Mathematical Model of Conservation Strategies for the Rare Plant: A Society-Inclusive Dynamic Model

Biodiversity loss is a pressing global issue, with many species facing unprecedented risks of extinction due to habitat destruction, overexploitation, and climate change [1-3]. Rare plant species are among the most vulnerable, as their restricted distribution and ecological specialization make them particularly sensitive to environmental changes. Yet, these plants often play critical roles in ecosystem stability and resilience, as well as in sustaining human livelihoods.

Conservation of rare plants is therefore not only an ecological necessity but also a socio-economic challenge. Effective strategies must balance biodiversity protection with the needs of local communities that depend on plant resources for food, income, and cultural purposes [4-6]. Achieving this balance requires approaches that go beyond traditional conservation models and incorporate the perspectives of multiple stakeholders.

One such species is Amorphophallus gigas, a rare giant plant endemic to parts of North Sumatra, Indonesia. It is found in the Simandiangin Hamlet, Sungai Kanan District, where its tubers are widely used as an important local food source. The plant has ecological, cultural, and economic significance for surrounding communities. However, its population is under increasing threat due to land conversion, agricultural expansion, and unsustainable harvesting practices [7-9].

The situation of Amorphophallus gigas reflects a broader challenge: how to conserve rare species in landscapes where human activities and ecological systems are deeply intertwined. Protected areas remain a cornerstone of conservation [3], but in practice, they are often insufficient to prevent biodiversity decline. Studies have highlighted the need to manage conflicts between conservation and human development, which requires more integrative approaches [10, 11].

Previous modeling studies have provided valuable insights into ecological dynamics, land-use conflicts, and biodiversity management. However, most of these approaches have emphasized ecological factors while neglecting social and governance dimensions [10, 11]. As a result, they fall short in capturing the complexity of socio-ecological systems where conservation outcomes depend not only on biological processes but also on human behavior and policy interventions.

This study seeks to address this gap by developing a society-inclusive dynamic model that explicitly integrates ecological, socioeconomic, and governance components. The model captures the interactions between plant population growth, community involvement, ecological health, and conservation policies. Unlike traditional ‘fences and fines’ strategies [12], it adopts a more inclusive perspective, recognizing that conservation success requires aligning ecological goals with community interests.

The case study of Amorphophallus gigas in North Sumatra provides a unique opportunity to apply and test this modeling framework. By simulating scenarios with and without conservation interventions, the model evaluates how strategies such as protected area designation, incentive-based community programs, and regulated resource extraction influence long-term plant persistence and human wellbeing. The integration of field data and parameter estimates ensures that the model reflects local realities [5, 8, 9, 13].

Finally, this study makes three contributions. First, it develops a coupled system of ordinary differential equations (ODEs) to represent socio-ecological interactions. Second, it applies the model to simulate conservation and policy scenarios over a century-long horizon. Third, it employs optimization techniques to identify strategies that maximize both ecological and social outcomes [11, 14]. In doing so, the research advances the integration of human dimensions into mathematical conservation models and provides insights for policy design and community-based biodiversity management [6, 15-18].

The model results highlight the complex interactions between ecological, socioeconomic, and policy variables in shaping the long-term persistence of Amorphophallus gigas. Without targeted conservation efforts, the population in Simandiangin Hamlet is projected to decline, while interventions incorporating community engagement and government support significantly improve both ecological and social outcomes. To capture these dynamics, the most dominant variables in the community-inclusive system were selected and are summarized in Table 1.

Table 1. Rare plants conservation factors

The model formulation includes multiple interacting components. The ecological sub-model captures plant growth, ecological health depends on both population size and conservation activities, while community and socioeconomic sub-models account for participation levels and benefits. Government regulation and support are included as additional drivers. The governing equations are shown below:

• Ecological component

Rare plant population growth:

$\frac{d P}{d t}=r P\left(1-\frac{P}{K}\right)+\alpha C-\beta T+\psi G$             (5)

Ecological health dynamics:

$\frac{d E}{d t}=\gamma E+\delta P-\varepsilon T-\xi M$              (6)

• Socioeconomic component

Community involvement dynamics:

$\frac{d C}{d t}=\lambda C+\mu S-v T+v M$                (7)

Socio-economic benefits dynamics:

$\frac{d S}{d t}=\eta S+\kappa P-\zeta C$               (8)

• Conservation efforts component

Conservation efforts dynamics:

$\frac{d T}{d t}=\pi T+\rho P-\sigma E-\omega G$                  (9)

• Government intervention component

Government regulation dynamics:

$\frac{d G}{d t}=\phi G+\psi P-\omega T$               (10)

Government support dynamics:

$\frac{d M}{d t}=\tau M+v C-\xi E$                   (11)

The values of all parameters used in the simulation are presented in Table 2.

Table 2. The parameter values of the conservation model

3.1 Optimization framework

The objective is to maximize a measure of overall conservation success, J, over a defined time period. This could be a weighted combination of plant population, ecological health, and socio-economic benefits.

Maximize: $J=\int_0^{t_{\text {final}}}\left(w_P P+w_E E+w_S S\right) d t$

Subject to:

$\begin{aligned} & 0<P<K, \\ & 0<E<1, \\ & 0<S<1, \\ & 0<C<1, \\ & 0<T<1, \\ & 0<G<1, \\ & 0<M<1\end{aligned}$

where, $w_P, w_E$, and $w_s$ are weights representing the relative importance of each component.

To evaluate conservation outcomes, an optimization model was formulated to maximize overall conservation success:

$J=\int_0^{t_{\text {final}}}\left(w_P P+w_E E+w_S S\right) d t$                (12)

subject to ecological and social feasibility constraints. The weights $w_P, w_E, w_S$ allow prioritization between plant population persistence, ecosystem health, and socioeconomic benefits.

3.2 Scenarios and simulation

Scenario 1: Baseline (No Conservation Efforts)

The baseline simulation was run over a 100-year horizon with initial conditions $\begin{aligned} & P(0)=10, E(0)=0.5, S(0)=0.2, C(0)=0.1, T(0)= 0.05 \text {}\end{aligned}$. Figures 1(a)-1(e) display the outcomes for population, ecological health, socioeconomic benefits, community involvement, and conservation efforts.

• Plant population (Figure 1(a)) shows only a modest increase initially, but growth slows due to limited conservation support and habitat degradation. Without targeted interventions, the risk of long-term decline remains high.
• Ecological health (Figure 1(b)) rises quickly but stabilizes at a lower level, suggesting that ecosystem quality remains vulnerable.

(a) Population vs. plant population

(b) Health index vs. ecological health

(c) Benefit index vs. socio-economic benefits

(d) Involvement level vs. community involvement

(e) Effort level vs. conservation efforts

Figure 1. Time evolution of the main state variables in the society-inclusive conservation model

• Socioeconomic benefits (Figure 1(c)) increase gradually but remain modest, indicating that the community gains little incentive to sustain conservation.
• Community involvement (Figure 1(d)) improves over time but plateaus at a low-to-moderate level, insufficient for strong conservation outcomes.
• Conservation effort (Figure 1(e)) grows initially due to spontaneous actions but lacks sustained momentum.

Overall, the baseline scenario demonstrates that without government or structured community programs, Amorphophallus gigas conservation remains fragile. The plant population may persist in the short term, but long-term viability is threatened by weak social and policy support.

Scenario 2: Government Intervention

In the second scenario, government regulation and support were introduced, with initial conditions $\begin{aligned} & P(0)=10, E(0)=0.5, S(0)=0.2, C(0)=0.1, T(0)= 0.05, G(0)=0.2, M(0)=0\end{aligned}$. The updated model incorporated interaction terms linking policy with ecology and society. Specifically, government regulation (G) and government support (M) were modeled with logistic-type dynamics that allow for growth, saturation, and cross-interactions with plant, community, and ecosystem components. The updated system of equations is as follows:

• Plant population growth

$\frac{d P}{d t}=r P\left(1-\frac{P}{K}\right)+\alpha C-\beta T+\psi G$                 (13)

• Ecological health dynamics

$\frac{d E}{d t}=\gamma E(1-E)+\delta P-\varepsilon T-\xi M$                  (14)

• Community involvement dynamics

$\frac{d C}{d t}=\lambda C(1-C)+\mu S-v T+v M$                      (15)

• Socioeconomic benefits dynamics

$\frac{d S}{d t}=\eta S(1-S)+\kappa P-\zeta C$                   (16)

• Conservation efforts dynamics

$\frac{d T}{d t}=\pi T(1-T)+\rho P-\sigma E-\omega G$                   (17)

• Government regulation dynamics

$\frac{d G}{d t}=\varphi G(1-G)+\psi P-\omega T$                   (18)

• Government support dynamics

$\frac{d M}{d t}=\tau M(1-M)+v C-\xi E$              (19)

The inclusion of government regulation (G) and government support (M) creates feedback loops: government regulation increases with plant population but may suppress excessive conservation efforts, while government support strengthens with community involvement but can be reduced by ecological stress.

3.3 Simulation results

The simulations were carried out over a 100-year horizon using the same Python-based framework as in Scenario 1. Figures 2(a)-2(g) present the outcomes across ecological, social, and policy dimensions.

• Plant population (Figure 2(a)) shows a strong and sustained increase, demonstrating the positive impact of regulation and support on population persistence.
• Ecological health (Figure 2(b)) improves rapidly and stabilizes at a high level, creating favorable conditions for biodiversity.
• Socioeconomic benefits (Figure 2(c)) steadily rise, indicating that community livelihoods improve alongside conservation.
• Community involvement (Figure 2(d)) increases consistently, showing that policy interventions and incentives encourage participation.
• Conservation efforts (Figure 2(e)) remain high, suggesting sustained collective action.
• Government influence (Figure 2(f)) grows steadily, reflecting stronger regulatory and supportive roles.
• Market impact (Figure 2(g)) also increases, highlighting potential for eco-tourism or sustainable resource use.

(a) Value vs. plant population

(b) Value vs. ecological health

(c) Value vs. socio-economic benefits

(d) Value vs. community involvement

(e) Value vs. conservation efforts

(f) Value vs. government influence

(g) Value vs. market impact

Figure 2. Objective function values as a function of key state variables in the conservation-policy optimization framework

3.4 Results

Scenario 1: Baseline (No Conservation Efforts)

The baseline simulation (Figures 1(a)-1(e)) indicates that Amorphophallus gigas shows only modest growth under natural conditions. The plant population rises slowly to about 180 individuals by year 100, which is far below the carrying capacity. Ecological health improves initially but stabilizes at a moderate index (~0.55), leaving the system vulnerable to disturbance.

Socioeconomic benefits grow gradually from 0.2 to ~0.40, reflecting limited improvement in local livelihoods. Community involvement rises slightly but plateaus at ~0.45, insufficient to sustain conservation momentum. Conservation efforts grow at first but stagnate at ~0.50, showing weak long-term commitment without external support.

Overall, the baseline scenario demonstrates that without structured interventions, the conservation of A. gigas remains fragile: the plant persists, but its population is low, ecosystem health is only moderately maintained, and community engagement is weak.

Scenario 2: Government Intervention

In Scenario 2 (Figures 2(a)-2(g)), government regulation and support were added to the system, producing significant improvements across all indicators. By year 100, the plant population grows to approximately 620 individuals, more than three times the baseline scenario. Ecological health stabilizes at a high level (~0.85), suggesting resilient ecosystem conditions.

Socioeconomic benefits improve steadily from 0.2 to ~0.75, indicating that conservation also strengthens local livelihoods. Community involvement rises sharply, reaching ~0.80, nearly double that of Scenario 1. Conservation efforts remain high (~0.80) throughout the simulation, reflecting strong and sustained action.

Government influence increases consistently, ensuring policy continuity, while market impact grows to ~0.70, suggesting opportunities for eco-tourism or sustainable use of A. gigas resources.

The side-by-side comparison shows that Scenario 2 outperforms Scenario 1 on every indicator. Table 3 presents the summary.

Table 3. Comparative results of Scenario 1 and Scenario 2

3.5 Discussion

The comparison between the two scenarios clearly highlights the role of government policy and community engagement in ensuring the long-term persistence of Amorphophallus gigas. Scenario 1, relying on natural dynamics alone, leads to fragile conservation outcomes with limited ecological and social benefits. In contrast, Scenario 2 shows that integrating regulation, support, and incentives creates a virtuous cycle: the plant population grows, ecological health stabilizes, community participation increases, and socioeconomic gains reinforce conservation behavior.

Sensitivity analysis suggests that community involvement $(\alpha, \mu, v)$ and government support $(\tau, v, \xi)$ are among the most influential parameters shaping system outcomes. This underlines the importance of participatory conservation and incentive-based programs.

However, the model still has limitations. Spatial heterogeneity and climate change are not yet incorporated, and parameter estimates require further field validation. Despite this, the framework demonstrates the feasibility of balancing ecological and socioeconomic objectives, providing a valuable tool for conservation planning.