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Direct habitat lossAccording to the global projections of urban expansion under five SSPs17 (Supplementary Note 3 and Supplementary Fig. 1), 36–74 million hectares (Mha) of land areas will be urbanized by 2100, representing a 54–111% increase compared with the baseline year of 2015. Among these, 11–33 Mha natural habitats (Supplementary Table 1) will become urban areas by 2100. Across SSP scenarios, the patterns of change in losses of total habitat, forest, shrubland, and grassland are consistent with the global projections of urban expansion (Fig. 1). In terms of urban encroachment on wetlands, wetland will undergo the largest loss under scenario SSP4 than under other scenarios. However, if the sustainable pathway of scenario SSP1 is properly implemented, this will enable us to conserve the global wetland. The greatest loss of other habitat will occur under scenario SSP3, but the minimal loss of other habitat will occur under scenario SSP1. Under the five different SSP scenarios, the United States, Nigeria, Australia, Germany, and the UK are consistently predicted to have greater habitat loss due to urban expansion (Supplementary Table 2).Fig. 1: Future direct habitat loss due to urban expansion under SSP scenarios.a The habitat loss by 2100 for each habitat type. Bars indicate the mean habitat loss area (five scenarios) for each habitat type. Error bars represent mean values ± 1 SEM for the loss of each habitat type under five scenarios, n = 5 scenarios. Points represent data in five scenarios. b The losses in total area, forest, shrubland, grassland, wetland, and other land.Full size imageThere are obvious disparities in the hot spots and cold spots of habitat loss under the five SSP scenarios (Fig. 2 and Supplementary Figs. 2–6). Potential hot spots of habitat loss are concentrated in regions such as the northeastern, southern, and western coasts of the United States, the Gulf of Guinea coastal areas, Sub-Saharan Africa, and the Persian Gulf coastal areas. Under scenario SSP5, parts of central and western Europe will also become hot spots. However, under other scenarios, the cold spots will be particularly concentrated in eastern and southern Europe. East Asia and South Asia, which are represented by China, India, and Japan, are dominated by cold spots (Supplementary Figs. 2–6), because these regions may experience a decline in urban land demand from 2050 to 2100 (for examples in China, see Supplementary Figs. 7–11), although they are currently the most populous regions in the world.Fig. 2: Future hot spots and cold spots of habitat loss due to urban expansion under SSP scenarios by 2100.Figures for the United States (a), Europe (b), Africa (c), and China (d) are presented separately. The Gi_Bin identifies statistically significant hot spots and cold spots. Statistical significance was based on the p-value and z-score (two-sided), and no adjustments were made for multiple comparisons.Full size imageOur scenario projections show that the largest natural habitat loss is expected to occur in the temperate broadleaf and mixed forests biome (except for scenario SSP3). In addition, many biomes will experience proportionate loss of natural habitat. These biomes include the tropical and subtropical coniferous forests biome, the temperate coniferous forests biome, the flooded grasslands and savannas biome, the Mediterranean forests, woodlands, and scrub biome, and the mangroves biome (Supplementary Table 3). Although the rate of future habitat loss is small at the global scale, it can be large in some areas. For example, the habitat in the temperate broadleaf and mixed forests may decrease by 1.4% under scenario SSP5. At the ecoregion scale, about 9% of 867 terrestrial ecoregions will lose more than 1% of habitat due to urban expansion (Supplementary Fig. 12). In the future, four ecoregions—the Atlantic coastal pine barrens, the coastal forests of the northeastern United States, and the Puerto Rican moist and dry forests—will experience more than 20% of habitat loss.Urban expansion threatens biodiversity prioritization schemesTo reflect the potential impact of urban expansion on protected areas (Supplementary Note 4), the analyses presented here were based on the assumption that urban expansion within protected areas is not strictly restricted and can even occur in the currently gazetted protected areas (Supplementary Note 5, Supplementary Figs. 13 and 14). In 2015, urban areas with a total area of 30,594 km2 were distributed in 28,152 protected areas, accounting for 12.6% of global protected areas (Supplementary Figs. 15 and 16). Moreover, 38% of the urban land-use changes within protected areas were due to the conversion of natural habitats into urban land between 1992 and 2015. If urban expansion continues without strict restrictions, 13.2–19.8% of the protected areas will be affected by urban land by 2100, and urban land will occur in 29,563–44,400 protected areas with a total urban land area of up to 46,705–89,901 km2 across the five SSP scenarios (the lowest and highest proportions of urban land in each protected area by 2100 under SSP3 and SSP5 scenarios are presented in Supplementary Figs. 17 and 18).We also found that 0.90% of all terrestrial biodiversity hotspots (Supplementary Note 6), which are the world’s most biologically rich yet threatened terrestrial regions24, were urbanized in 2015. And this proportion (0.90%) is higher than that located in the rest of the Earth’s surface (0.51%) in 2015. By 2100, the new urban expansion will additionally occupy 1.5–1.8% of hotspot areas under the five SSP scenarios (Supplementary Table 4). Five biodiversity hotspots are projected to suffer the largest proportion of urban land conversion: the California Floristic Province (6–11%), Japan (6–8%), the North American Coastal Plain (4–8%), the Guinean Forests of West Africa (4–8%), and the Forests of East Australia (2–6%). In contrast, the East Melanesian Islands and the New Caledonia are almost unaffected by urban expansion. Biodiversity hotspots (e.g., the Guinean Forests of West Africa, the Coastal Forests of Eastern Africa, Eastern Afromontane, and the Polynesia-Micronesia) with few human disturbances in 2015 are projected to experience the highest percentage of future urban growth. Compared with the urban areas in 2015, by 2100, the urban areas in these four biodiversity hotspots will experience a disproportionate increase of 281–708, 294–535, 169–305, and 33–337%, respectively.The World Wildlife Fund (WWF) selected the ecoregions that are most crucial to the conservation of global biodiversity as Global 20025 (Supplementary Note 7). However, about 93% of the Global 200 ecoregions will be affected by future urban expansion. Although the proportion of urban land in each ecoregion will be less than 1% in 2100, the urban area located in these ecoregions will experience an increase of 74–160% from 2015 to 2100 across the five SSP scenarios (Supplementary Table 4). Four ecologically vulnerable ecoregions that have the highest urban growth rates are the Sudd-Sahelian Flooded Grasslands and Savannas, the East African Acacia Savannas, the Hawaii Moist Forest, and the Congolian Coastal Forests. By 2100, the urban areas in these four ecoregions will increase by 877–9955, 527–646, 18–902, and 500–1037%, respectively.The five SSP scenarios showed that the urban area is expected to increase by only 73–213 km2 in the Last of the Wild areas26 (see Supplementary Note 8 for descriptions about the Last of the Wild areas) by 2100 (Supplementary Table 4).Impacts of urban expansion on habitat fragmentationThe increasing exposures of natural habitat to urbanized land use may cause long-term changes in the function and structure of the natural habitat that is adjacent to urban areas13. To examine this proximity effect, we investigated the impact of future urban expansion on the nearest distance between urban areas and natural habitat (i.e., the distance from patch edges of urban areas to patch edges of the nearest natural habitats) under different SSP scenarios. Although the global urban area is expected to increase by 36–74 Mha by 2100, the impacts of future urban expansion on adjacent natural habitat are disproportionately large. Future urban expansion will make urban areas much closer to patch edges of 34–40 Mha natural habitat, which will inevitably threaten the natural habitat and increase the risk of biodiversity decline. The effects of urban expansion on adjacent patch edges of natural habitats are remarkably different across different scenarios. Specifically, the area of affected adjacent natural habitat is expected to be 38.45, 34.24, 40.31, 37.84, and 39.42 Mha under SSP1 to SSP5 scenarios by 2100, with the smallest effect under scenario SSP2, and the largest effect under scenario SSP3. Moreover, the scale of urban expansion does not correspond directly with the size of the impact. Several countries, including Mauritania, Algeria, Saudi Arabia, Western Sahara, and the United States, will have a large change in the distance from future urban areas to natural habitats due to urban expansion (Supplementary Table 5). Such effects also varied across different natural habitat types. The distance from the patch edges of urban areas to patch edges of (a) wetland, other land, and forest, (b) grassland, and (c) shrubland will generally be shortened by ~2000, ~1500 and ~900 m, respectively.In addition to the effect on the distance to the habitat edge, urban-caused habitat fragmentation is also reflected in reducing mean patch size (MPS)13, increasing mean edge index (edge density (ED), i.e., edge length on a per-unit area)27, and enlarging isolation (mean Euclidean nearest neighbor distance, ENN_MN)28 (Fig. 3). Taking the global ecoregions as the analysis unit, we found that within a 5 km buffer of urban areas, the median of MPS of natural habitats tends to show an overall decline trend, and the segmentation and subdivision of habitats become more obvious as future urban land expands. The median of MPS is the largest under scenario SSP1, followed by SSP4, SPP2, and SSP3 with some fluctuations in between, and the smallest MPS is found with the most fragmented landscape under scenario SSP5. A smaller patch size indicates that the inner parts of the habitat are subject to higher risk of being influenced by external disturbance. Future urban expansion also tends to cause an increase in the ED of natural habitat, which is often linked with smaller patches or more irregular shapes, and therefore poses a threat to biodiversity that influences many ecological processes (e.g., the spread of dispersal and predation)13,27,28. Scenario SSP1 shows the best performance in maintaining a low habitat ED and a high level of biodiversity conservation. However, under scenario SSP5, ED will experience a rapid increase in the second half of the 21st century. Meanwhile, the ENN_MN will increase substantially in the future, suggesting that areas with the same habitat type will become increasingly isolated, irregular, dispersed, or unevenly distributed due to the barrier of urban land. This will affect the speed of dispersal and patch recolonization. Scenario SSP1 is also most conducive to maintaining the proximity of natural habitats with the same habitat type. Other scenarios show relatively similar performance.Fig. 3: Future urban expansion effects on habitat fragmentation under SSP scenarios.a Mean patch size (MPS), b edge density (ED), c mean Euclidean nearest-neighbor distance (ENN_MN).Full size imageImpacts of urban expansion on terrestrial biodiversityWe focus on biodiversity in three common vertebrate taxa (i.e., amphibians, mammals, and birds) in our analyses. Future land system conversion to urban land will cause an average of 34% loss in the overall relative species richness. Land conversion from dense forest, mosaic grassland and open forest, mosaic grassland, and bare and natural grassland to urban land will cause the highest overall relative biodiversity loss (48%, 95% confidence interval (CI): 34–59% on a 1 km grid). These land systems with a high risk of biodiversity loss are concentrated in the United States, Europe, and Sub-Saharan Africa (Supplementary Fig. 19). Overall, the negative effect of future urban expansion on the total abundance of species will be more pronounced than that on species richness. Urban land changes will result in an average of 52% overall loss in relative total abundance of species. In particular, the losses of dense forest, natural grassland, and mosaic grassland, due to conversion to urban land, will lead to a high risk of species loss (62%, 95% CI: 38–76%).In terms of the number of species (i.e., all amphibians, mammals, and birds), future urban expansion will cause an average loss of 7–9 species and a loss of up to ~197 species per 10 km grid cell by 2100 across the five SSP scenarios (Fig. 4 and Supplementary Fig. 20). Species loss is most likely to be concentrated in Sub-Saharan Africa (particularly the Gulf of Guinea coast), the United States, and Europe. In addition, southeastern Brazil, India, and the eastern coast of Australia are also relatively high-risk areas. However, the specific effects of urban expansion vary substantially across different SSP scenarios. For instance, under scenario SSP5, urban expansion will pose a fatal threat to the global species richness in areas with urban development potential (species richness loss will occur in ~740 Mha land areas), whereas under the divided pathway (SSP4) and regional rivalry pathway (SSP3) scenarios, urban expansion will threaten the richest biodiversity hotspots, such as Sub-Saharan Africa and Latin America (Supplementary Fig. 20).Fig. 4: Potential biodiversity loss due to future urban expansion under SSP scenarios.The biodiversity loss in terms of the number of terrestrial vertebrate species (amphibians, mammals, and birds) lost per 10 km grid cell in the North America (a), Europe (b), the Gulf of Guinea coast (c), and East Asia (d).Full size imageWe also found a loss of up to 12 species of threatened amphibians, mammals, and birds (including vulnerable, endangered, or critically endangered categories defined in the IUCN Red List), and a loss of up to 40 species of small-ranged amphibians, mammals, and birds (small-ranged species are species with a geographic range size smaller than the median range size for that taxon)29 due to future urban expansion by 2100. There are a few scattered areas that will be hotspots for the loss of threatened species, such as West Africa, East Africa, northern India, and the eastern coast of Australia (Supplementary Fig. 21). The loss of small-ranged species will concentrate in fewer areas (Supplementary Fig. 22). We have identified 30 conservation priority ecoregions with high risks of habitat loss and small-ranged species loss due to future urban expansion (Supplementary Table 6). These conservation priority ecoregions are all found in Latin America and Sub-Saharan Africa (Supplementary Fig. 23). However, some hotspots outside of these conservation priority regions, such as tropical Southeast Asia, the west coast of the United States, and northern New Zealand, will also be affected (Supplementary Fig. 23).The top 5% 10 km grid cells with the highest loss in species richness (28–38 species potentially being lost) scatter across adjacent urban areas. However, only 6.4–8.6% of these regions are covered by the current global network of protected areas. These areas are often overlooked, and thus receive relatively low conservation spending. Ecoregions in Sub-Saharan African, Central and South America, Southeast Asia, and Australia will be responsible for the top 43% of average species loss across the SSP scenarios (Fig. 5). Kenya, Swaziland, Brunei, Zambia, Republic of Congo, and Zimbabwe will face the largest potential species richness loss (approximately > 29 species lost per 10 km grid cell) under all five SSP scenarios (Supplementary Fig. 24 and Supplementary Table 7).Fig. 5: Average potential biodiversity loss per 10 km grid cell in ecoregions due to future urban expansion under SSP scenarios.The mean potential biodiversity loss represents the average number of terrestrial vertebrate species (amphibians, mammals, and birds) lost per 10 km grid cell.Full size image More

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If you are a scientist hoping to photograph and share your own research:
•    Don’t underestimate the power of modern media and social-media platforms. Content is changing the world and people’s lives, and it can easily change your life. Stay at the forefront of media technology, or at least be aware of developments. It’s a never-ending race, but it’s easy to get into.
•    If you plan to share your work with others, imagine what will be of interest to them. If you can excitingly describe your work to a 5-year-old, you won’t have any trouble getting anyone interested. Beautiful pictures help, but the story always comes first.

•    You will stand out much more if you have a niche and unique story. It could be your rare field of science or a special angle that you use to tell the story of your work. Being different is awesome.
•    Set the bar very high. You can find dozens of examples of truly high-quality content on the Internet. And you can almost always find resources that can help you to learn how to create work of the same calibre. With practice, your skills will inevitably rise — but at any given time, it’s important to know the level you should aim for.
•    Find people who are cooler than you. Don’t hesitate to ask them for advice or to shadow them. Have them share their experiences, stand behind them and observe their work if they’ll let you. Few things are more useful than real work experience, both your own and that of others.
•    Take on a project. This could be a an illustrated workbook for colleagues or students, a guide book, a lecture for schoolchildren with compelling visuals, a course for students or a documentary on your topic.
•    If you work in a team, you can raise the bar even higher. Use each other’s strengths, share experiences, make plans, apply for grants and take on challenging science-communication projects together. This multiplies the fun and the results. More

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ModelsConsider an IPD game with misperception such as implementation errors and observation errors22,23,31. Due to the misperception, the parameter in the real game changes from (omega _1=[T_1,R_1,P_1,S_1]) to (omega _2=[T_2,R_2,P_2,S_2]), and only player X notices the change. Thus, player Y’s cognition of the parameter is (omega _1), while player X’s cognition of the parameter is (omega _2). In each round, player X chooses a strategy from its strategy set (Omega _X={{mathbf {p}}=[p_{cc},p_{cd},p_{dc},p_{dd}]^T|p_{xy} in [0,1],xyin {cc,cd,dc,dd}}), e.g., (p_{xy}) is player X’s probability for cooperating with given previous outcome (xyin {cc,cd,dc,dd}). Similar to (Omega _X), player Y’s strategy set is (Omega _Y={{mathbf {q}}=[q_{cc},q_{dc},q_{cd},q_{dd}]^T|q_{xy} in [0,1],xyin {cc,dc,cd,dd}}). According to Press and Dyson7, this game can be characterized by a Markov chain with a state transition matrix (M=[M_{jk}]_{4times 4}) (see “Notations” for details). Denote ({mathbf {v}}=[v_{cc},v_{cd},v_{dc},v_{dd}]^T) as a probability vector such that ({mathbf {v}}^T M={mathbf {v}}^T) and (v_{cc}+v_{cd}+v_{dc}+v_{dd}=1). Let ({mathbf {S}}^{omega _i}_{X}=[R_i,S_i,T_i,P_i]^T), and ({mathbf {S}}^{omega _i}_{Y}=[R_i,T_i,S_i,P_i]^T,) (iin {1,2}). The expected utility functions of players are as follows:$$begin{aligned} begin{aligned} u_X^{omega _i}({mathbf {p}},{mathbf {q}})={mathbf {v}} cdot {mathbf {S}}^{omega _i}_{X}, u_Y^{omega _i}({mathbf {p}},{mathbf {q}})={mathbf {v}} cdot {mathbf {S}}^{omega _i}_{Y},iin {1,2}. end{aligned} end{aligned}$$Denote (G_1 = {{mathbf {P}}, {varvec{Omega }}, {mathbf {u}}, omega _1}), and (G_2={{mathbf {P}},{varvec{Omega }},{mathbf {u}},omega _2}), where ({mathbf {P}}={X,Y}), ({varvec{Omega }}=Omega _Xtimes Omega _Y), and ({mathbf {u}}={u_X^{omega _i},u_Y^{omega _i}}, iin {1,2}). Thus, the actual utilities of players are obtained through (G_2), and in the view of player Y, they are playing game (G_1). In the view of player X, they are playing game (G_2) but player X knows that player Y’s cognition is (G_1). (G_1) and (G_2) are shown in Table 2.Table 2 Utility matrices in IPD games with misperception.Full size tableLet ({mathbf {p}}_0=[1,1,0,0]^T). For (iin {1,2}), ({mathbf {p}}=alpha {mathbf {S}}^{omega _i}_{X} +beta {mathbf {S}}^{omega _i}_Y +gamma {mathbf {1}}+{mathbf {p}}_0), where (alpha ,beta ,gamma in {mathbb {R}}), is called a ZD strategy7 of player X in (G_i) since the strategy makes the two players’ expected utilities subjected to a linear relation:$$begin{aligned} alpha u_X^{omega _i}({mathbf {p}},{mathbf {q}})+beta u_Y^{omega _i}({mathbf {p}},{mathbf {q}})+gamma =0, end{aligned}$$for any player Y’s strategy ({mathbf {q}}). All available ZD strategies for player X in G can be expressed as (Xi (omega _i)={{mathbf {p}}in Omega _X|{mathbf {p}}=alpha {mathbf {S}}^{omega _i}_{X} +beta {mathbf {S}}^{omega _i}_Y +gamma {mathbf {1}}+{mathbf {p}}_0,alpha ,beta ,gamma in {mathbb {R}} }.) Also, the three special ZD strategies are denoted as:

(1)

equalizer strategy7,12: ({mathbf {p}}=beta {mathbf {S}}^{omega _i}_{Y}+gamma {mathbf {1}}+{mathbf {p}}_0);

(2)

extortion strategy7,13: ({mathbf {p}}=phi [({mathbf {S}}^{omega _i}_X-P_i{mathbf {1}})-chi ({mathbf {S}}^{omega _i}_Y-P_i{mathbf {1}})]+{mathbf {p}}_0,chi geqslant 1);

(3)

generous strategy14,15: ({mathbf {p}}=phi [({mathbf {S}}^{omega _i}_X-R_i{mathbf {1}})-chi ({mathbf {S}}^{omega _i}_Y-R_i{mathbf {1}})] +{mathbf {p}}_0,chi geqslant 1).

Based on the past experience, player Y knows that player X prefers ZD strategies, which has been widely considered in many IPD games7,9. To avoid that player Y notices the change, which may result in potential decrease of player X’s utility21 or collapse of the model28, player X keeps choosing ZD strategies according to (G_1), such that the strategy sequence matches player Y’s anticipation. To sum up, in our formulation,

the real game is (G_2);

player Y thinks that they are playing game (G_1), and player X thinks that they are playing game (G_2);

player X knows that player Y’s cognition is (G_1);

player Y believes that player X chooses ZD strategies;

player X tends to choose a ZD strategy according to (G_1) to avoid player Y’s suspicion of misperception.

In fact, player X can benefit from the misperception through the ZD strategy. For example, player X can adopt a generous strategy in (G_1) to not only promote player Y’s cooperation behavior, but also make player X’s utility higher than that of player Y, if the generous strategy is an extortion strategy in (G_2). A beneficial strategy for player X is able to maintain a linear relationship between players’ utilities or improve the supremum or the infimum of its utility in its own cognition. In the following, we aim to analyze player X’s implementation of a ZD strategy in IPD with misperception, and proofs are given in the Supplementary Information.Invariance of ZD strategyPlayer X’s ZD strategies may be kept in IPD games with misperception from implementation errors or observation errors. In particular, player X keeps choosing a ZD strategy ({mathbf {p}}) in (G_1) to avoid player Y’s suspicion about possible misperception. In the view of player X, it can also enforce players’ expected utilities subjected to a linear relationship if ({mathbf {p}}) is also a ZD strategy in (G_2). The following theorem provides a necessary and sufficient condition for the invariance of the linear relationship between players’ utilities.Theorem 1
Any ZD strategy ({mathbf {p}}) of player X in (G_1) is also a ZD strategy in (G_2) if and only if$$begin{aligned} frac{R_1-P_1}{2R_1-S_1-T_1}=frac{R_2-P_2}{2R_2-S_2-T_2}. end{aligned}$$
(1)

If (1) holds, player X can ignore the misperception and choose an arbitrary ZD strategy based on its opponent’s anticipation since it also leads to a linear relationship between players’ utilities, as shown in Fig. 1; otherwise, player X can not unscrupulously choose ZD strategies based on player Y’s cognition. There is a player X’s ZD strategy in player Y’s cognition which is not the ZD strategy in player X’s cognition. Further, because of the symmetry of (omega _1) and (omega _2), player X’s any available ZD strategy ({mathbf {p}}) in (G_2) is also a ZD strategy in (G_1) if and only if (1) holds. It indicates that (Xi (omega _1)=Xi (omega _2)) and player X can choose any ZD strategy based on its own cognition, which does not cause suspicion of the opponent since it is also consistent with player Y’s anticipation. Additionally, the slopes of linear relations between players’ utilities may be different, as also shown in Fig. 1, and player X can benefit from the misperception by choosing a ZD strategy to improve the corresponding slope.In fact, (1) covers the following two cases:

(1)

(2P_i=T_i+S_i), (iin {1,2}), is a sufficient condition of (1). Thus, when (2P_i=T_i+S_i), (iin {1,2}), player X’s any ZD strategy ({mathbf {p}}) in (G_1) is also a ZD strategy in (G_2). Actually, (2P_i=T_i+S_i), (iin {1,2}), means that the sum of players’ utilities when players mutual defect is equal to that when only one player chooses defective strategies.

(2)

(R_i+P_i=T_i+S_i), (iin {1,2}), is another sufficient condition of (1). Thus, when (R_i+P_i=T_i+S_i), (iin {1,2}), player X’s any ZD strategy ({mathbf {p}}) in (G_1) is also a ZD strategy in (G_2). Actually, (R_i+P_i=T_i+S_i), (iin {1,2}), means that the game has a balanced structure in utilities32. At this point, the relationship between cooperation rate and efficiency is monotonous, i.e., the higher the cooperation rate of both sides, the greater the efficiency (the sum of players’ utilities).

Furthermore, for the three special ZD strategies, player X can also maintain a linear relationship between players’ utilities in the IPD game with misperception.Figure 1Player X can also enforce a linear relationship between players’ utilities in its own cognition. Let (omega _1=[T,R_1,P_1,S]=[5,3,1,0]) and (omega _2=[T,R_2,P_2,S]=[5,frac{23}{7},frac{1}{7},0]), which satisfy (1). Consider that player X chooses two different ZD strategies in (a) and (b), respectively, and the red lines describe the relationships between players’ utilities in (G_1). We randomly generate 100 player Y’s strategies, and blue circles are ((u^{omega _2}_X,u^{omega _2}_Y)), correspondingly. Notice that blue circles are indeed on a cyan line in both (a) and (b).Full size imageEqualizer strategyBy choosing equalizer strategies according to player Y’s cognition, player X can unilaterally set player Y’s utilities, as shown in the following corollary.
Corollary 1
Player X’s any equalizer strategy ({mathbf {p}}) in (G_1) is also an equalizer strategy in (G_2) if and only if$$begin{aligned} frac{R_1-P_1}{R_2-P_2}=frac{R_1-T_1}{R_2-T_2}=frac{R_1-S_1}{R_2-S_2}. end{aligned}$$
(2)

(2) is also a sufficient condition of (1). If (2) holds, player X can unilaterally set player Y’s utility by choosing any equalizer strategy in (G_1) even though they have different cognitions; otherwise, player X can not unscrupulously choose an equalizer strategy based on player Y’s cognition since it may not be an equalizer strategy in player X’s cognition.Extortion strategyBy choosing extortion strategies according to player Y’s cognition, player X can get an extortionate share, as shown in the following corollary.
Corollary 2
For player X’s extortion strategy ({mathbf {p}}) with extortion factor (chi >1) in (G_1), ({mathbf {p}}) is also an extortion strategy in (G_2) if (1) and the following inequality hold:$$begin{aligned} begin{aligned} (S_1-P_1)(R_2-P_2)-(R_1-P_1)(T_2-P_2)-chi ((T_1-P_1)(R_2-P_2)-(R_1-P_1)(T_2-P_2))1) in (G_1), ({mathbf {p}}) is also a generous strategy in (G_2) if (1) and the following inequality hold:$$begin{aligned} begin{aligned}(S_1-R_1)(R_2-P_2)-(R_1-P_1)(T_2-R_2)-chi ((T_1-R_1)(R_2-P_2)-(R_1-P_1)(T_2-R_2))b^1_i, iin {1,2}, end{aligned} end{aligned}$$
(5)
where (a^1_i) and (b^1_i,iin {1,2}) are parameters shown in “Notations”.
Actually, when player Y chooses the always cooperate (ALLC) strategy35, i.e., ({mathbf {q}}=[1,1,1,1]^T), player X gets the supremum of the expected utility in (G_1) and player X’s utility is improved in the IPD game with misperception.Figure 4Player X can use either equalizer strategies and extortion strategies to raise the supremum of its expected utility or generous strategies to raise the infimum of its expected utility. (a) and (b) consider that (omega _1=[T,R_1,P,S]) and (omega _2=[T,R_2,P,S]), where (R_1ne R_2); (c) considers that (omega _1=[T,R,P_1,S]) and (omega _2=[T,R,P_2,S]), where (P_1ne P_2). The red lines in (a), (b), and (c) describe utilities’ relationships when player X chooses an equalizer strategy, an extortion strategy, and a generous strategy in (G_1), respectively; The yellow area contains all possible relationships between players’ utilities in (G_2) if player X does not change its strategy. In (a) and (b), r is the supremum of player X’s utility in (G_1), and (r’) is lower than the supremum of player X’s utility in (G_2); In (c), l is the infimum of player X’s utility in (G_1), and (l’) is lower than the infimum of player X’s utility in (G_2).Full size imageExtortion strategyBy choosing extortion strategies according to player Y’s cognition, player X can also improve the supremum of its expected utility.
Corollary 5
For player X’s extortion strategy ({mathbf {p}}) with extortion factor (chi >1) in (G_1), the supremum of player X’s expected utility in (G_2) is larger than that in (G_1) if$$begin{aligned} begin{aligned}a^2_ichi ^2+b^2_ichi +c^2_i1), the infimum of player X’s expected utility in (G_2) is larger than that in (G_1) if$$begin{aligned} begin{aligned}a^3_ichi ^2+b^3_ichi +c^3_i More

Comparison of T
ag calculated from publicly available data and actual measurement dataThe calculated Tag (m2/kg-FM) in each year is summarized for each species in Supplemental Table 1:

The geometric means (GMs) of Tag values calculated using the collected samples ranged from 8.1 × 10−6 to 2.5 × 10−2 m2/kg-FM; the minimum was for western bracken fern in 2019 and the maximum was for koshiabura in 2018 at Kawamata, Fukushima.

The GMs of Tag values calculated using the publicly available data ranged from 1.6 × 10−5 to 1.2 × 10−2 m2/kg-FM and thus were similar to the actual measurement data. The minimum GM was for udo in 2019 and the maximum was for koshiabura in 2019. The geometric standard deviation (GSD) range was 1.5–4.5.

Annual GMs of Tag values calculated from publicly available data and actual measurement data are compared in Fig. 1. The values for individual years are represented by different points. The Tag values were distributed close to the 1:1 line, which suggested that Tag values calculated from the publicly available data generally agreed with those calculated from actual measurements. Hence, an obvious overestimation of Tag from the publicly available data described above was not observed in the present data. We confirmed that Tag calculated from the publicly available food monitoring data and the total deposition data from the airborne survey are reliable surrogates for actual measurement samples. We discuss Tag calculated from the publicly available data hereafter.Figure 1Comparison of annual geometric means of the aggregated transfer factor (Tag) calculated from publicly available data and actual measurement data. Circles, diamonds, and triangles indicate deciduous perennial spermatophytes, deciduous tree spermatophytes, and deciduous perennial pteridophytes, respectively. Values for individual years are represented by different points. Error bars indicate the geometric standard deviation in cases where more than three samples were available.Full size imageRelationship between soil deposition and radioactivity in edible wild plants from publicly available dataWe confirmed the relationship between deposition and concentration of 137Cs for the publicly available data for butterbur scape, fatsia sprout, and western bracken fern in a year (Fig. 2), as a representative deciduous perennial and tree spermatophyte, and deciduous perennial pteridophyte, respectively, in the year of the maximum number of detections. Butterbur scape, fatsia sprout, and western bracken fern showed positive significant, nonsignificant, and weak negative significant correlations, respectively (Spearman’s rank correlation, butterbur scape, p = 0.001, rs = 0.45; fatsia sprout, p = 0.85, rs = − 0.03; western bracken fern, p = 0.03, rs = − 0.21). Among 29 subdata with more than 20 detections for each species in a year, in addition to the data shown in Fig. 2, 13 showed statistically significant positive correlations (Butterbur scape in 2014 and 2016; bamboo shoot in 2012, and 2014 − 2019; fatsia sprout in 2013 and 2016; koshiabura in 2013; and ostrich fern in 2012), and western bracken fern in 2017 showed a significant negative correlation. These weak correlations may be affected by uncertainty in the deposition data. We used a representative deposition value for each municipality and the original deposition data grid was of low resolution (see the “Methods” section Radiocesium deposition data from airborne survey). Especially for the cases lacking a clear positive correlation, the degree of radiocesium absorption by edible wild plants was largely different even in the same deposition. Radiocesium uptake by plants in an environment is also affected by other factors (e.g., soil characteristics25,26). The edible wild plants targeted in the present study were not cultivated but were collected in a variety of environments, such as forests with high organic matter content in the soil and paddy field margins with poorly drained soil high in clay content, although we cannot precisely confirm the growth environment of each species included in the present study.Figure 2Correlation between deposition and concentration of 137Cs in three edible wild plants. Circles, diamonds, and triangles indicate butterbur scape, fatsia sprout, and western bracken fern, respectively. The three species are representative deciduous perennial and tree spermatophyte, and deciduous perennial pteridophyte, respectively, in the year of the maximum number of detections.Full size imageTemporal change in T
ag
The time-dependence of Tag for each species in the period 2012–2019 is shown in Fig. 3. The Tag values of deciduous perennial spermatophytes and pteridophytes showed a decreasing trend with time. Given that the bioavailability of 137Cs in the soil in the plant root zone decreased with time, as observed in previous studies27,28, we also observed a decrease in Tag. The Tag of deciduous trees did not show a decreasing trend with time.Figure 3Temporal change in the aggregated transfer factor (Tag) in the period 2012–2019. Circles, diamonds, and triangles indicate deciduous perennial spermatophytes, deciduous tree spermatophytes (including bamboo shoot), and deciduous perennial pteridophytes, respectively. Single exponential fitted lines are shown. Solid lines indicate statistically significant parameters (see Table 2).Full size imageAfter the Chernobyl nuclear accident, radiocesium concentrations in deciduous tree leaves decreased with time owing to the effect of direct deposition at an early stage and the following root uptake effect29, and the Tag of tree leaves decreased accordingly. In previous studies conducted in orchards after the Chernobyl and Fukushima accidents, radiocesium concentrations in deciduous tree leaves showed a decreasing trend30,31. The lack of a declining trend for woody edible wild plants Tag in the present study may be due to a smaller effect of direct deposition at the early stage resulting from interception by tall tree canopies in the vicinity. The height of trees with edible wild plants is usually at eye level. The samples collected soon after the accident were possibly affected by direct deposition, whereas in the latter study period, many of the data were from trees grown after the accident. If the effect of direct deposition was large, a declining trend in Tag might have been observed as observed in orchards. Thus, the absence of a declining trend in Tag indicates that the effect of direct deposition was relatively small.As an additional possibility for the absence of a declining trend in tree Tag, the continuous supply of bioavailable radiocesium from the organic layer on the forest floor may affect the temporal change in Tag. Compared with the managed conditions in orchards of previous studies30,31, an organic layer develops on the soil surface in a forest and, therefore, reabsorption of radiocesium from the organic layer via the roots may be more active. Imamura et al.17 also observed a similar trend to that in the present study, namely that radiocesium concentrations in leaves of the canopies of the deciduous tree konara oak (Quercus serrata) did not show a temporal change from 2011 to 2015 in two Fukushima forests. These authors’ results included the effect of direct deposition on the tree bodies at an early stage of the accident, although the emergence of leaves was after the deposition. Nevertheless, a clear decreasing trend in the radiocesium concentration was not observed, which implies that a deciduous tree actively absorbs radiocesium via the roots in Fukushima forests, and a sufficient amount of radiocesium is absorbed to conceal a decline at an early stage owing to the effect of direct deposition.Single exponential fitted lines for each species are shown in Fig. 3. The estimated parameters and the Teff (year) calculated with Eq. (2) in “Methods” section are presented in Table 2. The Teff for Tag values that showed a decreasing trend was approximately 2 years, except for bamboo shoot. Tagami and Uchida10 reported that the Teff of the slow loss component for three edible wild plants of deciduous perennial spermatophytes was 970–3830 days. The 137Cs decline in pteridophytes, and deciduous shrub and herbaceous species on the floor of European forests was reported to be 1.2–8 years for Teff excluding the rapid loss component after the Chernobyl nuclear accident32. The present results are thus within the range of previous studies.Table 2 Estimated parameters and standard errors for correlations of Tag (m2/kg-FM) in the period 2012–2019 with time (day) calculated using Eq. (3) and effective half-lives [Teff, (year)] calculated using Eq. (2) for 11 parts of 10 edible wild plant species. A0 is estimated initial Tag, and λ (/day) is the 137Cs loss rate in edible parts of the plants.Full size tableFor bamboo shoot, applying a single exponential function, a relatively long Teff of 8.3 years was estimated. The Tag decreased between 2012 and 2014, and thereafter no notable change was observed. This observation may reflect the effect of rapid and a slow loss components. Indeed, we applied a two-component exponential function for bamboo shoot, and observed Teff of 0.7 years and − 7.8 years for the rapid and slow loss components, respectively. For edible wild tree species, statistically significant single exponential fitted lines were not observed, which reflected the absence of change in Tag with time, as discussed above in this section.The Tag varied for all species, varying by 1–3 orders of magnitude within a year that included more than two detections (Fig. 3, Supplemental Table 1). As demonstrated in previous studies5, the present study also showed substantial variation in Tag values, which may be for several reasons. Recently, Tagami et al.12 calculated Tag using the radiocesium concentration in edible wild plants measured by local municipalities from higher-resolution publicly available data (accurate to district level) for giant butterbur, bamboo shoot, fatsia sprout, and koshiabura. The municipalities in these authors’ study are located within the present study area. These authors’ results differed in being one or two orders of magnitude smaller than the present results. The lower resolution of the present deposition data may be one of the causes of the greater Tag variation. The other source of variation is the site dependency of radiocesium absorption by edible wild plants from the soil as described above. Clarification of factors that contribute to the variation in Tag other than 137Cs deposition, and its trends consistent with species, is necessary, which will decrease uncertainty and lead to more accurate estimation of Tag of 137Cs with wild plants.Summary of T
ag for estimation of long-term ingestion dose to the publicTo estimate long-term potential ingestion dose to the public, Tag with small temporal variability excluding high values at the early stage after the accident is required. However, for the edible wild plant species in the present study, no Tag information in an equilibrium condition from before the Fukushima accident is available. Therefore, average values of Tag for the period after the decrease in Tag has weakened and a certain number of samples is available would be appropriate. The Teff for Tag showing a decreasing trend was approximately 2 years except for bamboo shoot, which has not shown any temporal variation since 2014. The Tag for the other species, udo, uwabamisou, momijigasa, fatsia sprout, koshiabura and Japanese royal fern, has not shown temporal variation throughout 2012–2019 (see the “Results and discussion” section Temporal change in Tag). Therefore, Tag values since 2014 are applicable for estimation of long-term potential ingestion dose to the public. The GMs and GSDs of the Tag values for 2014–2019 for each species are shown in Table 3 listed in order of decreasing GM.Table 3 Aggregated transfer factor (m2/kg-FM) calculated from publicly available data for 2014–2019 for 11 parts of 10 edible wild plant species.Full size tableSignificant differences in Tag were observed among the species (one-way ANOVA with Tukey’s post hoc test, p  More