projected climate change conditions.The other characteristic one might expect from a robust project is its ability to take into account the multi-dimensionality nature of water-related infrastructure. Most notably, addressing the WEF security nexus must be a priority in water resources planning and management. Resultantly, any robust decision-making paradigm for water resources planning and management purposes should also account for the other pillars of the WEF nexus (i.e., energy and food sectors), as they would be consequentially affected by such decisions. It is also important to note that these sectors could be affected by the climate change phenomenon. The other crucial feature of a robust decision-making paradigm is that it should be able to account for the socio-economic, environmental, and technical factors that determine the overall quality of the project. Such a decision-making paradigm is depicted in Fig. 2. This notion in practice, however, can typically lead to a mega decision matrix composed of numerous criteria and alternatives that can be overwhelming if the subjective MADM methods are to be employed. This study, thus, employs an objective MADM framework (i.e., TOPSIS/Entropy) to help overcome the above-described problem. The basic idea is to promote a universal and practical decision support framework that enables the water resources planners and managers to account for the intricacies of the WEF security nexus while simultaneously taking the uncertainties of climate change projections into account. Figure 3 illustrates the flowchart of the proposed decision support framework.Figure 2Schematic diagram of the MADM problem.Full size imageFigure 3Flowchart of the proposed framework.Full size image More

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