Exploring the Evolutionary Traits of Post-Disaster Reconstruction Under the Background of Sustainable Development

Abstract

Sustainable development serves as a key gripper and critical juncture for disaster-stricken regions to achieve the goal of “building back better”, and post-disaster reconstruction (PDR) involves complex system engineering. To explore the evolutionary traits of PDR under the background of sustainable development, this paper analyzes the internal subsystems and the corresponding state variables of PDR based on complex system thinking, and establishes a nonlinear dynamic model to describe how PDR evolves by using synergetics and the Logistic equation. Through dissecting the model, this paper determines all the evolutionary equilibrium points of PDR as well as the conditions allowing these points to stabilize, identifies the evolutionary pathways of PDR, and verifies the proposed model and the results deduced from the model by simulations. Results show that the ideal steady state of PDR needs to balance multi-dimensional sustainability. This study also finds that the evolutionary pathways of PDR, along with the corresponding stage-specific measures, are mainly determined by the initial conditions of PDR, the resource inputs of each subsystem, and the interactions between subsystems.

1. Introduction

Natural or human-induced disasters—such as earthquakes, landslides, fires, toxic spills, tornadoes, and terrorist attacks—can strike human society with little or no warning and cause considerable environmental damage, immeasurable social losses, and substantial economic shocks, which seriously hinder disaster-stricken areas from developing sustainably [1,2,3]. In recent years, disasters have occurred with increasing frequency and intensity, creating a continuously growing need associated with the sustainable development of post-disaster reconstruction (PDR) [4]. After disasters, disaster-stricken regions would carry out disaster response and PDR in sequence. However, public attention and investment in disaster-stricken regions gradually diminish from the disaster response to PDR, yielding the result that PDR is often treated as the least surveyed and understood phase of disaster management [5]. Historic disasters have repeatedly corroborated the importance of PDR; for example, five years after the 2010 Haiti earthquake, about 65000 disaster victims were still homeless [6]. PDR is a complex system of systems, and its internal multifaceted complexity tends to overshadow the dynamic evolutionary laws of PDR [7]. As a result, it is difficult for relevant stakeholders to effectively manage PDR [8], let alone achieve sustainable recovery and reconstruction. Therefore, it is worth further surveying the evolutionary traits of PDR, which constitutes an important precondition to promote PDR with high quality under the context of sustainable development.

The existing literature prefers to qualitatively explore how PDR develops, relying on its inclusions from a local perspective [9,10]. For example, Ngulube et al. [11] surveyed what factors influenced participatory post-disaster housing reconstruction processes using qualitative methods such as observations and focus groups. Few studies apply quantitative methods to globally investigate how PDR evolves by taking all of its subsystems into consideration, let alone directly at sustainable PDR. Driven by the actual need to manage PDR as comprehensively as possible and the above research gap, this paper tries to establish matched holistic models to discuss the evolution of PDR in a more precise way when sustainable development is desired. Conceptual models are frequently proposed to grasp PDR, but they are unable to delineate the complex composition and dynamic properties of PDR due to their reductionism and mechanisms [12,13]. Complex systems thinking, which offers extensive quantitative analysis ranging from local to global levels, has been demonstrated as useful for describing PDR [14,15]. The existing studies based on complex systems thinking quantitatively dissect the complexity of PDR through designing holistic models [16,17,18]; however, these studies cannot show how PDR leverages its subsystems and inter-subsystem interactions to evolve. Fortunately, synergetics is born to research how all the subsystems within a complex system interact to give rise to macroscopic ordered behaviors [19]. PDR mainly includes an environmental system, a social system, and an economic system, which is widely adopted [5,20,21]. Given the environmental–social–economic nature of PDR, this paper establishes nonlinear dynamic models using synergetics and the Logistic equation to quantitatively probe the evolutionary traits of PDR in a bottom-up way.

The contributions of this study are presented as follows: (1) This study devises a three-dimensional nonlinear dynamic model to analyze the evolutionary stability and evolutionary pathways of PDR when sustainable development is emphasized, which overcomes current research inadequacies represented by conceptual and non-synergistic models along with local and qualitative-leaning analysis, provides a novel research paradigm in the area of PDR as a complex environmental–social–economic system, and enriches the applications of synergetics. (2) Numerical simulations on how PDR stabilizes at each equilibrium point, and how PDR evolves to the ideal equilibrium point through stepwise and leapfrogging pathways, are conducted to verify the proposed model and its theoretical results, which can provide scientific guidance to promote sustainable PDR by stages. The proposed methodology, as well as the consequent results, incorporates prior research to present a theoretically sound framework for PDR to realize the goal of “building back better” in the context of sustainable development.

The remainder of this study is organized as follows. Section 2 presents a review of relevant studies; Section 3 proposes and analyzes the nonlinear evolutionary model for PDR under the background of sustainable development; Section 4 performs a series of numerical simulations; and Section 5 concludes this paper.

2. The Literature Review

2.1. PDR Models

PDR presents many complex system characteristics, including dynamics, nonlinearity, emergence, and irreversibility, just to name a few [4]. To better understand PDR as a whole, related studies prefer to design varied conceptual frameworks, such as a decision-making framework for prioritizing PDR projects [22], a multi-dimensional framework to evaluate disaster recovery pathways [23], a framework for assessing the indirect economic impacts of PDR [24], a probabilistic recovery framework for modeling electric power networks [6], and a digital twin-based post-disaster risk management framework [25]. Such conceptual frameworks can provide guidance for PDR; however, most of these frameworks are limited by their reductionism and mechanisms and find it difficult to explain the complex behaviors and traits of PDR [26]. Therefore, many studies turn to constructing models for PDR based on complex systems thinking and have yielded several foundational results, such as agent-based or complex-network-based models of post-disaster recovery [14,17,27], fractional-order models of post-earthquake recovery [16], and low-carbon reconstruction models based on dissipative structure theory [28].

As can be seen in the above research, complex system theories, methods, and tools are very suitable for modeling PDR; however, these studies do not reveal how PDR relies on the competition and cooperation among its subsystems to evolve in an orderly manner. Synergetics, a key theory investigating complex systems, is unique in terms of uncovering how global evolution emerges through local competition and collaboration [19]. Nonlinear dynamics provides a solid mathematical foundation to understand the ideas of synergetics [29,30]. For this reason, this paper attempts to establish a nonlinear dynamic model for PDR based on synergetics, which enables the exploration of the dynamic evolutionary laws of PDR.

2.2. PDR Composition

PDR is a multi-coupled field that pre-disaster legacy elements and post-disaster emerging elements cohere closely [31]. Affected by the overlapping effects between system elements, PDR in developing countries and regions often takes years or even decades to reshape stable states [17,24]. Therefore, what PDR includes and how PDR develops depending on its inclusions warrants thorough investigation and is explored by some researchers. As to what PDR includes, no consensus has been reached, but the prevailing view holds that PDR mainly involves three typical elements, namely a social system, an environmental system, and an economic system [5,20,21]. The non-mainstream divisions of PDR are diverse. For example, Xiao et al. [32] divided PDR into a socioeconomic development subsystem and an ecological environment subsystem and examined the coupling coordination degree between these two subsystems; Xue et al. [14] modeled post-disaster recovery as a multilayer network, which is composed of a human layer, a social infrastructure layer, and a physical infrastructure. No matter how PDR is divided, all of its subsystems contribute to its development. However, existing research focuses on the local reconstruction of disaster-stricken regions, such as the reconstruction of electric power networks [6], shelters [33], and infrastructures [3]. It is necessary to globally study PDR through taking all the subsystems into account. With respect to how PDR develops depending on its inclusions, existing studies tend to carry out investigations in a qualitative way. For example, Ngulube et al. [34] investigated citizen participation in post-earthquake recovery using qualitative data and found that interpersonal interactions had been a great challenge for citizen participation during the recovery process; Okunola [35] explored how community stakeholders linked by varied interactions shaped disaster recovery by qualitative methods such as semi-structured interviews. Although these qualitative explorations are meaningful, they offer extremely limited utility when more accurate and objective PDR is needed. PDR is a complex system with nonlinear interactions linking its subsystems; hence, more attention needs to be paid to discussing how PDR evolves with nonlinear interactions inside itself in a quantitative way.

In the context of sustainable development, the environmental system, social system, and economic system are still the leading subsystems of PDR, and they jointly determine the evolutionary processes and the final states of PDR with nonlinear interactions between them [36,37]. Based on the prevailing view of PDR composition, this paper attempts to quantitatively survey how PDR relies on its environmental system, social system, and economic system to evolve in a bottom-up way when nonlinear interactions exist between the three subsystems.

2.3. Conceptualizing Sustainable PDR

The concept of sustainable development, which is often treated as synonymous with the concept of sustainability in the academic field [38,39,40], can be dated back to the late 18th century, when Thomas Malthus published his essay to elucidate that the world would encounter unbalanced relations between the exponential growth of population and the arithmetic growth of available food [41]. In 1987, the World Commission on Environment and Development presented the first formal and most politically significant definition of sustainable development: “sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs” [42]. This statement is criticized as an ambiguous and inaccurate definition, and many subsequent studies attempt to deepen the understanding of sustainable development by proposing sustainability principles, frameworks, indicators, metrics, tools, methodologies, and so on [43]. For example, the World Conservation Union proposed a composite of 88 indicators for 180 countries to measure sustainable development [44], and Ben-Eli [45] put forward five core principles to comprehend sustainability. After more than two centuries of evolution, sustainable development has been assigned diverse definitions and understandings, with different emphases on what is to be sustained, what is to be developed, how to balance present and future, etc. [46,47].

Although there is no unified definition associated with sustainable development to date [40], it is widely accepted that the concept of sustainable development should be grasped in terms of the three pillars of environmental, social, and economic aspects [48,49], including in the field of PDR [1,4]. Based on this broadly recognized viewpoint, this paper tries to explore the sustainability of PDR in light of environmental sustainability, social sustainability, and economic sustainability, which are viewed herein as the state variables of the environmental system, the social system, and the economic system of PDR, respectively.

2.4. The Position of This Paper

This paper tries to investigate evolutionary traits of PDR under the background of sustainable development. This study differs from previous research in two ways. On one hand, existing studies prefer to characterize PDR by constructing conceptual models based on mechanisms and reductionism, and such models are of limited utility in accounting for the complex evolution of PDR. Moreover, existing research has also put forward some complex systems models of PDR, although these models cannot reveal how the subsystems of PDR interact. Therefore, this paper attempts to fill this research gap by using synergetics to establish a nonlinear dynamic model for PDR when sustainable development is emphasized. On the other hand, this paper explores PDR composition in a global and quantitative way, which is different from the literature marked by local or qualitative exploration of PDR composition. In conclusion, through surveying the evolutionary stability and pathways of PDR under the background of sustainable development, we can provide more guidance for disaster-stricken regions to realize the goal of “building back better”.

Table 1. Comparison between previous papers and this paper.

Papers	PDR Models			PDR Composition
	Conceptual Models	Complex Systems Models		Local or Qualitative Exploration	Global and Quantitative Exploration
		Non-Synergetics-Based	Synergetics-Based
[6,12,13,22,23,24,25]	✓
[14,16,17,18,27,28]		✓
[3,6,33,34,35]				✓
This paper			✓		✓

compares the PDR models and PDR composition used in previous studies and those employed in this research. In Table 1, “✓” denotes that the relevant papers carry out the corresponding research.

3. Evolutionary Model of PDR

3.1. Model Assumptions

PDR is a dynamic super system in nature. In line with previous studies, PDR is divided into an environmental system, a social system, and an economic system in this study. This research makes three assumptions to lay the foundation for the following model construction and model analysis:

Assumption 1. The environmental system, the social system, and the economic system of PDR develop as a function of time marked as t. Under the influences of all the factors, these three subsystems change continuously and together depict the evolutionary processes of PDR in the context of sustainable development.

Assumption 2. If the environmental system develops solely, its evolution follows the growth law represented by the Logistic equation. The same is true for the social system and the economic system.

Assumption 3. During the PDR process, there exist complex nonlinear interactions between the environmental system, the social system, and the economic system. Such interactions can pose positive or negative impacts on the corresponding systems.

PDR is a time-varying complex system. From a complex systems perspective, how a complex system develops is determined by its subsystems and the nonlinear interactions between them when external conditions remain constant [50,51]. Such interactions may be beneficial or harmful to system growth. For example, competition and mutualism are two kinds of such nonlinear interactions; competition can inhibit the growth of interacting systems, while mutualism benefits the growth of interacting systems [52]. As a result, assumption 1 and assumption 3 are reasonable. Based on the biological principles of limited resources and intraspecific competition, a system cannot realize unrestrained growth and could experience S-shaped growth, which can be modeled by the Logistic equation [53]. Therefore, assumption 2 is also rational.

3.2. Model Construction

As mentioned above, PDR is an environmental–social–economic system. In accordance with nonlinear dynamics, the evolutionary laws of a system can be dynamically described by system features or attributes, which are defined as state variables [54]. In the context of sustainable development, the state variables of environmental systems, social systems, and economic systems are refined as environmental sustainability, social sustainability, and economic sustainability, respectively [2]. Environmental sustainability in PDR underscores minimizing the ecological footprint during recovery processes and enhancing the resilience of reconstructed areas against future environmental hazards [4]. Social sustainability in PDR centers on whether reconstruction efforts equitably consider the different needs of all stakeholders regarding social inclusion and enhance knowledge on disaster-risk reduction [55]. Economic sustainability in PDR emphasizes that reconstruction efforts should promote local economies, support local businesses, and create new employment opportunities in a cost-effective way [56].

An S-shaped function can be applied to depict how sustainability develops [1], and it is quantified as the Logistic equation in this paper. If all the subsystems of PDR develop in isolation, how PDR changes over time is denoted as follows:

$$\begin{array}{l}{\displaystyle \frac{\mathrm{d}{x}_1}{\mathrm{d}t}}={\alpha }_1{x}_1-{\beta }_{11}{x}_1^2\\ {\displaystyle \frac{\mathrm{d}{x}_2}{\mathrm{d}t}}={\alpha }_2{x}_2-{\beta }_{22}{x}_2^2\\ {\displaystyle \frac{\mathrm{d}{x}_3}{\mathrm{d}t}}={\alpha }_3{x}_3-{\beta }_{33}{x}_3^2\end{array}$$

(1)

where x₁, x₂, and x₃ are state variables that stand for environmental sustainability, social sustainability, and economic sustainability, respectively; α₁ represents the changing rate of environmental sustainability when the environmental system with constant inputs develops independently in PDR—the greater the value of α₁ is, the more inputs the environmental system has, and the same is true for α₂ and α₃. Let β_ij (i = j = 1, 2, 3) indicate the auto-inhibited effects of x_i. β_ij (i = j) generally satisfies β_ij > 0 because limited resources can be gradually consumed during PDR processes, and auto-inhibited effects emerge and hinder x_i from growing when resource consumption increases to a certain degree. In fact, there exist nonlinear relations between the environmental system, the social system, and the economic system of PDR, so that Equation (1) cannot represent the actual evolution of PDR. As a result, the evolutionary model of PDR is adjusted as follows:

$$\begin{array}{l}{\displaystyle \frac{\mathrm{d}{x}_1}{\mathrm{d}t}}={\alpha }_1{x}_1-{\beta }_{11}{x}_1^2-{\beta }_{12}{x}_1{x}_2-{\beta }_{13}{x}_1{x}_3\\ {\displaystyle \frac{\mathrm{d}{x}_2}{\mathrm{d}t}}={\alpha }_2{x}_2-{\beta }_{21}{x}_2{x}_1-{\beta }_{22}{x}_2^2-{\beta }_{23}{x}_2{x}_3\\ {\displaystyle \frac{\mathrm{d}{x}_3}{\mathrm{d}t}}={\alpha }_3{x}_3-{\beta }_{31}{x}_3{x}_1-{\beta }_{32}{x}_3{x}_2-{\beta }_{33}{x}_3^2\end{array}$$

(2)

where product x_ix_j (i ≠ j; i = 1, 2, 3; j = 1, 2, 3) represents the nonlinear relationships between x_i and x_j; β_ij (i ≠ j) shows the influences that the nonlinear relationships between x_i and x_j exert on the growth of x_i. If β_ij > 0, the nonlinear relationships between x_i and x_j restrain the growth of x_i; if β_ij < 0, the nonlinear relationships between x_i and x_j promote the growth of x_i; if β_ij = 0, the nonlinear relationships between x_i and x_j do not affect the growth of x_i.

3.3. Model Analysis

3.3.1. Evolutionary Stability Analysis

To survey the dynamics of PDR and to ensure that PDR can evolve to expected states, analyzing the evolutionary stability of PDR is required. Let ${\displaystyle \frac{\mathrm{d}{x}_i}{\mathrm{d}t}}=0\left(i=1,2,3\right)$ in Equation (2), then eight equilibrium points can be obtained. These equilibrium points are denoted as E₁(0, 0, 0), E₂$\left({\displaystyle \frac{{\alpha }_1}{{\beta }_{11}}},0,0\right)$, E₃$\left(0,{\displaystyle \frac{{\alpha }_2}{{\beta }_{22}}},0\right)$, E₄$\left(0,0,{\displaystyle \frac{{\alpha }_3}{{\beta }_{33}}}\right)$, E₅$\left({\displaystyle \frac{{\alpha }_1{\beta }_{22}-{\alpha }_2{\beta }_{12}}{{\beta }_{11}{\beta }_{22}-{\beta }_{12}{\beta }_{21}}},{\displaystyle \frac{{\alpha }_2{\beta }_{11}-{\alpha }_1{\beta }_{21}}{{\beta }_{11}{\beta }_{22}-{\beta }_{12}{\beta }_{21}}},0\right)$, E₆$\left({\displaystyle \frac{{\alpha }_1{\beta }_{33}-{\alpha }_3{\beta }_{13}}{{\beta }_{11}{\beta }_{33}-{\beta }_{13}{\beta }_{31}}},0,{\displaystyle \frac{{\alpha }_3{\beta }_{11}-{\alpha }_1{\beta }_{31}}{{\beta }_{11}{\beta }_{33}-{\beta }_{13}{\beta }_{31}}}\right)$, E₇$\left(0,{\displaystyle \frac{{\alpha }_2{\beta }_{33}-{\alpha }_3{\beta }_{23}}{{\beta }_{22}{\beta }_{33}-{\beta }_{23}{\beta }_{32}}},{\displaystyle \frac{{\alpha }_3{\beta }_{22}-{\alpha }_2{\beta }_{32}}{{\beta }_{22}{\beta }_{33}-{\beta }_{23}{\beta }_{32}}}\right)$, and E₈$\left({\displaystyle \frac{P_1}{P}},{\displaystyle \frac{P_2}{P}},{\displaystyle \frac{P_3}{P}}\right)$, where $P=\left|\begin{array}{ccc}{\beta }_{11}& {\beta }_{12}& {\beta }_{13}\\ {\beta }_{21}& {\beta }_{22}& {\beta }_{23}\\ {\beta }_{31}& {\beta }_{32}& {\beta }_{33}\end{array}\right|$, $P_1=\left|\begin{array}{ccc}{\alpha }_1& {\beta }_{12}& {\beta }_{13}\\ {\alpha }_2& {\beta }_{22}& {\beta }_{23}\\ {\alpha }_3& {\beta }_{32}& {\beta }_{33}\end{array}\right|$, $P_2=\left|\begin{array}{ccc}{\beta }_{11}& {\alpha }_1& {\beta }_{13}\\ {\beta }_{21}& {\alpha }_2& {\beta }_{23}\\ {\beta }_{31}& {\alpha }_3& {\beta }_{33}\end{array}\right|$, and $P_3=\left|\begin{array}{ccc}{\beta }_{11}& {\beta }_{12}& {\alpha }_1\\ {\beta }_{21}& {\beta }_{22}& {\alpha }_2\\ {\beta }_{31}& {\beta }_{32}& {\alpha }_3\end{array}\right|$.

PDR could stabilize at the above equilibrium points. The equilibrium point E₁ denotes that environmental sustainability, social sustainability, and economic sustainability all equal zero when the environmental system, social system, and economic system develop in unison, meaning that PDR fails in terms of sustainable development. E₂ represents that environmental sustainability, social sustainability, and economic sustainability, respectively, become ${\displaystyle \frac{{\alpha }_1}{{\beta }_{11}}}$, 0, and 0 in the end, indicating that PDR merely contains its environmental system; the same is true for E₃ and E₄. E₅ shows that environmental sustainability, social sustainability, and economic sustainability ultimately become ${\displaystyle \frac{{\alpha }_1{\beta }_{22}-{\alpha }_2{\beta }_{12}}{{\beta }_{11}{\beta }_{22}-{\beta }_{12}{\beta }_{21}}}$, ${\displaystyle \frac{{\alpha }_2{\beta }_{11}-{\alpha }_1{\beta }_{21}}{{\beta }_{11}{\beta }_{22}-{\beta }_{12}{\beta }_{21}}}$, and 0, respectively, signifying that PDR restores its environmental system and social system; the same is true for E₆ and E₇. E₈ suggests that environmental sustainability, social sustainability, and economic sustainability separately reach non-zero values, implying that PDR restores all the subsystems at the same time. It should be pointed out that this study only considers the case that equilibrium points possess non-negative values. Generally, PDR represented by equilibrium points E₁, E₂/E₃/E₄, E₅/E₆/E₇, and E₈ gradually improves.

To perform the local stability of PDR, it is necessary to further analyze the stability of Equation (2) at each equilibrium point by calculating the corresponding Jacobian matrix. The Jacobian matrix of Equation (2) at E_i (i = 1, 2, …, 8) is outlined below:

$$J\left(E_i\right)=\left|\begin{array}{ccc}{V}_1& V_2& V_3\\ V_4& V_5& V_6\\ V_7& V_8& V_9\end{array}\right|$$

(3)

where $V_1={\alpha }_1-2{\beta }_{11}{x}_1-{\beta }_{12}{x}_2-{\beta }_{13}{x}_3$, $V_2=-{\beta }_{12}{x}_1$, $V_3=-{\beta }_{13}{x}_1$, $V_4=-{\beta }_{21}{x}_2$, $V_5={\alpha }_2-{\beta }_{21}{x}_1-2{\beta }_{22}{x}_2-{\beta }_{23}{x}_3$, $V_6=-{\beta }_{23}{x}_2$, $V_7=-{\beta }_{31}{x}_3$, $V_8=-{\beta }_{32}{x}_3$, and $V_9={\alpha }_3-{\beta }_{31}{x}_1-{\beta }_{32}{x}_2-2{\beta }_{33}{x}_3$. The characteristic equation of the above Jacobian matrix J(E_i) is as follows:

$${\lambda }^3+{\delta }_1{\lambda }^2+{\delta }_2\lambda +{\delta }_3=0$$

(4)

where ${\delta }_1=-\left(V_1+V_5+V_9\right)$, ${\delta }_2=V_1{V}_5+V_1{V}_9+V_5{V}_9-V_2{V}_4-V_3{V}_7-V_6{V}_8$, and ${\delta }_3=V_1{V}_6{V}_8+V_2{V}_4{V}_9+V_3{V}_5{V}_7-V_1{V}_5{V}_9-V_2{V}_6{V}_7-V_3{V}_4{V}_8$.

If all the eigenvalues of the characteristic equation at an equilibrium point are negative or have negative real parts, then the equilibrium point is stable. According to the Routh–Hurwitz criterion [57], the necessary and sufficient conditions that all the eigenvalues of a characteristic equation are negative or have negative real parts are that all the Routh–Hurwitz determinants (RH_i, i = 1, 2, 3) are positive, where $R{H}_1=\left|{\delta }_1\right|$, $R{H}_2=\left|\begin{array}{cc}{\delta }_1& 1\\ {\delta }_3& {\delta }_2\end{array}\right|$, and $R{H}_3=\left|\begin{array}{ccc}{\delta }_1& 1& 0\\ {\delta }_3& {\delta }_2& {\delta }_1\\ 0& 0& {\delta }_3\end{array}\right|$. This paper discusses the stability of E_i (i = 1, 2, …, 8) based on the Routh–Hurwitz criterion.

3.3.2. Evolutionary Pathway Analysis

PDR can evolve to high-level states in both stepwise and leapfrogging ways, which means that PDR can develop through stepwise pathways and leapfrogging pathways. PDR needs to balance its three subsystems and maintain the corresponding sustainability at non-zero levels in order that it can realize its ideal stable states. Therefore, the equilibrium point labeled as E₈ is the only ideal state that PDR seeks to stabilize, and E_i (i = 1, 2, …, 7) can be the initial stable equilibrium points or intermediate stable equilibrium points before PDR reaches E₈.

The stepwise evolutionary pathways of PDR mean the pathways that PDR restores its environmental system, social system, and economic system to the ideal state, step by step. If PDR without intervention would stabilize at E₁, it can recover to the state represented by E₈ via six sequential evolutionary pathways, namely E₁→E₂→E₅→E₈, E₁→E₂→E₆→E₈, E₁→E₃→E₅→E₈, E₁→E₃→E₇→E₈, E₁→E₄→E₇→E₈, and E₁→E₄→E₆→E₈. The realization of these six stepwise pathways requires taking a minimum of three phase-specific measures at the right time. To be specific, the first-phase interventions must be implemented before PDR plateaus at E₁, otherwise PDR would maintain stability at E₁ forever; the second-phase measures can be deployed at an arbitrary time before, during, or after PDR stabilizes at E₂/E₃/E₄; and the third-phase actions can be taken at any point before, during, or after stabilization at E₅/E₆/E₇. The measures at all stages mainly concern the resource inputs of the environmental system, social system, and economic system, and the interactions between these subsystems, which are known as parameters α_i and β_ij (i = 1, 2, 3; j = 1, 2, 3; i ≠ j). Through depending on the above six stepwise pathways, disaster-stricken areas are able to identify and overcome the barriers to sustainable development in each subsystem one by one. For example, the evolutionary pathway E₁→E₂→E₅→E₈ denotes that PDR first reshapes its environmental system by varied measures such as ecological remediation and infrastructure construction, then recovers its social system by housing improvements, resource allocation optimization, etc., and finally adopts diverse economic approaches to reconstruct its economic system. Following the pathway, PDR can realize sustainable development.

If PDR without interference stabilizes at E₂, E₂→E₆→E₈ and E₂→E₅→E₈ are two available evolutionary pathways for PDR to reach E₈ step by step; if E₃ is the initial stable equilibrium point of PDR, E₃→E₅→E₈ and E₃→E₇→E₈ are able to gradually guide PDR to evolve towards E₈; if PDR remains steady at E₄ with its initial conditions, it can recover to E₈ by two stepwise evolutionary pathways, which are, respectively, denoted as E₄→E₇→E₈ and E₄→E₆→E₈; the achievement of the above six pathways needs to take at least two phased actions, and these two staged actions can be implemented at any time before, during, or after PDR stabilizes at the corresponding equilibrium point. If disaster- stricken areas are initially equipped with available resources to stabilize at E₅, E₆, or E₇, its reconstruction can directly evolve towards E₈ by E₅→E₈, E₆→E₈, or E₇→E₈, respectively; the successful fulfillment of these three evolutionary pathways needs to take at least one staged measure at any time before, during, or after PDR stabilizes at E₅/E₆/E₇. The stepwise evolutionary pathways of PDR are summarized in

Table 2. The evolutionary pathways of PDR.

Evolutionary Pathway Types	Initial Stable Point	Intermediate Stable Point	Ideal Stable Point	Pathways	Intervention Stages Required
Stepwise evolutionary pathways	E1	E2, E5/E6; E3, E5/E7; E4, E6/E7;	E8	E1→E2→E5→E8, E1→E2→E6→E8, E1→E3→E5→E8, E1→E3→E7→E8, E1→E4→E6→E8 and E1→E4→E7→E8	At least three
	E2	E5/E6	E8	E2→E5→E8 and E2→E6→E8	At least two
	E3	E5/E7	E8	E3→E5→E8 and E3→E7→E8
	E4	E6/E7	E8	E4→E7→E8 and E4→E6→E8
	E5/E6/E7	None	E8	E5→E8, E6→E8 and E7→E8	At least one
Leapfrogging evolutionary pathways	E1	E2/E3/E4/E5/E6/E7	E8	E1→E2→E8, E1→E3→E8, E1→E4→E8, E1→E5→E8, E1→E6→E8 and E1→E7→E8	At least two
	E1	None	E8	E1→E8	At least one
	E2/E3/E4	None	E8	E2→E8, E3→E8 and E4→E8

.

The leapfrogging evolutionary pathways signify the pathways by which PDR skips intermediate states and reaches its expected states in a non-consecutive manner, and they emphasize that PDR restores two or more subsystems simultaneously. If PDR stabilizes at E₁ with its initial conditions, it can evolve towards E₈ by seven leapfrogging pathways, namely E₁→E₂→E₈, E₁→E₃→E₈, E₁→E₄→E₈, E₁→E₈, E₁→E₅→E₈, E₁→E₆→E₈, and E₁→E₇→E₈. The former three pathways restore one subsystem of PDR with priority, and then recover the remaining two subsystems; conversely, the latter three pathways give priority to developing two subsystems of PDR, with the recovery of the remaining one subsystem to follow; realizing these six pathways needs to take at least two phased measures, of which the first-phase measures must be carried out before the stabilization at E₁, and the second-phase measures can be implemented at any time before, during, or after the stabilization at E_i (i = 2, 3, …, 7). For example, the evolutionary pathway E₁→E₅→E₈ represents a pathway in which PDR first recovers its environmental system and social system concurrently via countermeasures of environmental protection, societal governance, and so on, followed by the development of the economic system using effective economic measures. The pathway E₁→E₈ concurrently develops all the subsystems of PDR by taking one necessary action before PDR stabilizes at E₁. If PDR is qualified with proper initial conditions to stabilize at E₂, E₃, or E₄, its direct evolution to E₈ just needs one stage-skipping development, which is supported by at least one staged measure executed without time constraints. For instance, disaster-stricken regions with adequate environmental substrates can directly transit to E₈ with powerful economic and social backings. The leapfrogging evolutionary pathways of PDR are summarized in Table 2.

4. Numerical Simulations

This section conducts a series of numerical simulations to verify the above theoretical results and to visually show the evolutionary traits of PDR under the background of sustainable development.

4.1. Simulations on Evolutionary Stability

To validate the evolutionary stability of PDR represented by Equation (2), this section first sets values for parameters α_i and β_ij (i = 1, 2, 3; j = 1, 2, 3) in Equation (2), and the specific parameter settings are shown in

Table 3. Parameter settings.

Parameters (10−1)												Effective Equilibrium Points
α1	α2	α3	β11	β12	β13	β21	β22	β23	β31	β32	β33	Stable	Unstable
−1	−3	−2	10	−0.3	−0.4	4	10	−0.2	3	0.2	10	E1	None
5	1	1	10	−0.3	−0.4	4	10	0.2	4	0.2	10	E2	E1, E3, E4, E7
2	5	1	10	4.5	0.3	−0.3	10	−0.2	0.3	4	10	E3	E1, E2, E4, E6
1	1	5	10	0.3	4	4	10	0.4	−0.4	−0.2	10	E4	E1, E2, E3, E5
10	6	2	10	−0.3	−0.4	−0.3	10	−0.2	3	0.2	10	E5	E1, E2, E3, E4, E7
10	2	6	10	−0.3	−0.4	4	10	0.2	−0.2	−0.3	10	E6	E1, E2, E3, E4, E7
2	10	6	10	4.5	−0.2	−0.3	10	−0.2	3	0.2	10	E7	E1, E2, E3, E4, E6
10	10	10	10	−0.4	−0.3	−0.4	10	−0.2	−0.3	−0.2	10	E8	Ei(i = 1, 2, …, 7)

. According to Table 3, when α₁ = α₂ = α₃ = 1, β₁₁ = β₂₂ = β₃₃ =1, β₁₂ = β₂₁ = −0.04, β₁₃ = β₃₁ = −0.03, and β₂₃ = β₃₂ = −0.02, the equilibrium points of Equation (2) and the stabilizing conditions of these equilibrium points are summarized as follows: (1) The effective equilibrium points of Equation (2) are E_i (i = 1, 2, …, 8). (2) RH_i > 0 (i = 1, 2, 3) does not always hold for E_i (i = 1, 2, …, 7); therefore, these equilibrium points are unstable, and PDR cannot stabilize at these equilibrium points. (3) RH_i > 0 (i = 1, 2, 3) always holds for E₈ whatever the initial values of x_i (i = 1, 2, 3); thus, E₈ is stable, which means that PDR finally stabilizes at E₈ with environmental sustainability, social sustainability, and economic sustainability equaling 1.074, 1.064, and 1.054, respectively, as shown in Figure 1.

When α₁ = −0.1, α₂ = −0.3, α₃ = −0.2, β₁₁ = β₂₂ = β₃₃ =1, β₁₂ = −0.03, β₁₃ = −0.04, β₂₁ = 0.4, β₂₃ = −0.02, β₃₁= 0.3, and β₃₂ = 0.02, E₁ is the unique effective equilibrium point of Equation (2); moreover, PDR stabilizes at E₁ because RH_i > 0 (i = 1, 2, 3) always holds for E₁. When α₁ = 0.5, α₂ = 0.1, α₃ = 0.1, β₁₁ = β₂₂ = β₃₃ =1, β₁₂ = −0.03, β₁₃ = −0.04, β₂₁ = 0.4, β₂₃ = 0.02, β₃₁= 0.4, and β₃₂ = 0.02, E_i (i = 1, 2, 3, 4, 7) are the effective equilibrium points of Equation (2), RH_i > 0 (i = 1, 2, 3) holds exclusively for E₂, and PDR eventually converges to E₂ (0.5, 0, 0). Similarly, if parameter settings make an equilibrium point effective and enable all the Routh–Hurwitz determinants at this point to be positive, PDR would ultimately become stable at the equilibrium point, as shown in Table 3. For simplicity, this section omits simulating the evolutionary stability of PDR at E_i (i = 1, 2, …, 7).

4.2. Simulations on Evolutionary Pathways

This section conducts simulations to verify the evolutionary pathways proposed in Section 3.3.2 and to vividly present how PDR evolves from non-ideal states to ideal stable states through the proposed pathways. The iteration duration t is set as 50 or 100, depending on specific evolutionary pathways.

4.2.1. Simulations on Stepwise Evolutionary Pathways

This section focuses on simulating the stepwise evolutionary pathways of PDR. Since the stepwise pathways from E_i (i = 2, …, 7) to E₈ are covered in the stepwise pathways from E₁ to E₈, this section just simulates the evolutionary pathways starting from E₁ and omits the simulations on the stepwise pathways from E_i (i = 2, …, 7) to E₈. The detailed simulation results are displayed in the following figures.

If the initial parameter values of PDR precisely match those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.7, 0.4, 0.2), PDR along the stepwise evolutionary pathway E₁→E₂→E₅→E₈ could adhere to the procedures below: (1) When t = 3, PDR needs to increase the inputs of each subsystem, reinforce the inhibitory effects of social–economic interactions on the growth of social sustainability and the inhibitory effects of the environmental–economic couplings on the growth of economic sustainability; more precisely, parameters α_i (i = 1, 2, 3), β₂₃, and β₃₁ need to be increased to α₁ = 0.5, α₂ = 0.1, α₃ = 0.1, β₂₃ = 0.02, and β₃₁ = 0.4, respectively. By implementing these measures, PDR can gradually deviate from the original stable equilibrium point of phase 1 (i.e., E₁) and evolve towards E₂, as shown in Figure 2. (2) When t = 8, it is necessary to continuously increase the inputs of each subsystem, enhance the promoting effects of social–environmental and social–economic interactions on the growth of social sustainability, and weaken the inhibitory effects of environmental–economic interactions on the growth of economic sustainability; to be more exact, parameters α_i (i = 1, 2, 3) should be separately increased to α₁ = 1, α₂ = 0.6, and α₃ = 0.2, and parameters β₂₁, β₂₃, and β₃₁ should be decreased to β₂₁ = −0.03, β₂₃ = −0.02, and β₃₁ = 0.3, respectively. By these measures, PDR can progressively stray from the original stable equilibrium point of phase 2 (i.e., E₂) and evolve towards E₅, as shown in Figure 2. (3) When t = 13, PDR could keep adding the inputs of the social system and the economic system, adjust the promoting effects of environmental–social and environmental–economic interactions on the growth of environmental sustainability, strengthen the promoting effects of social–environmental interactions on the growth of social sustainability and the promoting effects of economic–environmental and economic–social interactions on the sustainable growth of the economic system; specifically, parameters α_i (i = 2, 3) should be individually increased to α₂ = 1 and α₃ = 1, parameters β₁₂ and β₁₃ should be separately adjusted to β₁₂ = −0.04 and β₁₃ = −0.03, and parameters β₂₁, β₃₁, and β₃₂ should be decreased to β₂₁ = −0.04, β₃₁ = −0.03, and β₃₂ = −0.02, respectively. After taking these steps, PDR can slowly move away from the original stable equilibrium point of phase 3 (i.e., E₅) and converge steadily to the stable equilibrium point of phase 4 (i.e., E₈), as shown in Figure 2.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.9, 0.5, 0.1), directing PDR to develop along the pathway E₁→E₂→E₆→E₈ can follow the procedures below: (1) Applying the measures proposed in the first procedure of the evolutionary pathway E₁→E₂→E₅→E₈ at t = 4 enables PDR to evolve from E₁ to E₂. (2) When t = 8, PDR can continuously increase the inputs of each subsystem, and reinforce the promoting effects of economic–environmental and economic–social interactions on the growth of economic sustainability; to be specific, α_i (i = 1, 2, 3) should be separately increased to α₁ = 1, α₂ = 0.2, and α₃ = 0.6, while β₃₁ and β₃₂ should be decreased to β₃₁ = −0.02 and β₃₂ = −0.03, respectively; by taking these actions, PDR can gradually move away from the original stable equilibrium point of phase 2 (i.e., E₂) and evolve towards E₆. (3) When t = 13, PDR can increase the inputs of the social system and the economic system, and adjust the promoting effects of inter-subsystem interactions on the sustainable growth of each subsystem; in other words, α_i (i = 2, 3) should be increased to α₂ = 1 and α₃ = 1, respectively, and β_ij (i ≠ j; i = 1, 2, 3; j = 1, 2, 3) should be individually adjusted to β₁₂ = −0.04, β₁₃ = −0.03, β₂₁ = −0.04, β₂₃ = −0.02, β₃₁= −0.03, and β₃₂ = −0.02. After executing the measures, PDR can slowly drift from the original stable equilibrium point of phase 3 (i.e., E₆) and converge to the stable equilibrium point of phase 4 (i.e., E₈). How PDR develops along the pathway E₁→E₂→E₆→E₈ is shown in Figure 3.

If the initial parameter values of PDR precisely match those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.15, 0.1, 0.2), guiding PDR to evolve along E₁→E₃→E₅→E₈ can follow the procedures below: (1) When t = 3, PDR could increase the inputs of each subsystem, strengthen the inhibitory effects of environmental–social and environmental–economic interactions on the sustainable growth of the environmental system, enhance the promoting effects of social–environmental interactions on the sustainable growth of the social system and the inhibitory effects of economic–social interactions on the sustainable growth of the economic system, and attenuate the inhibitory effects of economic–environmental interactions on the sustainable growth of the economic system; to be more exact, α_i (i = 1, 2, 3), β₁₂, β₁₃, and β₃₂ should be separately increased to α₁ = 0.2, α₂ = 0.5, α₃ = 0.1, β₁₂ = 0.45, β₁₃ = 0.03, and β₃₂ = 0.4, and β₂₁ and β₃₁ should be decreased to β₂₁ = −0.03 and β₃₁ = 0.03, respectively. By taking these measures, PDR can gradually stray from the original stable equilibrium point of phase 1 (i.e., E₁) and evolve towards E₃, as shown in Figure 4. (2) When t = 10, PDR could continuously increase the inputs of each subsystem, reinforce the promoting effects of environmental–social and environmental–economic interactions on the growth of environmental sustainability, increase the inhibitory effects of economic–environmental interactions on the sustainable growth of the economic system, and weaken the inhibitory effects of economic–social interactions on the sustainable growth of the economic system; in other words, α_i (i = 1, 2, 3) and β₃₁ should be individually increased to α₁ = 1, α₂ = 0.6, α₃ = 0.2, and β₃₁ = 0.3, and β₁₂, β₁₃, and β₃₂ should be decreased to β₁₂ = −0.03, β₁₃ = −0.04, and β₃₂ = 0.02, respectively. After applying these measures, PDR can slowly move away from the original stable equilibrium point of phase 2 (i.e., E₃) and evolve towards E₅, as shown in Figure 4. (3) Implementing the interventions applied in the last procedure of the evolutionary pathway E₁→E₂→E₅→E₈ at t = 14 enables PDR to gradually switch from E₅ to E₈, as shown in Figure 4.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.6, 0.8, 0.3), how PDR evolves along E₁→E₃→E₇→E₈ can be tackled by the procedures below: (1) When t = 3, adopting the measures executed in the first procedure of the evolutionary pathway E₁→E₃→E₅→E₈ enables PDR to evolve from E₁ to E₃, as shown in Figure 5. (2) When t = 8, it is necessary to enrich the inputs of the social system and the economic system and weaken the inhibitory effects of the economic–social interactions on the sustainable growth of the economic system. In addition, enhancing the promoting effects of environmental–economic interactions on the sustainable growth of the environmental system and reinforcing the inhibitory effects of the same interactions on the sustainable growth of the economic system are also essential. In other words, α_i (i = 2, 3) and β₃₁ should be separately increased to α₂ = 1, α₃ = 0.6, and β₃₁ = 0.3, while β₁₃ and β₃₂ should be decreased to β₁₃ = −0.02 and β₃₂ = 0.02, respectively. By taking these measures, PDR can gradually diverge from the original stable equilibrium point of phase 2 (i.e., E₃) and evolve towards E₇, as shown in Figure 5. (3) When t = 13, PDR can increase the inputs of the environmental system and the economic system, enhance the promoting effects of environmental–economic interactions on the growth of environmental sustainability and economic sustainability, reinforce the promoting effects of environmental–social interactions on the sustainable growth of the corresponding subsystems, and strengthen the promoting effects of economic–social couplings on the sustainable growth of the economic system; specifically, α_i (i = 1, 3) should be individually increased to α₁ = 1 and α₃ = 1, while β₁₂, β₁₃, β₂₁, β₃₁, and β₃₂ should be decreased to β₁₂ = −0.04, β₁₃ = −0.03, β₂₁ = −0.04, β₃₁ = −0.03, and β₃₂ = −0.02, respectively. Through the above interventions, PDR can gradually stray from the original stable equilibrium point of phase 3 (i.e., E₇) and evolve to the stable equilibrium point of phase 4 (i.e., E₈), as shown in Figure 5.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.8, 0.2, 0.3), PDR following E₁→E₄→E₆→E₈ can be divided into the procedures below: (1) When t = 3, PDR can increase the inputs of each subsystem, reinforce the inhibitory effects of environmental–social and environmental–economic interactions on the sustainable growth of the environmental system, enhance the inhibitory effects of social–economic couplings on the growth of social sustainability, and strengthen the promoting effects of economic–environmental and economic–social interactions on the sustainable growth of the economic system, which can be quantified as increasing α_i (i = 1, 2, 3), β₁₂, β₁₃, and β₂₃ to the respective values of α₁ = 0.1, α₂ = 0.1, α₃ = 0.5, β₁₂ = 0.03, β₁₃ = 0.4, and β₂₃ = 0.04 and decreasing β₃₁ and β₃₂ to β₃₁ = −0.04 and β₃₂ = −0.02, respectively. Relying on the above actions, PDR can slowly move away from the original stable equilibrium point of phase 1 (i.e., E₁) and evolve towards E₄, as shown in Figure 6. (2) When t = 8, increasing the inputs of each subsystem, enhancing the promoting effects of environmental–social and environmental–economic interactions on the sustainable growth of the environmental system, impairing the inhibitory effects of social–economic interactions on the sustainable growth of the social system, and adjusting the promoting effects of economic–environmental and economic–social couplings on the growth of economic sustainability are available measures; these can be precisely reflected as increasing α_i (i = 1, 2, 3) and β₃₁ to the respective values of α₁ = 1, α₂ = 0.2, α₃ = 0.6, and β₃₁ = −0.02 and decreasing β₁₂, β_13, β₂₃, and β₃₂ to β₁₂ = −0.03, β₁₃ = −0.04, β₂₃ = 0.02, and β₃₂ = −0.03, respectively. By implementing these measures, PDR can gradually deviate from the original stable equilibrium point of phase 2 (i.e., E₄) and evolve towards E₆, as shown in Figure 6. (3) In order to shift from E₆ to E₈, PDR can take the measures occurring in the last procedure of the evolutionary pathway E₁→E₂→E₆→E₈ at t = 12, as shown in Figure 6.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.2, 0.7, 0.6), steering PDR to develop along E₁→E₄→E₇→E₈ can adhere to the procedures below: (1) PDR can take the measures implemented in the first procedure of the evolutionary pathway E₁→E₄→E₆→E₈ at t = 4 to realize the evolution from E₁ to E₄, as shown in Figure 7. (2) When t = 9, increasing the inputs of each subsystem and adjusting the effects of inter-subsystem interactions on the sustainable growth of each subsystem are feasible measures which can be elaborated as increasing α_i (i = 1, 2, 3) to the respective values of α₁ = 0.2, α₂ = 1, and α₃ = 0.6 and adjusting β_ij (i ≠ j; i = 1, 2, 3; j = 1, 2, 3) as β₁₂ = 0.45, β₁₃ = −0.02, β₂₁ = −0.03, β₂₃ = −0.02, β₃₁ = 0.3, and β₃₂ = 0.02, respectively. After implementing the measures, PDR is able to gradually stray from the original stable equilibrium point of phase 2 (i.e., E₄) and evolve towards E₇, as shown in Figure 7. (3) When t = 14, implementing the measures applied in the last procedure of the evolutionary pathway E₁→E₃→E₇→E₈ makes the shift from the original stable equilibrium point of phase 3 (i.e., E₇) to the stable equilibrium point of phase 4 (i.e., E₈) possible, as shown in Figure 7.

The above simulations show how PDR evolves along the stepwise pathways that take E₁ and E₈ as the initial and final stable equilibrium points, respectively. Different stepwise pathways are coupled with distinct specific actions, and when to take the phased measures of a stepwise pathway could vary within a certain time window, in line with the actual situations of disaster-stricken areas.

4.2.2. Simulations on Leapfrogging Evolutionary Pathways

PDR also preserves the ability to leapfrog to E₈ with E_i (i = 1, 2, 3, 4) as the initial stable equilibrium points. Thus, this section simulates the leapfrogging pathways of PDR.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.3, 0.2, 0.1), PDR along the leapfrogging evolutionary pathway E₁→E₂→E₈ can be divided into the procedures below: (1) Taking the measures adopted in the first procedure of the pathway E₁→E₂→E₅→E₈ at t = 2 enables PDR to evolve from E₁ to E₂, as Figure 8 shows. (2) When t = 7, PDR can increase the inputs of each subsystem, enhance the promoting effects of environmental–social interactions on the growth of environmental sustainability and social sustainability, weaken the promoting effects of environmental–economic couplings on the sustainable growth of the environmental system, and reinforce the promoting effects of economic–environmental and economic–social interactions on the sustainable growth of the economic system; these actions can be refined as increasing α_i (i = 1, 2, 3) and β₁₃ to the respective values of α₁ = 1, α₂ = 1, α₃ = 1, and β₁₃ = −0.03, while decreasing β₁₂, β₂₁, β₃₁, and β₃₂ to β₁₂ = −0.04, β₂₁ = −0.04, β₃₁ = −0.03, and β₃₂ = −0.02, respectively. After taking these actions, PDR can gradually shift from the original stable equilibrium point of phase 2 (i.e., E₂) to the stable equilibrium point of phase 3 (i.e., E₈), as shown in Figure 8.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, and if the initial point of PDR is (0.1, 0.5, 0.7), PDR along the pathway E₁→E₅→E₈ can be split into the procedures below: (1) When t = 4, enriching the inputs of each subsystem and strengthening the promoting effects of social–environmental interactions on the growth of social sustainability are available options that can be quantified as increasing α_i (i = 1, 2, 3) to the respective values of α₁ = 1, α₂ = 0.6, and α₃ = 0.2 and decreasing β₂₁ to β₂₁ = −0.03. After using these measures, PDR can slowly move away from the original stable equilibrium point of phase 1 (i.e., E₁) and evolve to E₅, as Figure 9 shows. (2) Carrying out the interventions implemented in the last procedure of the pathway E₁→E₂→E₅→E₈ at t = 10 enables PDR to shift from the original stable equilibrium point of phase 2 (i.e., E₅) to E₈, as shown in Figure 9.

If the initial parameter values of PDR equal those listed in Table 3 and make E₁ stable, PDR can also directly leapfrog to E₈ without intermediate stable points. If the initial point of PDR is (0.4, 0.3, 0.8), PDR along the leapfrogging evolutionary pathway, denoted as E₁→E₈, can be realized through increasing the inputs of each subsystem, reinforcing the promoting effects of environmental–economic and economic–social interactions on the sustainable growth of the economic system, enhancing the promoting effects of environmental–social couplings on the growth of environmental sustainability and social sustainability, and impairing the promoting effects of environmental–economic interactions on the sustainable growth of the environmental system at t = 6. Specifically, these measures can be quantified as increasing α_i (i = 1, 2, 3) and β₁₃ to the respective values of α₁ = 1, α₂ = 1, α₃ = 1, and β₁₃ = −0.03 and decreasing β₁₂, β₂₁, β₃₁, and β₃₂ to β₁₂ = −0.04, β₂₁ = −0.04, β₃₁= −0.03, and β₃₂ = −0.02, respectively. After taking these actions, PDR can slowly deviate from the original stable equilibrium point of phase 1 (i.e., E₁) and converge to the stable equilibrium point of phase 2 (i.e., E₈), as shown in Figure 10.

The leapfrogging evolutionary pathways denoted as E₁→E₂→E₈, E₁→E₅→E₈ and E₁→E₈ are able to stand for all kinds of leapfrogging evolutionary pathways starting from E₁; thus, numerical simulations on the remaining leapfrogging pathways starting from E₁ are neglected in this section. In the following context, this section performs simulations on the leapfrogging evolutionary pathways starting from E₂, E₃, and E₄.

If the initial parameter values of PDR equal those listed in Table 3 and make E₂ stable, and if the initial point of PDR is (0.5, 1, 0.4), PDR along the leapfrogging evolutionary pathway E₂→E₈ can be realized by increasing α_i (i = 1, 2, 3) and β₁₃ to the respective values of α₁ = 1, α₂ = 1, α₃ = 1, and β₁₃ = −0.03 at t = 8; at the same time, β₁₂, β₂₁, β₂₃, β₃₁, and β₃₂ need to be decreased to β₁₂ = −0.04, β₂₁ = −0.04, β₂₃ = −0.02, β₃₁= −0.03, and β₃₂ = −0.02, respectively. By taking these actions, it is possible for PDR to stray from the original stable equilibrium point of phase 1 (i.e., E₂) and evolve towards the stable equilibrium point of phase 2 (i.e., E₈), as shown in Figure 11.

If the initial parameter values of PDR equal those listed in Table 3 and make E₃ stable, and if the initial point of PDR is (0.2, 0.1, 1), how PDR leapfrogs along the evolutionary pathway E₃→E₈ can be realized by increasing α_i (i = 1, 2, 3) to 1 and decreasing β₁₂, β₁₃, β₂₁, β₃₁, and β₃₂ to the respective values of β₁₂ = −0.04, β₁₃ = −0.03, β₂₁ = −0.04, β₃₁= −0.03, and β₃₂ = −0.02 at t = 9. After taking these measures, PDR can shift from the original stable equilibrium point of phase 1 (i.e., E₃) to the stable equilibrium point of phase 2 (i.e., E₈), as Figure 12 shows.

If the initial parameter values of PDR equal those listed in Table 3 and make E₄ stable, and if the initial point of PDR is (1, 0.2, 0.1), PDR leapfrogging along the pathway E₄→E₈ can be achieved by increasing α_i (i = 1, 2, 3) and β₃₁ to the respective values of α₁ = 1, α₂ = 1, α₃ = 1, and β₃₁ = −0.03 at t = 7; at the same time, decreasing β₁₂, β₁₃, β₂₁, and β₂₃ to the respective values of β₁₂ = −0.04, β₁₃ = −0.03, β₂₁ = −0.04, and β₂₃ = −0.02 is also necessary. After implementing the above interventions, PDR can slowly deviate from the original stable equilibrium point of phase 1 (i.e., E₄) and finally converge to the stable equilibrium point of phase 2 (i.e., E₈), as Figure 13 shows.

5. Conclusions

PDR is one of the key phases of disaster management and serves as a pivotal opening for sustainable development in disaster-stricken areas. Revolving around the evolutionary traits of PDR under the background of sustainable development, this paper divides PDR into an environmental system, a social system, and an economic system with respective state variables named as environmental sustainability, social sustainability, and economic sustainability; proposes three assumptions justified by ecological theory and complex systems theory; establishes an evolutionary dynamic model for PDR based on synergetics; and reveals the evolutionary stability and evolutionary pathways of PDR based on model analysis. The proposed model and its theoretical results are verified through numerical simulations, which in turn validate the reasonability of the proposed assumptions to a certain extent.

This study yields the following findings: (1) PDR under the background of sustainable development is able to stabilize at eight equilibrium points, of which E₁ has zero-value sustainability at all the three dimensions; E₂, E₃, and E₄ all exactly possess zero-value sustainability at two dimensions; E₅, E₆, and E₇ all exactly show zero-value sustainability at one dimension; and E₈ exhibits non-zero sustainability at each dimension. The PDR states represented by E₁, E₂/E₃/E₄, E₅/E₆/E₇, and E₈ gradually improve, and E₈ is the unique ideal stable state for PDR. The Routh–Hurwitz criterion determines the necessary and sufficient conditions for PDR to stabilize at each equilibrium point. (2) How PDR evolves to E₈ can follow stepwise pathways and leapfrogging pathways. The stepwise evolutionary pathways, which develop the unrecovered subsystems of PDR one by one, can take E_i (i = 1, …, 7) as the initial stable equilibrium points and include the pathways denoted as E₁→E₂→E₅/E₆→E₈, E₁→E₃→E₅/E₇→E₈, E₁→E₄→E₆/E₇→E₈, E₂→E₅/E₆→E₈, E₃→E₅/E₇→E₈, E₄→E₆/E₇→E₈, and E₅/E₆/E₇→E₈, respectively. The leapfrogging evolutionary pathways, which develop two or more unrestored subsystems of PDR at the same time, can take E_i (i = 1, 2, 3, 4) as the initial stable equilibrium points and cover the pathways denoted as E₁→E₈, E₂/E₃/E₄→E₈ and E₁→E₂/E₃/E₄/E₅/E₆/E₇→E₈, respectively. Different evolutionary pathways are supported by different interventions that pertain to the inputs of the three subsystems of PDR and the effects of inter-systems’ interactions on the sustainable growth of the corresponding systems. For a single evolutionary pathway of PDR, when to take the specific measures of a certain stage is dynamically adjustable within a specified time window based on the practical conditions of disaster-stricken areas.

To facilitate PDR to realize the goal of “building back better” in the context of sustainable development, it is necessary to clarify the initial resource endowment related to the environmental system, the social system, and the economic system of PDR, and to identify the interactions between these three subsystems in the first place. According to the actual situations and external surroundings of disaster-stricken areas, it is then essential to identify the initial evolutionary stable state enabled by the incipient conditions of disaster-stricken regions, determine the categories of evolutionary pathways, choose a specific evolutionary pathway for PDR, formulate concrete measures for each phase of the selected evolutionary pathway, and delineate clear timeframes for phase-specific measures. For example, if the initial conditions would make PDR stabilize at E₁, the related subjects should first assess whether to pursue a stepwise pathway (e.g., E₁→E₂→E₅→E₈) or a leapfrogging pathway (e.g., E₁→E₂→E₈); if a stepwise pathway denoted as E₁→E₂→E₅→E₈ is selected for PDR to follow, the relevant stakeholders should then determine the specific actions represented by the changes of α_i and β_ij (i = 1, 2, 3; j = 1, 2, 3) during the shifts of E₁→E₂, E₂→E₅ and E₅→E₈, respectively, and confirm the earliest and latest time to take the corresponding measures at each phase. It should be noted that the internal and external conditions of disaster-stricken regions could change. Hence, the related subjects should better prepare alternative reconstruction pathways for emergency use. Last but not least, concerned parties need to design phase-tailored supervisory mechanisms (e.g., public supervision mechanisms), safeguard mechanisms (e.g., organizational and living support systems), and contingency actions and measures for PDR in practice, thereby guaranteeing that PDR surely evolves to its ideal stable states by following predetermined pathways or timely adjusted pathways.

This study mainly has the following drawbacks. (1) This study models inter-subsystem interactions as x_ix_j (i ≠ j; i = 1, 2, 3; j = 1, 2, 3), but there remain other quantitative forms for inter-subsystem interactions, such as $x_i^2{x}_j^3$ and $x_i^3{x}_j$. In other words, the proposed model does not account for more complex forms of interactions that would provide a more accurate reflection of reality, and the quantification of inter-subsystem interactions in this study is not all-encompassing. Moreover, PDR may be bistable, tristable, cyclic, or even chaotic, which is not further studied in this paper. (2) The proposed assumptions may prevent the model from fully aligning with reality, and this study is not backed by factual data and actual case examples. (3) PDR might vary with the type of disaster, which is neglected in this paper. Therefore, future studies will develop more generalizable nonlinear dynamic models that would include a wider spectrum of interaction forms between the subsystems of PDR, and compulsory verification of these models with factual data and specific examples drawn from real-world crisis situations in various parts of the world will be performed in terms of mathematical–empirical studies or case empirical studies. In addition, PDR related to a certain type of disaster and the cases that PDR is bistable, tristable, cyclic, or even chaotic will be comprehensively discussed in our future research.