Optimal Control of Lantana camara: An Entropy-Based Sustainable Strategy

TL;DR

The paper addresses sustainable management of an invasive species, Lantana camara, using a minimal three-node GLV network (Lantana camara, control plant, soil microbes) with two control inputs. It combines Lie-algebraic controllability analysis and nonlinear optimization (MPC) to derive policies and uses Shannon entropy $H$ to quantify policy sustainability. The results demonstrate full controllability and accessibility (rank 3) and effective steering from diverse initial states to a target state $((x_1^d,x_2^d,x_3^d)=(0.1,0.75,0.15))$, with the sustainability metric favoring more uniform control-action distributions; sensitivity reveals two regimes governed by the ratio $\frac{r_1}{r_2}$ where costs increase with higher $r_1$ and depend on initial conditions. The framework provides a modular decision-support workflow for designing ecologically balanced, sustainable interventions to restore biodiversity and soil health.

Abstract

Framing control policies to mitigate the impact of invasive plants on indigenous biodiversity within the Sustainable Development Goals (SDG) framework is the primary objective of this work. Using reported ecological dynamics of the invasive species \textit{Lantana camara}, we develop a minimal three-species network model, where each node follows generalized Lotka-Volterra (GLV) dynamical equations. Employing Lie algebra and network control theory, we establish the model's controllability and accessibility criteria. Through nonlinear optimization programming, we derive sustainable policies for controlling abundances of \textit{Lantana camara}. We also have used Shannon entropy as an indicator to assess the sustainability of these optimal policies. The analysis of the sensitivity measured using this technique reveals that the control strategy is critically dependent on the ratio of the intrinsic growth rates of the \textit{Lantana camara} and the control plant. Thus, we get a modular algorithmic decision support mechanism for designing control policies to manage \textit{Lantana camara} abundances.

Figures (4)

• Figure 1: Schematic of the ecological system, its modeling, and controlled state evolution.(a) Shows the eco-system of the three interacting species :Lantana camara, control plants, and microbes.(b) A network representation of GLV type interactions among ecological agents. The abundances of Lantana camara$x_1(t)$, control plants $x_2(t)$, and microbes $x_3(t)$ as nodes in the network. Control inputs $u_1(t)$ (reducing Lantana camara) and $u_2(t)$ (enhancing control plants) act on $x_1$ and $x_2$ respectively, with inset plots indicating $u_j(t)$ .(c) A state-space plot displays the system’s trajectory from the initial (blue) to desired (red) state, alongside pie charts showing species proportions aimed at reducing Lantana camara while boosting control plants and microbes.
• Figure 2: Control dynamics of the system across various initial species abundances. Growth rates: $r_{1} = 0.85$, $r_{2} = 0.4$, $r_{3} = 0.3$. Desired state: $(x_1^d, x_2^d, x_3^d) = (0.1, 0.75, 0.15)$. Interaction matrix: $a_{11} = 0.6$, $a_{12} = 0.6$, $a_{13} = 0.5$, $a_{21} = 0.6$, $a_{22} = 0.3$, $a_{23} = 0.5$, $a_{31} = 0.7$, $a_{32} = 0.6$, $a_{33} = 0.7$. Cost parameters: (a) and (c): Control cost $C_{\text{control}} = 1000.0$, biodiversity penalty $C_{\text{biodiversity}} = 1.0$. (b) and (d): Control cost $C_{\text{control}} = 1000.0$, biodiversity penalty $C_{\text{biodiversity}} = 1000.0$.
• Figure 3: Shannon entropy $H$ for evaluating the sustainability of control policies. (a) Box plot of $H$ against $\log(c_{\text{control}}/c_{\text{biodiversity}})$ for the ratio of species abundances $\frac{x_1}{x_2}$,this gives the information about the distribution, in time, of the values of $\frac{x_1}{x_2}$ This indicates how this ratio behaves in time when initially it is $\frac{x_1}{x_2}>1$ and crosses over to $\frac{x_1}{x_2}<1$, across a set of varied initial conditions. (b) $H_{u_1}$ and $H_{u_2}$ from the distribution of the control signals in time for the same set of initial conditions as in (a). (c) $H_{x_1}$ and $H_{x_2}$ for species abundance distribution in time for the same set of initial conditions as in (a), reflecting control effectiveness and biodiversity outcomes.
• Figure 4: Sensitivity analysis: Mean control costs show an increasing trend with an increase in the growth rate of Lantana. (a) and (b) correspond to the initial condition of high abundance of Lantana with the representative figures as (0.85, 0.1, 0.05). Figures (c) and (d) show results for a moderate abundance of Lantana with the initial condition (0.6, 0.3, 0.1). The interaction matrix is fixed as: $a_{11} = 0.3 ,a_{12} = 0.1 , a_{13} = 0.4 , a_{21} = 0.8 ,a_{22} = 0.2 ,a_{23} = 0.5, a_{31} = 0.6 , a_{32} = 0.7 , a_{33} = 0.7$. The mean is taken over for different values of parameter $r_2$.