Research on Ecological Protection Mechanisms in Watersheds Based on Evolutionary Games-Inter-provincial and Intra-provincial Perspectives

Abstract

The wholeness of ecological resources in watersheds and the divisiveness of administrative regions make the protection of watersheds exceptional, and their environmental development has always been a hot issue of social concern. To realize the high-quality development of the watershed, this paper studies the ecological protection mechanism at the inter-provincial and intra-provincial levels. At the inter-provincial level, we constructed a game model for the evolution of ecological compensation between upstream and downstream governments in the watershed. We explored the changes in the strategy choices of both sides of the game under the condition of an unconstrained mechanism and the constraints of the reward and punishment mechanism of the central government. At the intra-provincial level, the local government, enterprise, and public ecological protection evolution game models are constructed to analyze the strategic choices of the three-party game subjects. The study results show that firstly, the combination of strategies (governance, compensation) cannot be realized by upstream and downstream governments alone, and the central government can intervene by introducing specific incentives and penalties. Secondly, constructing upstream and downstream government ecological compensation mechanisms under the central government constraint has prompted the local government, enterprises, and the public to achieve the ideal stable state of the tripartite game subjects (strict regulation, treatment, supervision). Thirdly, factors such as local government strategy choices and regulatory efforts can impact the rate of evolution of enterprises and the public.

Appendix

Table 5 Comparison of watershed protection research methods

1.1 Model analysis of upstream and downstream government games—under the central government's reward and punishment constraints

Referring to the calculation method of upstream and downstream government replication dynamic equations under the unconstrained mechanism, we obtain the upstream and downstream government replication dynamic equations $U^{\prime} (x),V^{\prime} (y)$ under the central government's reward and punishment mechanism, as shown in Eq. (8) and (9).

$$U^{\prime}\left( x \right) = x\left( {1 - x} \right)\left[ {R_{1} - C + G + Q + y\left( {F - Q} \right)} \right]$$

(8)

$$V^{\prime} \left( y \right) = y\left( {1 - y} \right)\left[ {G - P + Q + x\left( {F - Q} \right)} \right]$$

(9)

Make $U^{\prime} (x),V^{\prime} (y)$ equal to 0 to get the five equilibrium points $A\left( {0,0} \right),B\left( {1,0} \right),C\left( {0,1} \right),D(1,1),E(\frac{Q + G - P}{{Q - F}},\frac{{R_{1} + G + Q - C}}{Q - F})$ of the game matrix, and then analyze the Jacobi matrix equilibrium points of this evolutionary game system to check the stable state of the system. The replicated dynamic equations from the upstream and downstream governments form the following Jacobi matrix, shown in Eq. (10).

$$A^{\prime} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{\partial U^{\prime}\left( x \right)}{{\partial x}}} \\ {\frac{\partial V^{\prime}\left( y \right)}{{\partial x}}} \\ \end{array} } & {\begin{array}{*{20}c} {\frac{\partial U^{\prime}\left( x \right)}{{\partial y}}} \\ {\frac{\partial V^{\prime}\left( y \right)}{{\partial y}}} \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left( {1 - 2x} \right)\left[ {R_{1} - C + G + Q + y(F - Q)} \right]} \\ {y\left( {1 - y} \right)\left( {F - Q} \right)} \\ \end{array} } & {\begin{array}{*{20}c} {x\left( {1 - x} \right)\left( {F - Q} \right)} \\ {\left( {1 - 2y} \right)\left[ {G - P + Q + x\left( {F - Q} \right)} \right]} \\ \end{array} } \\ \end{array} } \right]$$

(10)

The eigenvalues of the Jacobi matrix for each equilibrium are obtained as shown in Table 6. (Pay attention to the premise.$Q > F$).

Table 6 Equilibrium points eigenvalue

where $\begin{gathered} M = ((g - p + q)(f + g - p)(f - c + g + r)(g - c + q + r))^{(1/2)} /(f - q) \hfill \\ N = - ((g - p + q)(f + g - p)(f - c + g + r)(g - c + q + r))^{(1/2)} /(f - q) \hfill \\ \end{gathered}$

The equilibrium points $E(\frac{Q + G - P}{{Q - F}},\frac{{R_{1} + G + Q - C}}{Q - F})$ have eigenvalues $M$ and $N$ of different signs, and the system is in an evolutionary unstable equilibrium state(Cheng et al. 2021), so the equilibrium point $E$ will not be analyzed in the following. As described above, there are six situations in which upstream and downstream governments game the system under the central government's incentives and disincentives mechanism.

Situation $1$:$R_{1} + G + Q < C;G + Q < P$,equilibrium point $A(0,0)$ is the system evolutionary stabilization strategy. Situation $2$:$R_{1} + G + Q < C;G + F < P < G + Q$, equilibrium point $C(0,1)$ is the system evolutionary stabilization strategy. Situation $3$: $R_{1} + G + F < C < R_{1} + G + Q;G + F < P < G + Q$, equilibrium point $B(1,0)$ and $C(0,1)$ is the system evolutionary stabilization strategy. Situation $4$: $R_{1} + G + F < C < R_{1} + G + Q;P < G + F$, equilibrium point $C(0,1)$ is the system evolutionary stabilization strategy. Situation $5$: $C < R_{1} + G + F;G + Q < P$, equilibrium point $B(1,0)$ is the system evolutionary stabilization strategy. Situation $6$: $C < R_{1} + G + F;P < G + F$, equilibrium point $D(1,1)$ is the system evolutionary stabilization strategy.

1.2 Model analysis of the game among local governments, enterprises, and the public

The replication dynamic equation $U^{1} \left( x \right)$ for local government is shown in Eq. (11).

$$U^{1} \left( x \right) = x(1 - x)[(1 - r)(zh_{1} + yzJ - C_{1} - yE - zJ) + yO + (1 - y)L]$$

(11)

The replication dynamic equation $V^{1} (y)$ for enterprises is shown in Eq. (12).

$$V^{1} (y) = y(1 - y)[ - C_{2} + H + rE + F + zh_{2} + x(1 - r)E]$$

(12)

The replication dynamic equation $W^{1} (z)$ for public is shown in Eq. (13).

$$W^{1} (z) = z(1 - z)[ - C_{3} + x(1 - y)(1 - r)J + yI - ryJ + rJ]$$

(13)

Making the replicated dynamic equation of the three-party game subject equal to 0, we can get 15 equilibrium points in the game process of local government, enterprises, and the public, among which there are eight pure species strategy equilibrium points: $A\left( {0,0,0} \right),B\left( {1,0,0} \right),C\left( {0,1,0} \right),D(0,0,1),E(1,1,0),F(1,0,1),G(0,1,1),H(1,1,1)$. In the context of multi-game subjects, only pure species strategy Nash equilibria can be asymptotically stable equilibria (Hewitt and Wainwright 1993; Hu et al. 2023). Using the derived replication dynamic equation, the Jacobi matrix of the game system can be obtained, and the corresponding eigenvalue expressions can be obtained by substituting each equilibrium point into the Jacobi matrix. The stability of each equilibrium point is further discussed according to the value range of each parameter in the expression, and the stability conditions of each equilibrium point are shown in Table 7.

Table 7 Equilibrium points stability conditions and stability analysis