Fig. 1.

A schematic diagram of the number and reduction rate of natural villages in China during 2000-2020. Data were collected from the China Statistical Yearbook (National Bureau of Statistics of China, 2000-2020)."

Fig. 2.

Mutual influence and restriction between local governments and residents."

Table 1

Descriptions of parameters used in the evolutionary game model."

Table 2

The payoff matrix for the evolutionary game model of traditional village protection."

Fig. 3.

Phase diagrams of replicator dynamics of local governments. (a), x=0 is the evolutionary stability strategy (ESS) when y<β, and x=1 is the ESS when y>β; (b), x=0 is the ESS when y>β, and x=1 is the ESS when y<β ($\beta =\frac{B}{A}$,where β is an equilibrium point only if 0≤β≤1). A represents Equation 10; B represents Equation 11"

Fig. 4.

Phase diagram of replicator dynamics of residents. (a), y=0 is the ESS when x<α, and y=1 is the ESS when x>α; (b), y=0 is the ESS when x>α, and y=1 is the ESS when x<α ($\alpha =\frac{D}{C}$, where α is an equilibrium point only if 0≤α≤1). C represents Equation 19; D represents Equation 20."

Table 3

Analysis of local asymptotic stability for the equilibrium points."

Table 4

Parameter assignments under different cases of evolutionary stability strategy (ESS)."

Fig. 5.

ESS analysis for the five different cases (a-e) in which the initial values of x and y are set to 0.2, 0.4, 0.6 and 0.8. t represents the length of a time unit, i.e., the number of times the game is played."

Fig. 6.

Impacts of λ1 (the intensity of local governments with support and supervision) and λ2 (the degree of residents with participation) on the evolutionary results of local governments (focusing on exploring different convergence directions and speeds of curves). (a), λ1=0.2 and λ2=0.2, 0.5, and 0.9; (b), λ1=0.5 and λ2=0.2, 0.5, and 0.9; (c), λ1=0.9 and λ2=0.2, 0.5, and 0.9."

Fig. 7.

Impacts of λ1 and λ2 on the evolutionary results of residents (focusing on exploring different convergence directions and speeds of curves). (a), λ1=0.1 and λ2=0.1, 0.5, and 0.9; (b), λ1=0.5 and λ2=0.1, 0.5, and 0.9; (c), λ1=0.9 and λ2=0.1, 0.5, and 0.9."

Fig. 8.

Impacts of m (the probability of penalties imposed by local governments with complete support and supervision on residents with partial participation) on the evolutionary results of local governments (a) and residents (b) (focusing on exploring different convergence directions and speeds of curves)."

Fig. 9.

The impact of δ (the proportion of subsidies provided by local governments to residents with complete support and supervision) (a) and D1 (the maximum remedial cost that needs to be paid by local governments with complete support and supervision) (b) on the evolutionary results of local governments’ strategies (focusing on exploring different convergence directions and speeds of curves)."

Fig. 10.

Impacts of R1(t) (the maximum local cultural development benefits obtained by local governments with full participation of the residents) (a) and S1(t) (the maximum local cultural development losses suffered by local governments with partial participation of residents) (b) on the evolutionary results of local governments’ strategies (focusing on exploring different convergence directions and speeds of curves)."

Fig. 11.

Impacts of φ (the taxation rates of residents) (a) and R3 (the direct economic benefits of residents with full participation) (b) on the evolutionary results of residents’ strategies (focusing on exploring different convergence directions and speeds of curves)."

Fig. 12.

Impacts of D2 (the cost of resistance for residents) (a) and S4 (the losses of resistance for local governments) (b) on the results of the evolutionary game model (focusing on exploring different convergence directions and speeds of curves)."