**3. Results**

#### *3.1. Identification of Human–Land Coupling System Equilibrium Point*

In order to make the application of the human–land coupling dynamics model meaningful in Yuxi City, it is necessary to find the sufficient conditions for the model's stability and discuss the stability of the model's equilibrium point and the sustainability of the model. Therefore, the model's equilibrium point is solved first, and the model's stability is further analyzed through the model's equilibrium point. When solving the fractional model, the Adama–Bashforth–Moulton predictive correction algorithm is applied [46]. To facilitate calculation, let the step size Δt = 0.01, *φ =* 0.9, substitute *r* = 0.0048 into the model, and then through data analysis and simulation, parameters *a* = 0.034, *b* = 0.012, *s* = 0.004, *h* = 0.05, *d* = 0.08 can be obtained. After a normalized processing of the data for various types of areas and the population of Yuxi City in 1995, the initial values are *F*(0), *R*(0), *U*(0), *P*(0) = (0.07, 0.02, 0.4, 0.75); substitute them into Equation (3). Then, the equation

$$\begin{cases} 0 = \frac{-dF(t)P(t-\tau)}{1+P(t-\tau)} + sLI(t) \\ 0 = \frac{dF(t)P(t-\tau)}{1+P(t-\tau)} - aR(t) + bLI(t) \\ 0 = aR(t) - sLI(t) - bLI(t) \\ 0 = rP(t)\left(1 - \frac{h}{R(t)}P(t-\tau)\right) \end{cases}$$

is obtained, and the solution to this equation is

$$\begin{cases} P^\* = 0.075617 \\ R^\* = 0.021169 \\ \mathcal{U}^\* = 0.44984 \\ P^\* = 0.423379 \end{cases} .$$

That is, the equilibrium point of the model is (0.075617, 0.021169, 0.44984, 0.423379), and the results are all positive. Therefore, it is consistent with the non-negative situation of land and population, which is systematic and meaningful and reflects the rationality of the model.

#### *3.2. Visual Output and Expression of Human–Land Coupling Relationship*

The basic reproduction number *R0* of the human–land coupling dynamics model is a very important parameter; it is said that in the state of balance, the amount of the increase in population brought by land-use changes is a sign that decides whether the land-use type changes or not, namely, only when *R0* > 1 does land-use-type transformation occur. If *R0* < 1, the transformation will tend to zero. Therefore, according to the calculation method of the basic regeneration number [47], the basic regeneration number of the model can be obtained after solving for the equilibrium point, *R*<sup>0</sup> = *bd*+*ds ash* = 18.8235 > 1, indicating that the land-use types' transformation is significant in the model. Through calculation, *ω*<sup>0</sup> = 0.002358, *τ<sup>0</sup>* = 728.403 [30]. An arbitrary *τ* value which is less than *τ<sup>0</sup>* was chosen arbitrarily, and MATLAB software was used for numerical simulation. Let *τ* = 700, and it can be seen through numerical simulation that the human–land change trend over time and the three-dimensional evolution of the human–land relationship can be obtained after a period of damped oscillation (Figures 3 and 4). As *τ* = 700 < *τ0*, both the theoretical and numerical simulation show that the equilibrium point (0.075617, 0.021169, 0.44984, 0.423379) is locally asymptotically stable, that is, under the influence of the population, the development and change value of all kinds of land use fluctuates, but eventually tends to be stable around the equilibrium point of the model.

**Figure 3.** Evolution trend of people and land over time in Yuxi City when *τ* = 700 ((**a**) change trends of forest and grass land over time; (**b**) change trends of production and living land over time; (**c**) change trends of unused land over time; (**d**) change trends of population over time).

**Figure 4.** Three-dimensional map of human–land evolution in Yuxi City when *τ* = 700 ((**a**) threedimensional evolution of the unused land, forest and grass land and production and living land over time; (**b**) three-dimensional evolution of the population, unused land, and forest and grass land over time; (**c**) three-dimensional evolution of the population, production and living land and forest and grass land over time; (**d**) three-dimensional evolution of the population, unused land and production and living land over time).

As long as the time delay *<sup>τ</sup>* does not exceed *<sup>τ</sup><sup>0</sup>* = 728.403, after a *<sup>t</sup>* = 8 × <sup>10</sup><sup>4</sup> simulation running time, the land-type area and population in Yuxi City will reach a balance point with time, and the system tends to be in a dynamic stable state. As can be seen from Figure 3a, the forest and grass land in the study area keeps changing with time, increasing and decreasing, but eventually tends to a stable value (0.075617). Similarly, under the system's equilibrium state, the production and living land tends to 0.021169 (Figure 3b), and the unused land tends to 0.44984 (Figure 3c). Under the three land-type transformation and constraint conditions, the population also changes with time, and the final population tends to 0.423379 (Figure 3d). Figure 4 shows the human–land three-dimensional evolutionary relationship at *τ* = 700, which explains the dynamic characteristics of the human–land coupling system in a stable state. It can be seen that the system is in a stable state when the time delay is less than *τ0*.

Similarly, a value of *τ* that is larger than *τ<sup>0</sup>* was arbitrarily selected for numerical simulation with MATLAB software. Let *τ* = 740, and after a period of oscillation, the trend of human–land changes over time and the three-dimensional evolution of the human–land relationship is obtained (Figures 5 and 6). It can be seen that when *τ* = 740 > *τ<sup>0</sup>* = 728.403, the equilibrium point (0.075617, 0.021169, 0.44984, 0.423379) is no longer stable, that is, under the influence of population, the development of different land types fluctuates greatly at first; however, after the time delay is greater than a certain value, the values of all land-use types will tend to oscillate within a certain period, that is, Hopf bifurcation occurs. At this time, the area of the forest and grass land, the area of the production and living land and the area of the unused land will all show a periodic decline and increase.

**Figure 5.** Evolution trend of people and land over time in Yuxi City when τ = 740 ((**a**) change trends of forest and grass land over time; (**b**) change trends of production and living land over time; (**c**) change trends of unused land over time; (**d**) change trends of population over time).

**Figure 6.** Three-dimensional map of human–land evolution in Yuxi City when τ = 740 ((**a**) threedimensional evolution of the unused land, forest and grass land and production and living land over time; (**b**) three-dimensional evolution of the population, unused land, and forest and grass land over time; (**c**) three-dimensional evolution of the population, production and living land and forest and grass land over time; (**d**) three-dimensional evolution of the population, unused land and production and living land over time).

When the time delay *τ* exceeds *τ<sup>0</sup>* = 728.403, after the simulation running time of *<sup>t</sup>* = 8 × <sup>10</sup>4, the population and land types change from the initial large fluctuations to periodic changes around the equilibrium points (0.075617, 0.021169, 0.44984, 0.423379) (Figure 5); that is, in an unstable state, it is significantly different from the situation when the time delay *τ* is less than *τ<sup>0</sup>* (Figure 3). When the time delay becomes larger, the area of forest and grass land *F*(*t*), the area of production and living land *R*(*t*) and the area of unused land *U*(*t*) will change periodically. The forest and grass land change periodically around 0.075617 (Figure 5a), the production and living land change periodically around 0.021169 (Figure 5b), and the unused land changes periodically around 0.44984 (Figure 5c), and will not tend to a stable value. Figure 6 shows the three-dimensional evolution of the human–land relationship when *τ* = 740, which explains the dynamic characteristics of the human–land coupled system in the unstable state. It can be seen that when the time delay is greater than *τ0*, the system exhibits an unstable periodic oscillation phenomenon.

Comparing Figure 4 with Figure 6, we could see that when *τ* > *τ0*, the human–land evolution trend is more consistent with the actual situation, that is, with the periodic change in the population, the area of land types changes periodically. However, in order to tend to a stable state at *τ* < *τ0*, that is, the equilibrium state of the human–land system, relevant policies can be formulated and implemented by the government, that is, parameters can be controlled and adjusted.

#### *3.3. Coupling Spatiotemporal Parameters of Mountain–Basin Human–Land Relationship*

With the migration of the population from mountainous areas to basin areas, the agglomeration of the population leads to the occupation of agricultural land and the expansion of urban land [45,48]. Early on, due to the limited population, the urbanization level is not high, and coupled with the influence of the policy constrains, the migration of the population is also rare. The main way of life is farming, and the ecological and environmental effects caused by population migration are not obvious. However, with the rapid improvement in the urbanization level and the acceleration of the migration of the population from rural to urban areas, land-use changes are accelerating. In mountainous areas, the arable land and construction land have been abandoned [49] and turned into a wasteland, that is, unused land. However, the wasteland will naturally recover into forest and grass land after a certain period of time. In basin areas, with the migration of the population, the demand for land is increasing, and the unused land will be gradually transformed into forest and grass land and farmland and construction land.

With the rapid increase in the population of the basin area, the land's resource-carrying capacity, industrial-supporting capacity and infrastructure are facing more challenges, driving the changes in the land-use pattern to meet the needs of population agglomeration. At the same time, the population in mountainous areas is decreasing, farmland and construction land will be abandoned, and ecological land has been restored. According to Model (3), two subsystems of mountainous areas and basin areas are distinguished, and human–land coupling evolution and development models of mountainous areas and basin areas are constructed, respectively:

$$\begin{cases} D^{\Phi}F\_{M}(t) = \frac{-d\_{M}f\_{M}(t)P\_{M}(t-\tau\_{M})}{1+P\_{1}(t-\tau\_{1})} + s\_{M}\mathcal{U}\_{M}(t) \\ D^{\Phi}R\_{M}(t) = \frac{d\_{M}f\_{M}(t)P\_{M}(t-\tau\_{M})}{1+P\_{M}(t-\tau\_{M})} - a\_{M}R\_{M}(t) + b\_{M}\mathcal{U}\_{M}(t) \\ D^{\Phi}\mathcal{U}\_{M}(t) = a\_{M}R\_{M}(t) - s\_{M}\mathcal{U}\_{M}(t) - b\_{M}\mathcal{U}\_{M}(t) \\ D^{\Phi}P\_{M}(t) = r\_{M}P\_{M}(t) \left[1 - \frac{h\_{M}}{R\_{M}(t)}P\_{M}(t-\tau\_{M})\right] \end{cases} \tag{4}$$
 
$$\begin{cases} D^{\Phi}F\_{B}(t) = \frac{-d\_{B}F\_{B}(t)P\_{B}(t-\tau\_{B})}{1+P\_{B}(t-\tau\_{B})} + s\_{B}\mathcal{U}\_{B}(t) \\ D^{\Phi}R\_{B}(t) = \frac{d\_{B}P\_{B}(t)P\_{B}(t-\tau\_{B})}{1+P\_{B}(t-\tau\_{B})} - a\_{B}R\_{B}(t) + b\_{B}\mathcal{U}\_{B}(t) \\ D^{\Phi}\mathcal{U}\_{B}(t) = a\_{B}R\_{B}(t) - s\_{B}\mathcal{U}\_{B}(t) - b\_{B}\mathcal{U}\_{B}(t) \\ D^{\Phi}P\_{B}(t) = r\_{B}P\_{B}(t) \left[1 - \frac{h\_{B}}{R\_{B}(t)}P\_{B}(t-\tau\_{B})\right] \end{cases} \tag{5}$$

In the formula, *φ*∈[0, 1] is the fractional order, and the relevant parameters are as follows: *FM*(*t*), *RM*(*t*), *UM*(*t*) and *FB*(*t*), *RB*(*t*), *UB*(*t*) respectively represent the forest and grass land, production and living land, unused land in mountainous areas and basin areas, and *FM*(*t*) + *FB*(*t*) = *F*(*t*), *RM*(*t*) + *RB*(*t*) = *R*(*t*), *UM*(*t*) + *UB*(*t*) = *U*(*t*), *N* = *F*(*t*) + *R*(*t*) + *U*(*t*) = 1. *FM*(*t*), *RM*(*t*), *UM*(*t*) and *FB*(*t*), *RB*(*t*), *UB*(*t*) are the population density of mountainous areas and basin areas at time *t*, respectively. The production and living land [*RM*(*t*) and *RB*(*t*)] will change into unused land [*UM*(*t*) and *UB*(*t*)] during the period of 1/*aM* and 1/*aB*, and then become forest and grass land [*FM*(*t*) and *FB*(*t*)] through natural succession or ecological restoration after the interval of 1/*sM* or 1/*sB*. The unused land can also be reclaimed or developed into production and living land after an interval 1/*bM* and 1/*bB*. The average reclamation capacity of mountainous areas and basin areas is described by the constants *dM* and *dB* respectively. *rM* and *rB* are the natural growth rate of the population in mountainous areas and basin areas, respectively.

MATLAB software was used to further fit the land-type area and population for different periods of mountainous areas and basin areas, and the parameters of the human–land coupling dynamics model in mountainous areas and the human–land coupling dynamics model in basin areas were obtained, respectively (Table 2).

According to the existing data analysis, the conversion time parameters for production and living land to unused land of mountainous areas and basin areas in Yuxi City are *aM* = 0.0486 and *aB* = 0.0126, respectively, indicating that it takes about 20 years for production and living land in mountainous areas to be abandoned and then converted to unused land, while it takes about 80 years for basin areas. *aM* is greater than *aB*, indicating that the conversion time of productive and living land to unused land in the basin area is slower than that in the mountainous area, mainly because the population growth in the basin area is faster than that in the mountainous area, and the demand for productive and living land for economic construction is larger. This trend will continue for a long time in the future, making the conversion time of productive and living land to unused land much longer than that in the mountainous area.


**Table 2.** Fitting parameters of human–land coupling dynamics model in mountainous areas and basin areas of Yuxi City.

The time parameters of conversion from unused land to productive and living land in mountainous areas and basin areas are *bM* = 0.0062 and *bB* = 0.0139, respectively, indicating that unused land in mountainous areas will be reclaimed or developed into productive and living land again after about 160 years, while the time of conversion in basin areas is shorter, about 70 years. *bM* is less than *bB*, indicating that the conversion time of unused land to productive and living land in basin areas is faster than that in mountainous areas. With the rapid development of the society and economy and the growth of the population in the basin area, the demand for construction land is always on the rise. Under the background of the policy of vigorously protecting cultivated land, the demand for construction land is mainly solved by the transformation of forest and grass land and unused land, which allows the unused land in the basin area to be transformed quickly and in a shorter time than that in the mountainous area.

The conversion time parameters of unused land to forest and grass land for mountainous areas and basin areas are *sM* = 0.0051 and *sB* = 0.0028, respectively, indicating that it takes about 200 years for unused land to convert to forest and grass land by natural succession or ecological restoration in the mountainous area, while it takes about 350 years for the basin area. *sM* is greater than *sB*, showing that the conversion time from unused land to forest and grass land in the basin area is longer than that in the mountainous area. The main reason is that the social and economic development of the basin area has a radiation effect on the population and economy in the mountainous area, which makes the mountainous area gradually decline. The production and living land, such as farmland and construction land, is transformed into unused land and further transformed into forest and grass land due to their abandonment and extensive management, and the transformation speed is fast.

The land area parameters required for individual maintenance are *hM* = 0.0335 and *hB* = 0.0165, respectively. *hM* is greater than *hB*, indicating that the land area required for each unit in basin areas is smaller than that in mountainous areas. The main reason is that the land-use intensity in mountainous areas is significantly lower than that in basin areas, and the land yield rate is far lower than that in the basin area, so that the land area required by each unit is larger than that in the basin area. The average reclamation or development capacity in mountainous areas and basin areas are *dM* = 0.03 and *dB* = 0.05, respectively. *dM* is less than *dB*. This shows that the average reclamation and development capacity of the basin area is higher than that of the mountainous area, mainly because the economic development level, investment capacity and natural conditions of the basin area are better than those of the mountainous area.

#### **4. Discussion**

Under mathematical and geostatistical semantics, the order in a fractional differential equation can not only affect the dynamic characteristics of the fractional differential model, but also advance or delay the occurrence of the stability of the fractional differential model [50,51]. Therefore, the stability of the model can be improved by adjusting the model parameters. The influence of the time delay on the fractional order model is also dual. On the one hand, the time delay can make the fractional order model lose stability and lead to bifurcation. On the other hand, under certain conditions, the stability of the fractional order model can be improved with an appropriate time delay, and the occurrence of bifurcation can be further delayed [43]. Under the existing conditions of Yuxi City, *τ<sup>0</sup>* = 728.403 is the critical point between the stability and instability of a human–land coupling system. The state of the human–land coupling system includes a stable equilibrium state and periodic oscillation state. The time delay can be used to determine the two states and how to adjust from periodic change to a stable state, or how to adjust to a stable state when periodic change is presented. Then, the time delay can be changed by changing the parameters *a*, *b*, *s*, *h*, *d* and *r*. The time delay is like an invisible hand, which can not only optimize the allocation of land resources, but also give a warning signal when the production and living land area tends to be unstable, reminding people to make the land-use evolution stay stable by regulating the average reclamation or development capacity of individuals (*d*) and the inherent growth rate of population changes (*r*). The rate by which the unused land is recultivated or developed (*b*) and an individual's average reclamation or development capacity (*d*) can be artificially controlled. When the human–land evolutionary system tends to be stable, *b*∈[0.012, 1], *d*∈[0.08, 1] and *d* is greater than *b*, indicating that the average reclamation or development capacity has a greater impact on the evolutionary stability of land use than the transformation rate of unused land. The optimal state of balancing a human–land coupling system is the optimal state of sustainable development, which is an ideal state. By adjusting these parameters, the system can be as close to the ideal state as possible.

Chen [47] constructed an integer order land dynamics model with a time delay in 2017, simulated the data of population and land-use change over the years in China, and found that the service life of subsistence land in China is about 25–70 years (*a* = 0.04), the time for reclaiming or developing the wasteland into subsistence land is about 100 years (*b* = 0.011), and the time for restoring the wasteland to original land is about 1000 years (*s* = 0.005). By comparison, there are similarities and differences with the parameters in this study, which are highlighted by the conversion time parameter *s* of unused land to forest and grass land. Theoretically, it will take a long time for the degraded land to be restored to the original forest and grass land through natural succession, but with the progress of science and technology, this time will be greatly shortened, especially in the mountainous area of Yuxi City with its better ecological environment. Of course, these time parameters are obtained from existing data analysis. The reclamation or development rate of unused land (*b*) and the average reclamation or development capacity (*d*) are both controllable factors, and the average reclamation or development capacity has a greater impact on the stability of human–land systems.

This study based on systems integration thought that by analyzing the relationship between population and land based on coupled differential equations of forest and grass land, production and living land, unused land and population, it could build a fractional order human–land coupling dynamics mode with a Holling-II-type land conversion rate and time delay. The human–land coupling mechanism of the mountain–basin system is quantitatively described, and a new simulation direction is provided for coordination and optimization. However, the application of the method still needs to be improved in the future. First, since the human–land coupling relationship is a long-term process, the accuracy of parameter estimations depends on long-term sequence data. However, the existing collected data are only 24 years old, which is too small compared to the land conversion cycle. The model built is a high-dimensional and nonlinear differential system, and the

estimation of its parameters is inherently difficult. In addition, the lack of data makes it more difficult to achieve accurate model parameters such as for population growth rate and land conversion rate, etc. Secondly, in the classification process, there are problems such as the inaccuracy of various land types. For example, forest and grass land are assumed as primitive land in the model, but in fact, the existing forest and grass land have been transformed by human beings for a long time and lost their original nature. Finally, the coupling relationship between humans and land is complex. Although population is the main influencing factor for land-use changes, there are still more disturbance variables. The differential effects of special regional policies (such as urbanization in nearby areas, relocation of poor people from inhospitable areas, rural revitalization, development of plateau agriculture, ecological protection, etc.), technical means and micro-farmer behaviors on the coupled evolution of mountain–basin human–land systems should also be considered. Therefore, how to further build a multi-factor land-use dynamics model for specific regions and explore the coupled evolution mechanism of mountain–basin human–land systems should be the future direction of efforts.
