**1. Introduction**

Human–land relationship research is of great significance in geography, contributing to the duality of geography and the development of human geography [1]. For a long time, the study of "humans" and "land" has been carried out separately. However, as a series of environmental problems and food security problems brought about by human activities on the earth continue to affect the human system [2–4], the academic circle is paying more and more attention to the comprehensive research of "people" and "land" [5]. A variety of new comprehensive methods, including statistical methods, GIS and spatial

**Citation:** Wu, L.; Yang, Y.; Xie, B. Modeling Analysis on Coupling Mechanisms of Mountain–Basin Human–Land Systems: Take Yuxi City as an Example. *Land* **2022**, *11*, 1068. https://doi.org/10.3390/ land11071068

Academic Editor: Xiaoyong Bai

Received: 9 June 2022 Accepted: 11 July 2022 Published: 13 July 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

analysis methods, simulation methods and hybrid methods have been applied [6–9]. As the questions raised by researchers increasingly involved the intersection of human activities and the earth's environmental system [10,11], the academic circle further recognized that the modeling of feedback between humans and the natural environment has become an urgent requirement [12,13]. In the 1980s, the modeling concept of coupled natural systems and human socio-economic systems was proposed, and the two-way coupling of positive and negative feedback and the coupling with human activities in the earth system model became the research object of academic community [14]. Synthetic integrated models that carried out bidirectional coupling and exchanges of information in certain forms have increasingly become a research hotspot [15]. Since the 1990s, 11 different Integrated Assessment Models (IAMs) have been developed worldwide [16]. The Integrated Model to Assess the Global Environment Framework (IMAGE) model developed by the Netherlands Environmental Assessment Agency is one of the representative models of comprehensive integration, in which the impacts of agricultural land expansion and changes in land-use types on the environment were evaluated by considering population density, resources, topography, etc. [17]. In addition, there are some models based on multi-agents [18] that analyze and explain the complex human–land coupling relationship and its coupling degree. Meanwhile, with the continuous development of computer technology, multisource data-model fusion has made new progress, and the uncertainty of the human–land system coupling relationship has been further quantified [19,20].

With the deepening of studies on the human–land relationship, regional spatiality has attracted more and more attention [21,22], but most relevant studies on this complex issue focus on a single factor [23]. Mountainous areas and basin areas, as special geomorphic spaces in Yunnan Province, have not been strictly subdivided in existing studies, and the relationship between humans and land is rarely involved. The concept of "coupling" in geography originated from physics, which refers to the synergy of two or more systems through various interactions, or the dynamic relationship between the elements of the system [24]. Mountainous areas and basin areas mainly include flat land between mountains and surrounding mountains [25]. The two have a close genetic relationship in topography and geomorphology. Relying on their geographical proximity, they form a complex coupling system of mutual cooperation and constraints through continuous material circulation, energy flow and information transmission, including the two coupling relationships of near-range coupling and remote coupling [11,26,27]. In order to deeply reveal the interactions and feedback mechanisms between human activities and the natural environment in the mountain–basin human–land coupled system, it is necessary to conduct coupling simulations and predictions around the human–land system and build a comprehensive integrated human–land system dynamics model. By analyzing the interaction of element coupling and process coupling between two different geographic spaces, the complexity and dynamics of human–land systems coupling are revealed, and the mechanism and feedback paths of human activities such as social and economic development on land-use changes are explored. The human–land coupling system for mountain–basin has the nonlinear dynamics and chaotic characteristics of complex systems. To obtain a quantitative expression in the structure and function process, it is necessary to refer to a mature paradigm of the existing research and actively explore more integrated multivariate coupling models to dynamically resolve the interaction coupling relationship and dynamics mechanisms within the complex system and among subsystems based on an interdisciplinary perspective. An outstanding feature of human–land system dynamics models is that it can deal with nonlinear, complex, long-term and dynamic system coupling problems, and it is one of the main models to simulate human–land systems and other complex giant systems [28].

In 1997, Dobson published "Hopes for the Future: Restoration Ecology and Conservation Biology" in the journal Science [29], and proposed a dynamic land model to describe the transformation and restoration of natural habitats, which can explain the driving mechanism of increasing populations' agricultural demand on natural habitat transformation. However, when a mathematical model needs to be established to solve many specific problems in reality, the time delay cannot be ignored, and it is also one of the essential characteristics of the evolution and interaction results of the human–land systems' elements. From the point of the dynamic system, the existence of a time delay can induce the stability of the system to switch, resulting in complex dynamic behaviors such as periodic oscillation and chaos. Therefore, it is quite necessary to consider the dynamic properties of the land dynamics model with a time delay [30]. In addition, fractional order calculus is an arbitrary generalization of integer order calculus in order, and calculus is widely used in the study of complex dynamic systems, such as the regulation of various ecosystems [31,32], secure communication [33,34], system controls [35,36] and stability issues [37]. Compared with the classical integer order model, fractional order calculus is more suitable for describing systems or processes with memory and hereditary characteristics, and can more accurately describe the physical and ecological phenomena in nature [38,39], which has attracted great attention from scholars at home and abroad [40–43].

Based on this, according to the relatively closed mountain–basin human–land system in Yuxi City, this study took advantage of the limitations on population density and introduced an appropriate land-use conversion rate to focus on analyzing the differences in land-use conversion and population changes over time in two different geographical spaces. On this basis of the land dynamics model and fractional calculus theory constructed by Dobson, a fractional human–land coupling dynamics model with a time delay was established to analyze the evolution mechanism of regional land-use systems and other issues, which is helpful and has important theoretical significance and a practical application value for the in-depth interpretation of the land-use system change mechanism with population development. It also provides reference for the differential human–land countermeasures of mountainous areas and basin areas in different development stages.

#### **2. Materials and Methods**

#### *2.1. Study Area*

Yuxi City is located in the central part of Yunnan Province, on the Yunnan Plateau at low latitudes. It belongs to the subtropical plateau monsoon climate, ranging from 23◦19' to 24◦53' north latitude and 101◦16' to 103◦09' east longitude (Figure 1). Yuxi is located in the core position of Yunnan Province, connecting the east to the west and connecting the north to the south. It is adjacent to the provincial capital, Kunming, which is to the northeast; Chuxiong Autonomous Prefecture in the north; Pu'er city in the southwest and Honghe Autonomous Prefecture in the southeast. The city covers an area of 15,285 km2 and has jurisdiction over 75 townships (towns and streets) in 7 counties and 2 districts [44]. The terrain of Yuxi City is high in the northwest and low in the southeast. The western part is mainly deep-cut alpine and valley landforms, the central and eastern parts belong to the mountainous areas of central Yunnan and are dominated by mid-mountain landforms, and the eastern part is mainly plateau lake basin landforms. The Chengjiang, Jiangchuan and Tonghai lacustrine basins are formed around three plateau rifted lakes, the Fuxian Lake, Xingyun Lake and Qilu Lake, with flat and open terrain [45]. According to its special topography, combined with administrative regions, it can be divided into two types of geographical spaces: mountainous areas and basin areas [44]. Due to the complex terrain and large height difference, the mountainous area generally has more rainfall than the basin area. The cultivated land in the mountainous area is shallow and the soil fertility is low, but the basin area has fertile soil and more farmland with high and stable yields. From 1995 to 2018, the urban population growth and economic development in the basin area were significantly higher than those in the mountainous area, and the land-use change and social and economic development status differed significantly between the mountainous area and the basin area [27].

**Figure 1.** Location and elevation of the study area.

#### *2.2. Data Sources*

The land-use survey data in this study are mainly from the annual change survey data based on the second national land-use survey data (Table 1). The social and economic data involved are mainly from the statistical yearbook of Yunnan Province (1996–2019), the statistical yearbook of Yuxi City (1995–2018), the statistical yearbook of all counties and districts of Yuxi City and the statistical bulletin of national economic and social development from 1995–2018 (Table 1). The role of these data in the research is mainly to train and fit the parameters of human–land coupling dynamics models based on long time series data.

**Table 1.** Land-type area and population changes in mountainous and basin areas of Yuxi City from 1995 to 2018 (unit: hm2, person).



**Table 1.** *Cont.*

## *2.3. Human–Land Coupling Model Construction*

When discussing land-type transformation, Dobson only considered the direct transformation from the natural habitat to the agricultural land, but did not consider the direct transformation from the natural habitat to the construction land. According to the status of land-use changes in Yuxi City, this study has different definitions of land types based on the original model. Through the combination of land-use types in Yuxi City, it can be divided into the following three types: (1) Forest and grass land: they mainly represent the natural habitat and are set as the original state of land. The forest and grass land in this study are mainly the combination of forest and grass land. (2) Land for production and living: both the agricultural land and construction land transformed from the natural habitat under the current situation are taken into account. Therefore, the farmland, construction land and other necessary land for production and living are combined and collectively referred to as the production and living land. (3) Unused land: the land that cannot be used temporarily due to bad conditions, or after artificial reclamation or productive and living utilization, or long-term unmanaged and barren land. Let the area of forest and grass land in Yuxi be *F*, the production and living land be *R*, and the unused land be *U*, and *N* = *F* + *R* + *U* = 1. At the same time, the following assumptions are made:


**Figure 2.** Evolution between different land-use types.

Since the unit of land-use-type area is not consistent with that of the population, the data are transformed into dimensionless data after normalization in the process of model construction and analysis. Through analyzing the data in Yuxi City over the years, it can be seen that the transformation function of population affecting land use is a nonlinear function, and the following function can be obtained through fitting:

$$f(t, P, F) = \frac{dPF}{1 + P} \tag{1}$$

where *P* and *F* are the population density and forest and grass land area at time *t*, respectively, and *d* is the average reclamation or development capacity of the land. This function is generally interpreted as a Holling-II functional response function in mathematical definitions. Assuming that population growth conforms to the Logistic Retarded Growth model:

$$\frac{dP}{dt} = rP\left(1 + \frac{P(t)}{P\_{\text{max}}}\right) \tag{2}$$

where *r* is the inherent growth rate of the population, and *Pmax* (greater than 0) is the maximum population that the environment can carry.

Based on the transformation mechanism mentioned above, in the interval (*t*, *t* + Δ*t*), the forest and grass land, production and living land and unused land change with time *t*, and with the help of the population retardation growth model, the population also changes with time *t* under the constraints of the production and living land. Due to the need for both survival and population growth, more food is needed during population growth than during saturation, that is, the inherent growth rate of the population is a function of a time delay *t*-*τ*. Considering the coupling relationship between the population, forest and grass land, the production and living land, and unused land comprehensively, and only discussing the impact of population development on the time delay, the following fractional time delay human–land coupling dynamics model with a Holling-II functional response function can be obtained [30]:

$$\begin{cases} \begin{aligned} D^{\phi}F(t) &= \frac{-dF(t)P(t-\tau)}{1+P(t-\tau)} + sLI(t) \\ D^{\phi}R(t) &= \frac{dF(t)P(t-\tau)}{1+P(t-\tau)} - aR(t) + bLI(t) \\ D^{\phi}I(t) &= aR(t) - sLI(t) - bLI(t) \\ D^{\phi}P(t) &= rP(t) \left[1 - \frac{h}{R(t)}P(t-\tau)\right] \end{aligned} \end{cases} \tag{3}$$

where *φ*∈[0, 1] is the fractional order, and *r* is the inherent growth rate of the population.
