Ecology

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As a starting point, we compare the vulnerability of four districts in Lyon, Paris (France), Firenze (Italy) and New York (US). These cities were chosen as emblematic of different topologies, resulting from different historical urban layering. The historic center of Firenze (panel b in Fig. 1) is mainly characterized by a dense urban fabric with a medieval signature of narrow and winding streets24. In Paris (panel c), Haussman’s renovation plan at the end of the 19th century supplemented the North–South and East–West ancient crossroad by a second network of concentric large avenues25. The rectilinear grid of Manhattan, New York, originates from 181126,27 and extends along the spine of Manhattan island (panel d). Despite the significant difference in size, a similar regular pattern is found in the modern urban area of Lyon (panel a), developed in the second half of the 19th century. In the insets of Fig. 1, we report for each city a polar histogram of the orientation of the streets. Although greater variability is observable for the orientation of the streets in the urban areas of Firenze (panel b) and Paris (panel c), two main orthogonal axes are found in the spatial structure of each city.The urban networks analysed in this work were delimited in order to be large enough to include the distinctive patterns of these four cities. The edges of the areas were traced along physical boundaries (e.g., rivers, parks, railways, large avenues) which act as elements of discontinuity in the dispersion process. Where not possible, the break was forced along wide streets.We promptly computed vulnerability maps for the selected urban areas by means of the centrality metric we derived in20 and recall in the “Methods” section. The nodes with the highest centrality values (V) are the most vulnerable as they correspond to the best spreading locations in the urban fabric. The spreading potential of a node is evaluated based on the extent of the area that is contaminated when the release takes place in this same node.We report in Fig. 1 the vulnerability maps of the four urban areas for the indicative scenario of a wind blowing at an angle (phi =45^circ). In the insets of Fig. 1, the wind direction is indicated with a red arrow. Given the different orientation and structure of the street networks, (phi) is defined as a clockwise angle with respect to the main axis of the city, which is identified as the longest bar in the polar histogram of street orientation.To extend the analysis to multiple meteorological scenarios, we estimated the vulnerability of each node (seen as a spreading source) for eight different wind directions ((phi =0^circ), (45^circ), (90^circ), (135^circ), (180^circ), (225^circ), (270^circ), (315^circ)). In this way, for each city, we obtained an extended dataset of vulnerability values that we represent in a compact way by means of a cumulative distribution function, as shown in Fig. 2a. The intercept of the cdf represents the nodes with null vulnerability. These are mostly located along the physical edges of the domain where the pollutant gas is blown away by the wind without affecting other streets. Where the delimitation of the network is forced (for example on the sides of central park as regards Manhattan), the interruption of the propagation, in the vulnerability model, is also constrained. This does not result in any artificial effect when the boundary is located upwind with respect to the network (propagation carries on from the boundary towards the considered urban area). On the other hand, when the boundary is downwind, vulnerability can be there underestimated. Considering the multiple wind directions simulated and the small number of nodes belonging to these edges (1% of the total number of network nodes), this effect has been calculated negligible to the purposes of this work.According to the mean values (vertical dashed lines) of the distributions reported in Fig. 2, New York is the most vulnerable city on average, while Firenze is the most protected. The vulnerability of New York and Lyon are the most sensitive to changes in wind direction, as shown by Fig. 2.b, where a polar histogram reports the mean vulnerability for each city for the eight directions of the approaching wind. In general, the spreading potential is more effective when the wind is oblique ((phi =45^circ ,) (135^circ), (225^circ), and (315^circ)) to the main orthogonal axes of the street network, as evidenced by the higher vulnerability observed for the dark gray sectors of Fig. 2b. We also notice that vulnerability for parallel ((phi =0^circ ,) (phi =180^circ)) and perpendicular ((phi =90^circ ,) (phi =270^circ)) wind directions is quite similar. This seems counterintuitive as previous studies (e.g.,28,29) have reported that a perpendicular wind is much more unfavorable for the dispersion of pollutants in a street. In this regard, we underline that (phi) is here defined with respect to the main axis of the city, so for (phi =90^circ) not all streets will be perpendicular to the wind direction. For example, in the regular network of Manhattan we expect the number of perpendicular streets to be similar to that of parallel streets, when (phi =90^circ).Figure 1Vulnerability maps for (a) Lyon, (b) Firenze, (c) Paris, and (d) New York for a wind direction of (45^circ) with respect to the main axis of the urban fabric. The polar histograms in the insets report the distribution of street orientation, while the red arrows represent the wind direction with respect to the street network. Panels a1–d1 show the urban pattern in a rectangular area of 0.5 km(^2) (reported in panels a–d) for the cities of Lyon, Firenze, Paris, and New York, respectively. Background images made with QGIS 2.18 (https://qgis.org).Full size imageFigure 2Vulnerability distribution for different cities and wind directions. (a) Cdf of node vulnerability for the different cities under eight different wind directions. The mean vulnerability is shown as a dashed line and reported numerically together with the standard deviation (in parentheses). (b) Mean vulnerability of city networks for each wind direction. Colors blue, yellow, green and magenta correspond to the urban networks of Lyon, Firenze, Paris and New York, respectively.Full size imageThe reasons for the different resilience of cities (and their patterning) to gas propagation are embedded in the centrality metric adopted to compute urban vulnerability. The key factors for node vulnerability can then be analytically recognized in the metric definition (Eqs. 4–5 in “Methods”): the highest vulnerabilities are achieved when the set of reachable nodes ((mathcal {V})) from the source node is large, and the paths connecting the source and the reachable nodes ((d_{sr})) are short, i.e. the propagation cost ((omega)) along the paths is minimal. In other words, the spots in a city (i.e. nodes in a network) with the highest spreading potential are those from which a toxic plume can reach many other locations with significant concentration. Going beyond the vulnerability results, we aim here to decompose the aforementioned elaborate and meaningful quantities (the set of reachable nodes, the shortest paths, the propagation cost) in elementary properties of the urban area in order to link the vulnerability of a city to its tangible characteristics.We start by disassembling the propagation cost associated to each street. Given a source node, a pollutant plume will propagate along the streets downwind the node. The propagation cost of each street (Eq. 4) describes the decay of concentration that the plume undergoes when it propagates along the street. Neglecting physico-chemical transformations, this cost depends on the transport processes within the streets and is a function of two dimensionless quantities: a geometrical ratio between the length (l) and height (h) of the street canyon, and a dynamic ratio between the exchange rate of pollutants towards the atmosphere above roof level (v) and the advective velocity along the longitudinal (u) axis of the street. According to30 and31, these two velocities can be parametrized as a function of the external wind intensity, the cosine ((theta)) of the angle between the wind direction and the orientation of the street, the geometry of the street canyon (its length l, height h and width w) and the aerodynamic roughness of building walls. As detailed in the Methods, the dependence of the propagation cost on the external wind intensity disappears as both velocities u and v scale linearly with it. Assuming constant aerodynamic resistance of the surfaces, the parameters l, h, w, (theta), remain the relevant building blocks for the propagation cost along a street.We underline that the parametrizations adopted here for the transport mechanisms in a street are based on the up-to-date literature and are currently employed in operational models (see the Methods section for mode details). Any refinements to this transport model may be included in the future. In this case, the cost associated to each street may depend on additional parameters that, however, we expect to be of second-order importance to those listed above.While pollutant transport in a single street canyon (i.e. the propagation cost) has been easily broken down into its basic elements, the information enclosed in the shortest paths ((d_{sr})) and in the set of reachable nodes from the source ((mathcal {V})) is much more challenging to trace back to evident properties of the city. These quantities depend on the sequence of streets that must be traveled to connect a source node to the surrounding nodes, i.e. on the way the streets are interconnected. The information is thus primarily topological. However, we point out that the interconnectivity of the network is not frozen, but dynamic, as it is given by the reaction of the urban structure to the direction of the external wind. In fact, the links of the street network are directed according to the orientation of the approaching wind. Moreover, the connectivity between the nodes is limited by the decay of the concentration along the streets. Although a target node may be reached from the source node by means of a path across the network, the two nodes may not actually be connected by a propagation path as the pollutant concentration may vanish along the path. For these reasons, traditional descriptors of network topology cannot be applied directly to describe the topological component of the vulnerability. Instead, we have to look for tailored and simple indicators that can express the wind-driven interconnectivity of the street network and the reachability potential between the nodes.Focusing on a node as spreading source, we infer that the number of links in its downwind area gives a first estimate of the potential for a release in the node to affect many other locations in the network. To delimit this downwind area, we adopt the concept of n-hop neighborhood32,33. Two nodes are n hops apart if it is possible to reach the target node from the source node by traveling n links. We identify the downwind area of the source node as the subnetwork composed by the nodes that are reachable from the source via at most n hops along the directed links. We propose the number of links in this neighborhood (k) as a suitable measure of reachability from the node. This reachability depends upon three features: (i) the local structure of the street network, (ii) the direction of the wind, and (iii) the topological distance n. This latter parameter is intuitively correlated to the intensity of the release. More precisely, it depends on the ratio between the magnitude of the toxic release at the source and the threshold value for pollutant concentration to be significant. In this work, n is taken as constant and its value is obtained from an optimization analysis detailed in the “Methods” section (Fig. 6) .Once the (n-hop) neighborhood of a node is delimited, the number of links k is not exhaustive in giving information about the properties of the paths connecting the source to the other nodes of the neighborhood. For the same k, different structures of the neighborhood can take place (see Fig. 6b), with consequent different outcomes for the propagation process that we are breaking down to basic components. The higher the number of links outgoing each node of the neighborhood, the higher the potential concentration for the k links, as they are topologically closer to the source. This feature can be accounted for by means of a simple branching index (b) for the node neighborhood, defined in Eq. 8 as the average outdegree for the nodes belonging to the neighborhood34.The disassembling analysis presented above suggests that the spreading potential of a node, and thus its vulnerability, mainly depends on the topological parameters k and b and on the geometrical characteristics of its neighborhood, i.e. L, H, W, (Theta), where the capital letters are used to indicate the local average (over the n-hop neighborhood) for the length (l), height (h), width (w), and orientation ((theta)) of the street canyons.In adopting averaged geometrical properties, we are assuming that these characteristics are rather homogeneous in the surroundings of a node. While the height, width and length of the street canyons are actually quite uniform on a local scale, especially in European city centers, the same does not apply to the orientation of the streets. The streets of a neighborhood intersect each other at different angles (e.g., at (90^circ) in grid plans), and the intensity of the wind in the streets changes strongly with their orientation. Low wind streets act as bottlenecks in the propagation paths, thus strongly influencing the spreading dynamics. For this reason, the standard deviation of street orientation in the neighborhood ((sigma)) is expected to be an additional topological index of node vulnerability.To assess whether the identified parameters are valuable basic elements of node vulnerability, we perform a regression analysis adopting a simple (but versatile) non linear model of the form:$$begin{aligned} V_{pred}=alpha L^beta H^gamma W^delta Theta ^epsilon k^zeta b^eta (1-sigma )^lambda . end{aligned}$$
(1)
We estimate the coefficients (alpha) to (lambda) by means of a nonlinear least square technique (namely the fitnlm function in Matlab) that minimizes the sum of the squares of the residuals between the predicted vulnerability (V_{pred}) and the vulnerability V obtained from the centrality metric (Eq. 5 in “Methods”). The regression is performed considering all the scenarios presented in this study: four different urban networks and eight different wind directions. The p-values for the coefficients (alpha) to (lambda) tend to zero, indicating that the relationships between the independent variables and the observations (V) are statistically significant. Note that in Eq. (1) we adopt (1-sigma) as predictor, instead of (sigma), to avoid null entries, as (sigma) takes value in [0 1). To explain the reason for this range for (sigma), we point out that the angle between the wind direction and the street axis is defined in [(-90^circ) (90^circ)]. As a consequence, the cosine ((theta)) of the angle varies in [0 1] and the standard deviation of (theta) (i.e. (sigma)) varies in [0 1).The scatter plot in Fig. 3 compares (V_{pred}) against V. Points correspond to the nodes of the four urban networks in the eight wind scenarios. The figure suggests that 80% of the spreading capacity (V) of a spot in a city can be grasped from the basic geometrical and topological characteristics of its neighborhood. To identify the most influential parameters in the regression, we evaluate the gain in the coefficient of determination (R^2) as they are progressively included in the model (red circles in the inset). The quantities are entered in order to optimize (R^2) at each addition. Alternatively, the role of each parameter can be evaluated adopting the concept of unique contribution (triangles in the inset), i.e. the loss in the coefficient of determination induced by the exclusion of the parameter from the model35. Both analyses reveal k and (sigma) as the main indicators for the vulnerability of a node. Actually, more than 60% of the total variance (inset in Fig. 3) is explained by these two parameters, unveiling the crucial role of topology in governing the dynamics of pollutants in urban areas. The effect of the geometrical properties (L, H, W, (Theta)) of the street canyons is secondary. Among these, the contribution of the building height (H) is the most remarkable as its contribution, combined with that of the two topological parameters k and (sigma), brings the correlation to almost its maximum value.Given these results, it is enlightening to show some tangible examples of how the three simple indicators k, (sigma) and H dominate urban vulnerability. We wonder which of these properties determine the distinct vulnerability of neighboring areas belonging to the same district, and which ones differentiate the resilience of cities with a different urban history.Figure 3Correlation of node vulnerability with basic geometrical and topological parameters of the street network. Color (blue to red) is associated to point density. Left y-axis of inset: trend of the coefficient of determination (R^2) as the urban indicators are progressively included in the model. Right y-axis of inset: unique contribution of the indicators.Full size imageFigure 4a shows the spatial distribution of the key parameters k, (sigma) and H and of node vulnerability, for Manhattan and a wind direction (phi =45^circ). In panel b, high street reachability (k) is observed in the central part of Midtown, in the heart of Downtown, and near Wall Street. An homogeneous distribution in the orientation of the streets with respect to the incident wind (low values for (sigma) and thus high (1-sigma)) is especially found in Midtown (panel c). Finally, in panel d, high-buildings (H) distinguish the Financial District and East Midtown. A perfect match between the four layers is not expected as vulnerability is given by the synergistic contribution of the different parameters. However, in line with the results of the regression model shown in Fig. 3, a positive correlation is observable between the most vulnerable areas (circled areas comprising the nodes with highest V in panel a of Fig. 4) and those with the highest values for the three indicators. In these areas, high buildings inhibit the vertical exchange of pollutants between the streets and the atmosphere above, as largely discussed in literature (see e.g.36,37). This inhibition limits the concentration decay along the propagation paths and facilitates large-scale contamination. Moreover, the great number (k) of streets topologically close to the node increase the impact of the release. The effect of (sigma) is significant especially for the vulnerability of Midtown. Here, since (phi =45^circ) and the street network is regular, (theta) (the cosine of the wind-street angle) is almost the same for all the streets. Therefore, the standard deviation of (theta) ((sigma)) is low and the predictor (1-sigma) is high. Physically, this means that the external wind approaches all the streets with almost the same angle. As a consequence, the intensity of the longitudinal wind in the streets is similar (the street aspect ratio is also similar) and the propagation takes place equally along both the dominant and lateral segments of the street network38, thus favoring the spread over large areas. Although high values of H and (1-sigma) can be detected in the North-East corner of Midtown too, here the vulnerability is mitigated by a higher discontinuity in the urban pattern (low k). This feature, together with the great overlapping of red areas in panel a with those in panel b, evidences the key role of street reachability (k) in the heterogeneity of vulnerability between areas of the same urban district.Figure 4Street network of Midtown and Downtown Manhattan. Node color is associated to node vulnerability (V), and to its key indicators: street reachability (k), inhomogeneity in street orientation ((sigma)), and average height of buildings in the node neighborhood (H).Full size imageFrom these observations, we move to a broader view and investigate the structural fragility of a city as a whole. In Fig. 5a–c, we report the probability density function (pdf) of the key parameters k, (1-sigma) and H. For each city, the pdf is calculated over all the network nodes and for the different wind directions. So, each pdf is representative of eight different networks for the same urban area. In panel a, the distributions for the four cities are quite similar but the tails of the pdfs highlight that the highest values for street reachability (k) occur in the street networks of Lyon and New York. The homogeneity in street orientation with respect to the wind (panel b), expressed by (1-sigma), exhibits a bimodal distribution and a slightly higher mean for the regular street network of New York. The two peaks are associated to distinctive wind scenarios, as will be discussed below. Also in this case, the observation of the tails of the pdfs reveals that high values of the vulnerability indicator are more probable in Lyon and New York. Finally, the distribution of building height (panel c) presents the most marked difference between the considered architectures, with high-rise buildings contributing to the heavy pdf tail of Manhattan. Comparing these results with those in Fig. 2a, Manhattan’s greatest vulnerability appears to be due to the greater depth of the urban canyons (high H) and the greater homogeneity, on average, in wind-street orientation (high (1-sigma)). Conversely, the medieval structure of Firenze, with higher heterogeneity in street orientation (low (1-sigma)) and low buildings (low H), enhances street ventilation and hinders propagation over long distances. Moreover, the tails of the pdfs for k and (1-sigma) reveal the role of topology in the higher variability of vulnerability values (given by the standard deviation of the pdfs in Fig. 2) for the street networks of Lyon and Manhattan.After discussing the behavior of the single parameters, we assess the synergistic contribution of the three quantities. To this aim, we define a simple correlation index (rho =widehat{k} cdot (1-widehat{sigma }) cdot widehat{H}), where the hat denotes a min-max normalization of the parameters, i.e. the range of values of each parameter is rescaled in [0, 1]. For the urban areas of Manhattan, Lyon, Paris, and Firenze, (rho) gives 0.039, 0.017, 0.012, and 0.011, respectively. This ranking complies with the ranking inferable in Fig. 2 for the average vulnerability of the cities. This result confirms that vulnerability occurs when the three parameters are correlated, as already evidenced in Fig. 4.To make the picture even more fascinating, it is worth noting that the role of topology, shown above as key, is dynamic as it varies according to the direction of the wind impacting the urban fabric. In panels d to f of Fig. 5, the pdfs of k, (1-sigma) and H are distinguished for four wind directions ((phi =0^circ), (45^circ), (90^circ) and (135^circ)). For each angle, the statistics are calculated over the examined cities, together. Although wind orientation alters the direction of the network links, and thus the delimitation of the n-hop neighboring area of each node, street reachability (panel d) and building height (panel f) remain statistically invariant for the different wind directions, suggesting a rather isotropic structure of the urban fabric. On the other hand, the variability in street orientation with respect to the wind (panel e) presents two distinctive trends for wind directions aligned with or oblique to the main axes of the street network. To explain this behavior, we refer to the simple case of a grid-like urban plan, like Manhattan’s plan. When (phi =0^circ) or (90^circ), (theta) (the cosine of the angle between the street and the wind direction) mostly switches between 0 (for the streets aligned with the wind) and 1 (for the orthogonal streets), resulting in a high standard deviation over the neighborhood (low (1-sigma)). When (phi =45^circ) or (135^circ), instead, the incident angle (theta) mainly takes intermediate values, leading to higher values for (1-sigma). This distinctive behavior is clearly detectable in the two peaks that we have observed in panel b for the regular grid of Lyon and New York. The left peak of the bimodal distribution corresponds to the scenarios with aligned wind directions, while the right peak occurs for oblique wind directions over the city. A more irregular street pattern in Firenze and Paris adds random contributions to the way the wind approaches the street, thus altering this bimodal shape. Going back to panel e, the greater homogeneity in wind-street orientation (higher (1-sigma)) for (phi =45^circ) or (135^circ) gives insights into the higher vulnerability found for the scenarios with these wind directions in almost all cities (dark gray sectors in Fig. 2b). This result is confirmed by the correlation ((rho)) between the three rescaled parameters ((widehat{k}), (1-widehat{sigma }), (widehat{H})). The correlation (rho) is estimated separately for the different wind directions, but considering the nodes from the four urban areas together. For oblique wind directions, (rho) is about twice ((rho =0.035)) the value found for the aligned wind directions ((rho =0.018)).Figure 5Probability density function of the key parameters k, (1-sigma), H. In the first row, each curve refers to a city and includes vulnerability data from eight different wind directions. In the second row, each curve corresponds to a specific wind direction and includes vulnerability data from the four cities, together.Full size image More

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Well-mixed populationAs shown in the Methods section, in a well-mixed population, the model can be described in terms of the replicator-mutator dynamics. I begin, by a case where the quality of the two resources are similar, r1 = r2 = r, and plot the frequency (solid blue), the average payoffs from the game (dashed red), and the amplitude of fluctuations (dotted blue) for different strategies in Fig. 1a–d. Here, the replicator dynamic is solved starting from a uniform initial condition in which all the strategies’ initial frequency is equal. The results of simulations in finite populations are in good agreement with the replicator dynamics results (see Supplementary Note 2 and Supplementary Figs. 1, 2, 3, and 4 for comparison to simulations). Throughout this manuscript I fix c = 1.Fig. 1: The frequency, game payoffs, and the amplitude of fluctuations for different strategies.The frequency (solid blue), game payoffs (dashed red), and amplitude of fluctuations (dotted blue) of costly cooperators (a), costly defectors (b), non-costly cooperators (c), and non-costly defectors (d), as a function of the enhancement factor, r. As r increases, above a first threshold (r* = 1 + cg), cooperation in the costly institution evolves, and above a second threshold (approximately r = 2) cooperation in both the costly and the free institutions evolves. For medium r, the system shows periodic fluctuations. Parameter values: g = 5, nu = 10−3, π0 = 2, cg = 0.398. The replicator dynamic, derived in the Methods Section, is solved for 9000 time steps, and the time averages are taken over the last 2000 time steps.Full size imageFor small enhancement factors r, the dynamics settle in a fixed point where only defectors in the free institution survive. The advantage of the costly institution becomes apparent as r increases beyond r* = 1 + cg, such that the maximum possible payoff of the costly institution, which is achieved when nobody defects and is equal to r − 1 − cg becomes positive. As shown in the Methods, a focal defector in a group composed of ({n}_{C}^{1}) costly cooperators and ({n}_{D}^{1}) other costly defectors (and ({n}_{C}^{2}+{n}_{D}^{2}) individuals who prefer the free resource), obtains a payoff of ({n}_{C}^{1}r/({n}_{D}^{1}+{n}_{C}^{1}+1)-{c}_{g}). This payoff becomes negative for a small enough value of ({n}_{C}^{1}) (or a large enough value of ({n}_{D}^{1})). Since groups with a small number of costly cooperators are drawn with a high probability when ({rho }_{C}^{1}) is small (these probabilities can be derived in terms of multinational coefficients, see the Methods), the average payoff of a costly defector remains negative in this regime (for instance, given at the transition ({rho }_{C}^{1}approx nu), a mutant costly defector finds itself in a group with no costly cooperator with probability ({(1-{rho }_{C}^{1})}^{g-1}approx {(1-nu )}^{g-1}), which is close to 1 for low mutation rates, and pays a pure cost of −cg). On the other hand, a costly cooperator’s payoff is equal to (({n}_{C}^{1}+1)r/({n}_{D}^{1}+{n}_{C}^{1}+1)-{c}_{g}-1), which for small enough ({n}_{D}^{1}) becomes positive. As in this region, the frequency of costly defectors, ({rho }_{D}^{1}), is small, such group compositions occur with a high probability (at the transition, ({rho }_{D}^{1}approx nu), and thus the probability that a costly cooperator joins a group with no costly defectors is ({(1-{rho }_{D}^{1})}^{g-1}approx {(1-nu )}^{g-1}), which is close to 1 for low mutation rates). Consequently, the average payoff of costly cooperators from the game becomes positive, and thus, larger than the dominant non-costly defectors’ payoff (who receive a payoff of zero). Consequently, the frequency of costly cooperators rapidly increases at r*. However, due to the rapid increase in the frequency of costly cooperators at r*, the probability of formation of such mixed groups increases, and costly defectors start to appear in the system. Further increasing r in this region, the frequency of costly cooperators and costly defectors increases at the expense of non-costly defectors.As r increases above a second threshold, cyclic fluctuations set in, and the dynamics settle in a periodic orbit. An example of this periodic orbit is presented in Fig. 2a, b. Interestingly, the average payoff of costly cooperators, costly defectors, and non-costly defectors in this region remains close to zero despite the evolution of cooperation. Although individuals constantly update their strategy to overcome others, no strategy wins in the evolution. Instead, individuals engage in a winnerless red queen dynamic. The game payoffs of costly cooperators and costly defectors fluctuate around zero (which is equal to the game payoff of non-costly defectors). The dynamics of the system in this regime resembles the frequency-dependent selection in the host-parasite evolution, coined the red queen dynamic based on the fact that no matter how much they run, all end up in the same place53,54. On this basis, I call this periodic orbit the red queen periodic orbit.Fig. 2: Red queen and black queen orbits.The frequency of different strategies (a) and the game payoffs (b) in the red queen, and the black queen (c, d) periodic orbits. In the red queen orbit, cooperators in the costly institution survive. However, the payoff of the surviving strategies fluctuates around zero, and none dominate others. In contrast, cooperators in both institutions evolve in the black queen orbit, and cooperators of each type suppress defection in their opposite institution. Consequently, the payoff of all the strategies starts to deviate from zero. Parameter values: g = 5, nu = 10−3, π0 = 2, and cg = 0.398. In (a, b) r = 1.7, and in (c, d) r = 2.2.Full size imageThe existence of a costly institution can facilitate the evolution of cooperation in its competing free institution too. As the amplitude of fluctuations increases, episodes where most of the individuals prefer the costly institution occur. During these episodes, ({rho }_{D}^{2}) drops to a small value. Consequently, the probability that a mutant non-costly cooperator finds itself in a group devoid of non-costly defectors ({(1-{rho }_{D}^{2})}^{g-1}), increases. In such groups, non-costly cooperators receive a payoff of r−1, which is larger than the payoff of all the other strategies and outcompete other strategies. At this point, a second periodic orbit emerges in which cooperation in both the costly and free institutions evolves. The evolution of cooperation in the free institution can, in turn, have a positive impact on cooperation in the costly institution. This is the case because above the point where cooperation in the free institution evolves, the frequency of individuals who prefer the free institution starts to increase by increasing r. This effect decreases the frequency of those who prefer the costly institution and its effective size. This decreases the mixing probability between costly cooperators and costly defectors and increases the costly cooperators’ payoffs. Consequently, a functional complementation between cooperators with different game preferences emerges, which is reminiscent of a black queen dynamics in which different types crucially depend on each other for performing vital functions36,55. While vulnerable to defectors in their own institution, cooperators complement each other by beating defectors in their opposite institution. Synergistically thus, they can suppress defection in the population and alternately dominate the population (see Fig. 2c, d). At this stage, the game payoff of all the strategies starts to increase beyond zero. I call this periodic orbit the black queen orbit.The picture depicted above is the typical behavior of the model for large enough values of the cost. To see this, in Fig. 3a, I plot the phase diagram of the model in the cg − r plane. Here, the frequency of cooperators in the population, ({rho }_{C}={rho }_{C}^{1}+{rho }_{C}^{2}), is color plotted as well (see Supplementary Fig. 1 for the frequency of different strategies). Red dashed lines show the boundary of the region where the system settles in a periodic orbit. For high costs, as r increases, the system shows a series of successive cross-overs from a defective fixed point to the red queen periodic orbit, black queen periodic orbit, and finally a cooperative fixed point. On the other hand, for small costs, the system possesses a bistable region where both the red queen and black queen periodic orbits (or a partially cooperative fixed point and black queen periodic orbit to the left of the red dashed line in the bistable region) are stable, and the system shows a discontinuous transition between these two orbits. Orange circles show the lower boundary of the bistable region, below which the black queen orbit is unstable. Its upper boundary, above which the red queen orbit becomes unstable, is plotted by red squares, in Fig. 3. The transition between the two periodic orbits becomes a continuous transition at a single critical point (see Supplementary Fig. 4).Fig. 3: Evolution of cooperation.a Time average total frequency of cooperators, ({rho }_{C}={rho }_{C}^{1}+{rho }_{C}^{2}) in the r − cg plane is color plotted. The dynamics can settle in fixed point (FP) (small and large enhancement factors), or two different periodic orbits, red queen periodic orbit (RPO) where cooperation only in the costly institution evolve and black queen periodic orbit (BPO) where cooperation in both institutions evolve. b Time average difference between the probability that an individual in the costly institution is a cooperator from the probability that an individual in the free institution is a cooperator, (gamma ={rho }_{C}^{1}/({rho }_{C}^{1}+{rho }_{D}^{1})-{rho }_{C}^{2}/({rho }_{C}^{2}+{rho }_{D}^{2})). Individuals are more likely to be cooperators in a costly institution. c The time average total frequency of cooperators in the r − cg plane under pure selection dynamic (ν = 0). Red queen and black queen periodic orbit can occur for, respectively, small and large enhancement factors. In other regions, the dynamics settle in a fixed point where either non-costly defectors (small enhancement factors), costly cooperators (inside the region marked with dashed black line), or non-costly cooperators survive. Parameter values: g = 5, and π0 = 2. In (a, b) ν = 10−3, and in (c) ν = 0. In (a, b) the replicator dynamic is solved for 8000 time steps, and the time average is taken over the last 2000 steps. In (c) the replicator dynamic is solved for 200,000 time steps, and the time average is taken over the last 150,000 time steps.Full size imageExamination of the overall cooperation in the population shows that an entrance cost has a contrasting effect on population cooperation for large and small enhancement factors. An entrance cost keeps free-riders away from a costly institution. This fact makes the relative frequency of cooperators to defectors higher in the costly institution than that in the free institution. To see that defectors are less likely to join the costly institution, I plot the difference between the probabilities that an individual in the costly institution is a cooperator and the probability that an individual in the free institution is a cooperator, (gamma ={rho }_{C}^{1}/({rho }_{C}^{1}+{rho }_{D}^{1})-{rho }_{C}^{2}/({rho }_{C}^{2}+{rho }_{D}^{2})) in Fig. 3b, where it can be seen it is always positive. Intuitively, as a costly defector’s payoff in a group with ({n}_{C}^{1}) cooperators and ({n}_{D}^{1}) other defectors is equal to (r{n}_{C}^{1}/({n}_{C}^{1}+{n}_{D}^{1})-{c}_{g}), a costly defector can reach a positive payoff only when ({n}_{C}^{1}) is large. Otherwise, costly defectors are better off hedging the risk of obtaining a negative payoff by joining the free institution, where their payoff is necessarily non-negative. Consequently, the expected number of cooperators in the costly institution, ({rho }_{C}^{1}g), sets a bound for the frequency of costly defectors. This fact increases a costly institution’s profitability, especially for small enhancement factors, and positively impacts cooperation in the population. On the other hand, for high enhancement factors, a large entrance cost is detrimental to cooperation. This is because, although the frequency of defectors in the costly institution remains close to zero, fewer individuals are willing to choose a costly institution with a high cost. This increases the effective size of the free institution and the mixing between cooperators and defectors in the free institution. Since defectors can better exploit cooperators in well-mixed groups, the increased mixing between cooperators and defectors in the free institution hinders cooperation.As shown in the Supplementary Note 3, while the phenomenology of the model remains the same for lower mutation rates, lower mutation rates increase the size of the region where the dynamics settle in a periodic orbit (see Supplementary Fig. 5). Regarding the dependence of the dynamics on the mutation rate, an interesting case is the zero mutation rate, where selection is the sole driver of the dynamics. The time average cooperation for zero mutation rate, starting from an initial condition where all the strategies are equal, is plotted in Fig. 3c (See Supplementary Fig. 6 for the frequency of different strategies). Both the red queen (for small enhancement factors) and the black queen (for large enhancement factors) periodic orbits are observed in this case. However, for zero mutation rate, both the amplitude and period of fluctuations increase: The fluctuating dynamics go through periods where one of the surviving strategies reaches a frequency close to 1 only to be later replaced by another strategy (see Supplementary Fig. 7). The dynamics can also settle in different fixed points. For cg = 0, depending on the enhancement factor, either cooperators or defectors in both institutions survive in equal densities. For nonzero cg, however, only one of the strategies survives. For small enhancement factors, non-costly defectors dominate the population. For larger enhancement factors, either costly cooperators (the region marked with a dashed black line) or non-costly cooperators dominate the population.In the Supplementary Note 2, I consider a case where the two institutions have different productivities, i.e., different enhancement factors, and show that similar phases are at work in this case (see the Supplementary Figs. 2 and 3). For instance, I show that a large entrance cost destabilizes full defection, removes the system’s bistability, and ensures the evolution of cooperation starting from all the initial compositions of the population. In addition, I study the continuous replicator dynamics and show similar phenomenology is at work in this case (see Supplementary Notes 1.4 and 4, and Supplementary Figs. 8 and 9).Finally, I note that a similar phenomenology is at work in a context where instead of a costly and a cost-free institution, two costly institutions interact. To see this, assume institution 1 has a cost cg and institution 2 has a cost ({c}_{g}^{0}). Without loss of generality, assume ({c}_{g} , > , {c}_{g}^{0}). Writing ({c}_{g}=({c}_{g}-{c}_{g}^{0})+{c}_{g}^{0}), it is easy to see that it is possible to absorb ({c}_{g}^{0}) in the base payoff b (as all the individual pay a cost ({c}_{g}^{0}) irrespective of their institution choice). Thus, the model is equivalent to a context where resource 2 has zero cost, resource 1 has a cost of ({c}_{g}-{c}_{g}^{0}), and all the individuals receive a shifted base payoff of (b-{c}_{g}^{0}) (see Supplementary Note 5 and Supplementary Fig. 10).Structured populationIn contrast to the well-mixed population, the model shows no bistability in a structured population, and the fate of the dynamics is independent of the initial condition. To see why this is the case, I note that in a well-mixed population, a situation where all the individuals are defectors, and randomly prefer one of the two institutions, is the worst case for the evolution of cooperation, as in this case, mutant cooperators are in a disadvantage in both institutions. However, in a structured population, starting from such an initial condition, blocks of defectors, most of whom prefer the same institution, form due to spatial fluctuations. A mutant cooperator who prefers the minority institution in these blocks obtains a high payoff and proliferates. This removes the bistability of the dynamics in a structured population.To study the model’s behavior in a structured population, I perform simulations starting from an initial condition in which all the individuals are defectors and prefer one of the two institutions at random. The model shows similar behavior in a structured population to that in a well-mixed population. This can be seen in Fig. 4a–d, where the densities of different strategies are color plotted in the cg − r plane (see Supplementary Note 6 and Supplementary Figs. 11 and 12 for further details). As was the case in a well-mixed population, cooperation does not evolve for too small values of r. As r increases beyond a threshold, cooperation does evolve in the costly institution but not in the free institution. In this region, for a fixed enhancement factor, an optimal cost, approximately equal to cg = r − 1, optimizes the cooperation in the population. On the other hand, cooperation in both the costly and the free institutions evolves for large enhancement factors. In this region, increasing the cost can slightly increases defection in the free institution and have a detrimental effect on the evolution of cooperation, but not as much as it does in a well-mixed population.Fig. 4: The frequency of different strategies in the cg − r plane in a structured population.The time average frequencies of costly cooperators (a), costly defectors (b), non-costly cooperators (c), and non-costly defectors (d) in the cg − r plane are color plotted. The system shows a red queen dynamic in which cooperators only in the costly institution survive in large numbers (for smaller enhancement factors), or a black queen dynamic, where cooperators in both institutions survive and help each other to suppress defection (for larger enhancement factors). Parameter values: g = 5, nu = 10−3, and π0 = 2. The population resides on a 200 × 200 first nearest neighbor square lattice with von Neumann connectivity and periodic boundaries. The simulation is performed for 5000 time steps starting from an initial condition in which all the individuals are defectors and prefer one of the two institutions at random. The time average is taken over the last 2000 steps.Full size imageInstead of periodic orbits observed in the well-mixed population, on a spatial structure the model’s dynamic is governed by the cyclic dominance of different strategies through spatiotemporal fluctuations manifested by traveling waves. In Fig. 5, I present snapshots of the population’s stationary state in different phases. In this figure, I consider a model in which individuals reproduce with a probability proportional to the exponential of their payoff, π, times a selection parameter, β, (exp (beta pi )) (see the Supplementary Note 1.3), with β = 5. The situation in the model where individuals reproduce with a probability proportional to their payoff is similar. In Fig. 5a, I have set r1 = r2 = 1.7, and cg = 0.6. This phase corresponds to the red queen periodic orbit in the well-mixed population case. Here, the majority of the population are non-costly defectors. Costly cooperators experience an advantage over the former and can proliferate in the sea of non-costly defectors. However, costly cooperators are vulnerable to both costly defectors and non-costly cooperators. The former can only survive in small bands around costly cooperators, as they rapidly get replaced by non-costly defectors once they eliminate costly cooperators. This phenomenon shows that spatial competition between defectors with differing institution preferences can positively impact the evolution of cooperation. Non-costly cooperators, in turn, can survive by forming compact domains where they reap the benefit of cooperation among themselves. However, as the effect of network reciprocity is too small to promote cooperation in this region, non-costly cooperators get eliminated by non-costly defectors once costly cooperators are out of the picture. Consequently, the system’s dynamic is governed by traveling waves of costly cooperators followed by small trails of costly defectors and non-costly cooperators in a sea of non-costly defectors (see the Supplementary Video, SV.156, and Supplementary Note 7 for an illustration of the dynamics in this regime).Fig. 5: Snapshots of the population in the stationary state for different parameter values.Different strategies are color codded (legend). In (a,) r1 = 1.7, r2 = 1.7, in (b,) r1 = 3.5 and r2 = 3.5, and in c, r1 = 3, and r2 = 1.8. In all the cases cg = 0.6. For small enhancement factors (a), the red queen dynamics in which cooperators only in the costly institute survive in large numbers occur. For larger enhancement factors (b), the black queen dynamics in which cooperators in both institutions survive and help each other suppress defection occur. By increasing the enhancement factors (c), non-costly cooperators dominate. However, a small frequency of costly cooperators survives and purge the population from defectors by moving along the bands of non-costly defectors. Here, individuals reproduce with a probability proportional to the exponential of their payoff with a selection parameter equal to β = 5. The population resides on a 400 × 400 square lattice with von Neumann connectivity and periodic boundaries. Parameter values: g = 5 and ν = 10−3.Full size imageFigure 5b shows a snapshot of the population for r1 = r2 = 2.2. This phase corresponds to the black queen periodic orbit in a well-mixed population. In this phase, cooperators in both the costly and free institutions evolve. Cooperators are vulnerable to defectors in their institution and lose their territory to defectors of similar type. Defectors are in turn vulnerable to cooperators in their opposite institution and are replaced by them. Consequently, the dynamic of the model is governed by traveling waves of cooperators, chased by defectors of similar type, who are in turn extincted by cooperators of the opposite type. Thus, while cooperators of different types on their own either can not survive (in the case of non-costly cooperators) or are doomed to a winnerless competition with defectors (in the case of costly cooperators), they complement each other to efficiently suppress defection in the population (see the Supplementary Video, SV.256, for an illustration of the dynamics in this regime).Another manifestation of functional complementation between cooperators of different types can be seen in the regime of large enhancement factors. An example of this situation is plotted in Fig. 5c. Here, r1 = r2 = 3.5 and cg = 0.6. In this region, non-costly cooperators dominate the population. However, non-costly defectors can survive in small bands in the sea of non-costly cooperators. While at a disadvantage in the sea of non-costly cooperators, costly cooperators beat non-costly defectors. Consequently, small blocks of costly cooperators are formed within the bands of non-costly defectors. These blocks of costly cooperators move along the bands of non-costly defectors and purge the population from non-costly defectors. In this way, although costly cooperators exist only in small frequency, they play a constructive role in helping non-costly cooperators to dominate the population.In summary, the analysis of the spatial patterns reveals that competition or synergistic relation between individuals with different institution preferences plays an essential role in the evolution of cooperation in the system. Defectors with different institution preferences always appear as competitors who compete over space. By eliminating each other, they play a surprisingly constructive role in the evolution of cooperation. Cooperators, on the other hand, while having direct competition over scarce sites, can also act synergistically and help the evolution of cooperation in their opposite institution since they can eliminate defectors in their opposite institution. In this way, by purging defectors with an opposite game preference, cooperators help fellow cooperators with an opposite game preference. Consequently, cooperators with different game preferences can engage in a mutualistic relation to efficiently suppress defection in the population.Finally, as shown in the Supplementary Note 6, the spatial model shows similar phases in the case where the two public resources have heterogeneous profitability, that is, when r1 ≠ r2 (see the Supplementary Fig. 12). More

Data for all analysed radionuclides are presented in the “Supplementary Material”. Cryoconite samples were collected on Nalli Glacier (Supplementary Fig. S1) on Sept. 25, 2017 (samples 1701–1714) and on Sept. 10, 2018 (samples 1801–1814) at 28 spots (Fig. 2, Supplementary Table S1). Gamma spectrometric analysis of samples showed the presence of anthropogenic radionuclides 137Cs, 241Am, and 207Bi. All quoted radioactivity values were recalculated for the sampling date, except those for 241Am since the concentration of the parent 241Pu isotope is unknown. However, for this isotope, the correction for decay is negligible. The activity of 137Cs reached 8093 (± 69) Bq kg−1 of dry weight, that of 241Am reached 58.3 (± 2.3) Bq kg−1 and that of 207Bi reached 6.3 (± 0.6) Bq kg−1. The natural radionuclides 210Pb and 7Be were also present in all samples. The activity of 210Pb varied in the range of 1280–9750 Bq kg−1. In addition, in the investigated samples, a significant amount of short-lived cosmogenic radionuclide 7Be was found, whose specific activity reached 2418 (± 76) Bq kg−1 (Fig. 3, Supplementary Table S2). To evaluate the contribution of atmospheric components to the total 210Pb activity, 226Ra activity was determined and found to be 17–27 Bq kg−1 (Supplementary Table S2). Based on the 210Pb/226Ra ratio, we conclude that more than 98% of 210Pb was of atmospheric provenance.Figure 2Location of sampling points on Nalli Glacier. A—137Cs activity zone  95%) of corresponding rocks and numerous outcrops likely promoted entrapment of these elements into explosion clouds, and their subsequent precipitation with radionuclides. This feature of the geological structure of the area explains the extremely high enrichment of surface waters in elements such as Zn, Pb, Sr, Ni, As, Cr, Co, Se, Te, Cd, W, Cu, Sb, and Sn; for many of them, the excess reaches 10-fold with respect to the Clrake values51. This hypothesis is supported by obvious correlations between the concentrations of Bi, Ag, Sn, Sb, Pb, Cd, W, and Cu and the activity of anthropogenic radionuclides 137Cs, 241Am and 207Bi. This relationship is obviously related to the simultaneous release of elements and radionuclides from the contaminated ice layer and their entrapment in cryoconite holes. More

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For the past five years, I’ve studied oysters — a commercially and environmentally important species in southeast England. My research is very practical: I help to solve problems by working with oyster growers (known locally as oystermen), regulators and other community members. Resulting papers are evidence of work I’ve already done.Most oysters in this area are a non-native species (Crassostrea giga). Locally, it’s well established and has been since the 1960s, but allowing it to spread to nearby estuary systems has been controversial: there are concerns that it could become an invasive species.Working with aquaculture producers, I help to guide efforts to restore the native oyster (Ostrea edulis), populations of which declined owing to overfishing, habitat destruction, pollution and disease. Crassostrea giga oysters have provided enough income for oyster growers to spend time and effort restoring the local species. We’ve done some cool things, including creating one of the largest coastal marine conservation zones in the United Kingdom — more than 284 square kilometres — and all for an unseen brown invertebrate that lacks the charisma of a dolphin.This picture is from a typical day in the field. During high tides, we go out in a boat to take sonar readings to map potential oyster habitats; at low tide, we put on waders and go out on the mud flats to look for juvenile oysters. We focus our conservation efforts on spots where juvenile oysters are already trying to get established.Amazingly, these filter feeders don’t require feeding by humans, and they clean the water as they grow. Bivalve aquaculture such as this has become a cornerstone of the ‘blue economy’ — using marine resources sustainably for economic growth while preserving ocean health. It will take more work to determine how the balance can be reached, but oysters will be part of that conversation.

Nature 600, 182 (2021)
doi: https://doi.org/10.1038/d41586-021-03573-5

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