Philippa Kaur

Assume n countries ((nge 2)) are located in a transboundary river basin and they are the players of an evolutionary game in which the countries’ strategies concerning water sharing in the basin evolve over time. Each country can choose between a cooperative strategy or a non-cooperative strategy. The game’s interactions and players’ payoffs vary with the number n and the location of the countries within the river basin, specifically, in relation to whether they are upstream-located or downstream-located countries within the river basin. The probability of country i choosing a cooperative strategy is herein denoted by ({x}_{1}^{(i)}), (i=1, 2, dots , n), and there are ({2}^{n}) payoff sets for all the combinations of the countries’ strategies. This paper assesses the interactions between three countries sharing a transboundary river basin.Problem descriptionLet 1, 2, and 3 denote three countries sharing a transboundary river basin. Country 1 is upstream and countries 2 and 3 are located downstream. Country 1 can use maximum amount of the water of the river and choose not to share it with the downstream countries. This strategy, however, may trigger conflict with the two other countries of political, social, economic, security, and environmental natures. Instead, Country 1 can release excess water to be shared by Countries 2 and 3. Countries 2 and 3 are inclined to cooperate with Country 1 unless other benefits emerge by being non-cooperative with Country 1.There are two types of benefits and one type of cost in the payoff matrix of the assumed problem that are economic in nature. The first is a water benefit earned by a country from receiving the water from the transboundary river. The set of benefits related to water use includes economic benefits earned from agricultural, urban, and industrial development benefits. It should be noted that the water benefit for Country 1 means the economic benefit of consuming more water than its water right from the river. So, water benefits of Country 2 and 3 are the economic benefit of consuming excess water of upstream which is released by Country 1.The second is a potential benefit earned from the cooperative strategy of a country. Cooperation benefits stem from sustainability conditions like social interests, environmental benefits and political conjunctures such as international alliances and harmony from amicable interactions with neighboring countries. The parameters F and E (water benefit and potential benefit, respectively) encompass a number of benefit parameters; nevertheless, parameters were simplified to two benefit parameters to simplify the complexity of the water-sharing problem. Costs forced on other countries from non-cooperation by a country involves commercial, security, political, diplomatic, military, and environmental costs. Figure 1 displays the locations of three countries and their shifting interactions in a transboundary river basin.Figure 1Schematic of the transboundary river and riparian countries with their shifting interactions.Full size imageBasic assumptionsThe evolutionary game model of interactions between riparian countries in the transboundary river basin rests on the following assumptions:
Assumption 1

There are three countries (i.e., players) in the game of transboundary water sharing, each seeking to maximize its payoff from the game.

Assumption 2

Country 1 has two possible strategies. One is for Country 1 to release a specified amount of water to the downstream countries (this would be Country 1’s cooperative strategy). The cooperative strategy by Country 1 would produce benefits F2 and F3 to Countries 2 and 3, respectively. By being cooperative Country 1 would attain a benefit E1 called the potential benefit from cooperative responses from the downstream countries. The other strategy is for Country 1 to deny water to the downstream countries (this would be Country 1’s non-cooperative strategy), in which case Country 1 would earn the water benefit F1 from using water that would otherwise be released, but would forego the potential benefit E1. Moreover, by pursuing a non-cooperative strategy Country 1 would inflict a cost C1m to the downstream countries.

Assumption 3

There are two possible strategies for Country 2. One is for Country 2 to accept the behavior of Country 1 (this would be Country 2’s cooperative strategy), which would cause earning a potential benefit E2 to Country 2. Recall that if Country 2 acquiesces to Country 1’s cooperative behavior it would receive a benefit F2. Or, Country 2 may disagree with Country 1 (this would be Country 2’s non-cooperative strategy), in which case, Country 2 would lose benefit E2, and it would inflict a cost C2m to the other countries.

Assumption 4

Similar to Country 2, Country 3 has two possible strategies. One is for Country 3 to agree Country 1’s behavior (this would be Country 3’s cooperative strategy) attaining a potential benefit E3. Recall that if Country 3 agrees with Country 1’s cooperative behavior it would gain a benefit F3. Another strategy for Country 3 is to oppose Country 1 (this would be Country 3’s non-cooperative strategy) missing the benefit E3 and forcing a cost C3m to the other countries.
Table 1 defines the benefits and costs that enter in the transboundary water-sharing game described in this work. The payoff to country (i=mathrm{1,2},3) depends on its own strategy and on the strategies of the other countries, and each country may choose to be cooperative or non-cooperative. The strategies of country (i) are denoted by 1 (cooperation) and 2 (non-cooperation). The probabilities of country (i)’s strategies are denoted by ({x}_{1}^{(i)}) and by ({x}_{2}^{(i)}), in which the former represents cooperation and the latter represents non-cooperation. Clearly, ({x}_{1}^{(i)})+ ({x}_{2}^{(i)}) = 1. The payoff to country (i=1, 2, 3) when the strategies of Countries 1, 2, 3 are (j, k,l), respectively, where (j, k,l) may take the value 1 (cooperation) or 2 (non-cooperation) is denoted by ({U}_{jkl}^{left(iright)}). Thus, for instance, the payoff to country (i=2) is represented by ({U}_{212}^{(2)}) when Countries 1 and 3 are non-cooperative and Country 2’s strategy is cooperative. Evidently, there are 23 payoffs to each country given there are three countries involved and each can be cooperative or non-cooperative. Table 2 shows the symbols for the payoffs that accrue to each country under the probable strategies.Table 1 Benefits and costs.Full size tableTable 2 Payoff matrix under cooperation or non-cooperation.Full size tableFormulation of the transboundary water-sharing strategies as an evolutionary gameThe expected payoff to country (i) is expressed by the following equation:$${U}^{(i)}=sumlimits_{j = 1}^2 {sumlimits_{k = 1}^2 {sumlimits_{l = 1}^2} } {x}_{j}^{(1)}{x}_{k}^{(2)}{x}_{l}^{(3)} {U}_{jkl}^{(i)} quad i=1, 2, 3$$
(1)
The following describe the expected payoffs of Country 1 when it acts cooperatively (({U}_{1}^{(1)})) or non-cooperatively (({U}_{2}^{(1)})):$${U}_{1}^{(1)}={x}_{1}^{(2)}{x}_{1}^{(3)}{U}_{111}^{left(1right)}+{x}_{1}^{(2)}{x}_{2}^{(3)}{U}_{112}^{left(1right)}+{x}_{2}^{(2)}{x}_{1}^{(3)}{U}_{121}^{(1)}+{x}_{2}^{(2)}{x}_{2}^{(3)}{U}_{122}^{(1)}$$
(2)
$${U}_{2}^{(1)}={x}_{1}^{(2)}{x}_{1}^{(3)}{U}_{211}^{left(1right)}+{x}_{1}^{(2)}{x}_{2}^{(3)}{U}_{212}^{left(1right)}+{x}_{2}^{(2)}{x}_{1}^{(3)}{U}_{221}^{left(1right)}+{x}_{2}^{(2)}{x}_{2}^{(3)}{U}_{222}^{left(1right)}$$
(3)
Therefore, the expected payoff of Country 1 is ({U}^{(1)}) which is equal to:$${U}^{(1)}={x}_{1}^{(1)}{U}_{1}^{(1)}+{x}_{2}^{(1)}{U}_{2}^{(1)}= sumlimits_{j = 1}^2 {sumlimits_{k = 1}^2 {sumlimits_{l = 1}^2 } }{x}_{j}^{(1)}{x}_{k}^{(2)}{x}_{l}^{(3)} {U}_{jkl}^{(1)}$$
(4)
The expected payoffs of Countries 2 and 3 can be similarly obtained as done for Country 1. The cooperative and non-cooperative expected payoffs of all countries can be expressed in terms of the payoffs listed in Table 1. The results are found in Appendix A.Replication dynamics equationsThe replication dynamics equations describe the time change of the probabilities of a player’s strategies. The replication dynamics equation of Countries (i) is denoted by ({G}^{(i)}left({x}_{1}^{(i)}right)) which is as follow22:$${G}^{(i)}left({x}_{1}^{(i)}right)=frac{d{x}_{1}^{(i)}}{dt}={x}_{1}^{(i)}left({U}_{1}^{(i)}-{U}^{(i)}right)$$
(5)
The replication dynamics equations of Countries 1, 2 and 3 are presented in Appendix B according to the benefits and costs showed in Table 1.Stability analysis of a country’s strategiesUnder the assumption of bounded rationality each country does not know which strategies may lead to the optimal solution in the game. Therefore, the countries’ strategies change over time until a stable (i.e., time-independent) solution named evolutionary stable strategy (ESS) is attained. The evolutionary stable theorem for replication dynamics equation states that a stable probability of cooperation ({x}_{1}^{(i)}) for country (i) occurs if the following conditions hold25: (1) ({G}^{(i)}left({x}_{1}^{(i)}right)=0), and (2) (d{G}^{(i)}left({x}_{1}^{(i)}right)/d{x}_{1}^{(i)} More

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Conjugative plasmids confer important traits to microbial communities, with both deleterious and beneficial effects on human health, the environment, and biotechnology [1,2,3]. The spread of virulence and resistance to antimicrobial agents [1, 2] and the facilitation of specific pollutant biotransformations exemplify the importance of conjugative plasmids [3, 4]. Understanding the mechanisms governing the transfer and spread of conjugative plasmids is therefore critically important. Although substantial research efforts have been made toward understanding the molecular mechanisms and biological determinants of plasmid conjugation [5], the underlying driving forces from physical and ecological aspects remain unclear.Many microbial communities exist in environments that are periodically or continuously exposed to unsaturated water conditions. For example, the communities residing in the vadose zone of soils are periodically exposed to saturated conditions after rainfall events and irrigation, and thereafter to unsaturated conditions upon soil draining. The microbial communities inhabiting the outer surfaces of various hosts such as skin, teeth, leaves, or roots also experience frequent hydration dynamics. The air-water interfaces of such soil particles or host surfaces are subject to water evaporation when the ambient relative humidity (RH) is More

Climate dataThermal profiles showed considerable diel variation at each field location (Fig. 3). Soil temperatures on the ground surface fluctuated the most, reaching maximum temperature (Mt Ginini, 45.5 °C; Piccadilly Circus first season, 46.5 °C; Piccadilly Circus second season, 46.0 °C; Cooma, 42.1 °C; Dartmouth, 47.5 °C) during the day (1330–1730 h) and dropping to low level (Mt Ginini, 4.0 °C; Piccadilly Circus first season, 8.5 °C; Piccadilly Circus second season, 6.5 °C; Cooma, 7 °C; Dartmouth 12.5 °C) at night (2330–0400 h) (ANOVA, F4, 6521 = 279.4, P  More

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