We identify sustainability with the idea that any action we take must not cause further depletion of resources. For example, if an action plan is designed such that one type of action is taken over a long period, it results in a cost-effective but not sustainable policy. Thus, we used Shannon entropy as an indicator of the degree of uniformity in the actions taken, i.e., we need to classify actions that use the maximum possible combinations in the control input space.

We have applied Shannon entropy to evaluate the degree of the sustainability of the control policies of the model. Shannon entropy is a fundamental information measure used to quantify the diversity in a distribution (Lin, 1991). It is mathematically defined as: H=−∑i=1npilog (pi), where pi represents the proportion of data in the i-th bin, and n is the total number of bins (Shannon, 1948). The entropy reaches its minimum value, H=0, when all the data is concentrated in a single bin ( pi=1 for one bin and pj=0 for all j≠i), indicating no diversity or uncertainty. Conversely, the entropy is maximized when the data is evenly distributed across all bins ( pi=1n for all i), in which case H=log(n), representing maximum diversity. For example, with six bins, the maximum entropy is H=log(6)≈1.7918. Policies with higher entropy indicate a more robust application of interventions, such as non-monotone cutting of Lantana ( u1) and planting of control plant species ( u2), which ensures ecological stability and sustainability.

We employ a model predictive control (MPC) framework to address the optimal control problem for managing the Lantana population dynamics defined by our three-species ecological system. The control objective is formulated as a variational problem that seeks to minimize a cost function while satisfying system dynamics and constraints. The cost function, given by

balances two competing objectives: Ccontrol=1000 penalizes aggressive control efforts (harvesting u1 and augmentation u2), while Cbiodiversity=10 penalizes deviations from the desired ecological equilibrium state (x1d,x2d,x3d)=(0.1,0.75,0.15). The optimal control problem was implemented in Python using the CasADi optimization framework (Andersson et al., 2019). The nonlinear dynamics were discretized over a time horizon of T=25 units using the fourth-order Runge-Kutta (RK4) method with step size Δt=0.03125, divided into N=40 control intervals. The discretized problem was formulated as a nonlinear programming (NLP) problem with state constraints maintaining populations within [0.0,1.0] and terminal constraints ensuring convergence to the target equilibrium. The NLP was solved using IPOPT (Wächter and Biegler, 2006). Control bounds were imposed as u1∈[−5,0] for harvesting the invasive species and u2∈[0,5] for augmenting native species, with two scenarios investigated: simultaneous application of both controls and selective harvesting only ( u2=0). The optimization was performed for various initial conditions, confirming successful convergence in all cases as shown in the results section.

We would like to point out that the form of the cost function chosen here is not exhaustive and one of the choice can be incorporating the entropy term which would require further detailed investigation.

The controllability analysis of the three-species system (2) establishes the theoretical foundation for our optimal control framework and clarifies the extent to which desired system states can be reached. Under the ecological constraint that all interaction coefficients satisfy aij∈(0,1) for all i,j∈{1,2,3} (Appendix, Assumption 1), representing non-negligible but bounded species interactions, the system is locally accessible at every point in the positive orthant ℝ++3 where all three species coexist (Appendix, Theorem 1). Importantly, this accessibility property holds uniformly across the entire parameter regime aij∈(0,1)—it is not confined to specific parameter configurations but applies universally to all biologically realistic interaction strengths.

The proof involves computing Lie brackets [f₀, f₁] and [f₀, f₂] between the drift vector field (natural ecosystem dynamics) and control vector fields (management interventions), which reveal additional directions in state space accessible by combining natural dynamics with control actions. The accessibility distribution matrix C(x) (Appendix, Equation 12) has full rank equal to three throughout ℝ++3, satisfying the LARC. This accessibility depends critically on non-zero interaction coefficients a31 and a32, representing how Lantana camara and native plants influence soil microbes (Appendix, Remark 1). These strictly positive coefficients provide the indirect control pathway: the control parameters u1 and u2 acting directly on plant populations can indirectly influence microbes through ecological coupling. If either of the coefficients a31 or a32 were zero, the accessibility matrix rank would reduce to 2, destroying controllability. Local accessibility guarantees that from any initial state x0∈ℝ++3, the reachable set has non-empty interior (Appendix, Theorem 2), meaning we can reach a neighborhood of any target state through appropriate control trajectories.

However, the system is not globally controllable. Forward-invariant manifolds exist at coordinate planes where a species has zero population (Appendix, Lemma 1, Theorem 4), and this limitation persists uniformly across all parameter values aij∈(0,1). This reveals fundamentally different dependencies on parameters versus initial conditions.

Control efficacy is universal across the parameter regime, and accessibility holds for all values in aij∈(0,1) without exception. However, control efficacy exhibits a sharp dichotomy with respect to initial conditions: if all species are initially present ( x0∈ℝ++3), any target state can be approached.

We also investigated small-time local controllability (STLC) using Sussmann’s sufficient condition. The system fails this condition uniformly across aij∈(0,1) (Appendix, Theorem 3): the Lie brackets [f0,f1] and [f0,f2], essential for controlling microbes, cannot be expressed as combinations of control vector fields f1, f2, which span only the plant population plane. This failure is a structural feature persisting across all parameter values. While the condition is sufficient but not necessary, the failure suggests potential time-scale limitations on restoration speed (Appendix, Remark 3). Target states are eventually reachable from ℝ++3, but arbitrarily rapid transitions may not be achievable, aligning with ecological reality where restoration unfolds over extended periods. The optimal trajectories (Figures 2–4) reflect these natural time-scale constraints.

Hence, control efficacy is universal across the biologically realistic parameter regime aij∈(0,1) and accessibility holds for all parameter values within this range, and controllability is not confined to specific parameter configurations. However, control efficacy is fundamentally constrained by initial conditions, i.e., desired states are reachable if and only if the initial state contains all three species at positive densities ( x0∈ℝ++3). These results provide rigorous justification that computed optimal trajectories are theoretically achievable across diverse ecosystems with varying ecological characteristics, provided management begins early to maintain all species at positive densities throughout restoration. Given this analysis, we proceed to solve numerically the control problem and present the results in subsequent sections.

Figure 5 presents a comprehensive comparison of system dynamics under different control scenarios. Figure 5a illustrates the uncontrolled state-space trajectories in the (x1,x2,x3) for multiple initial conditions, showing the dismal state of Lantana invasion if no control is applied. Figure 5b depicts the controlled trajectories under two distinct control strategies: single control with u1≠0,u2=0 (red curves) and dual control with u1≠0,u2≠0 (blue curves). Our simulations reveal that the case where only u2 is active, while u1=0, fails to drive the system to the desired state within the specified time horizon, indicating that the control input u2 alone is insufficient for effective system stabilization. Hence, we have omitted this scenario. We observe that both the single-control strategy ( u1≠0,u2=0) and the dual-control strategy ( u1≠0,u2≠0) successfully guide the system trajectories to the target state xd for all initial conditions examined, but they demonstrate different control input signal structures. Figure 5c provides a quantitative comparison of the total costs associated with these two successful strategies, where the cost function balances state deviation from the target and control effort. The box plots reveal a striking difference: the single-control strategy incurs approximately double the cost (median ≈ 1300) compared to the dual-control strategy (median ≈ 700), demonstrating a substantial cost reduction of nearly 50%. The insets display representative time profiles of the control inputs, where inset (i) shows that in the single-control case, u1(t) must exhibit large variations to compensate for the absence of u2, while inset (ii) illustrates that in the dual-control case, the control effort is distributed between u1(t) and u2(t). Collectively, these results establish that while the single-control strategy can stabilize the system, the dual-control policy provides superior performance by reducing intervention costs by approximately 50% and also producing smoother control actions that are more practical to implement in ecological management contexts, where resource constraints and minimized ecosystem disruption are critical considerations.

Since we find that the case of double controls incurs significantly lower cost than the single control one, all subsequent analyses are performed for the case of double control only.

Selection of a sustainable policy requires a sieve, which we now elucidate. When designing control strategies for managing the invasive species Lantana camara, optimizing policies solely based on a fixed cost function is insufficient to ensure ecological sustainability. To address this limitation, we introduced Shannon entropy as an additional metric to evaluate and select control policies. Shannon entropy quantifies the diversity and balance of control efforts, providing insights into how evenly the management inputs are distributed over time. We show in Figure 4 how on changing the two parameters of the cost function, i.e., the control cost penalty parameter ( CControl) and the Bio-Diversity Penalty parameter ( Cbiodiversity) used in the optimization technique, we observed the different trajectories available in Figure 4. This also illustrates various optimal control policies, each with distinct ecological implications. For example, the policy in Figure 4 employs a non-monotone application of control inputs—cutting Lantana camara ( u1) and planting control plant species ( u2)—throughout the management period. This approach promotes a gradual replacement of Lantana camara with control plants, fostering ecological balance while minimizing disruptions to soil health, microbial communities, and the broader environment. In contrast, the policy in Figure 4 adopts a more aggressive strategy, concentrating intensive cutting and planting efforts at the beginning and end of the management period, with prolonged intervals of inaction in between. The abrupt removal of Lantana without consistent planting risks leaving the land vulnerable to degradation, which can lead to soil erosion, nutrient depletion, microbial imbalance, and other cascading ecological issues. While this strategy may seem cost-effective, it compromises long-term stability and sustainability.

We report the impact of changes in the parameters of the cost function on the sustainability characterization of the feasible optimal control policies. Fixing the initial conditions as shown in Figures 5a, b and then using the defined cost function J in (Equation 7), and keeping the interaction matrix fixed(as given in the caption of Figure 4), we computed the optimal control trajectories for various parameter pairs: (Ccontrol,Cbiodiversity)=(103,103),(103,102),(103,10),(103,1), across multiple initial conditions. For each trajectory, we have used six bins for discretization (We also found that on increasing the number of bins, the distribution did not change in shape). The Shannon entropy ( H) was calculated for the time series of x1, x2, u1, u2, and the ratio of the Lantana and control plant abundances x1x2. As can be seen in the distribution of the control inputs shown in Figure 2b, in the case of the example trajectory in Figure 4 corresponds to the ratio of log10(CcontrolCbiodiversity)=0 having a lower value of Shannon entropy as compared to the trajectory in Figure 4 corresponding to the ratio of log10(CcontrolCbiodiversity)=3 which has a higher entropy value. Based on these results and analysis, we report that there is a meaningful way to design a sustainable policy for a desirable ecological state by maximizing the Shannon entropy as the basis for selecting the action plan to control abundances. This trend promotes a more uniform distribution of control efforts over time, as indicated by higher entropy values. Policies with higher entropy condition align with ecological sustainability principles by ensuring continuous and evenly distributed management efforts, ultimately promoting a balanced and desirable resilient ecosystem.

We study sequentially how sensitive the proposed control policies are to the changes in model parameter values. There are two types of parameters regulating our proposed model: firstly, we have the dynamical systems parameters, and secondly, we have the parameters in the optimization of the control cost function. We checked and systematically analyzed the sensitivity of the control cost to both types of parameters, as shown in the previous sections. Now for the dynamical systems parameters, since the precise values of model parameters are unknown, we analyzed the trends of control costs by uniformly varying intrinsic growth rates ( r1,r2,r3) across the range [0.2,0.8], generating 125 tuples while keeping the values of the elements of the interaction matrix fixed. Simulations were conducted for two initial conditions: Case I as shown in Figures 3a, b, where, the abundance of the Lantana camara is high over the landscape ( x1(0),x2(0),x3(0)=0.85,0.1,0.05), and Case II, where its abundance is moderate ( x1(0),x2(0),x3(0)=0.6,0.3,0.1) as shown in the Figures 3c, d. The analysis revealed two distinct regimes based on the ratio of Lantana camara’s growth rate ( r1) to that of the control plants ( r2), which matches the intuitive idea that we can use only certain plants to outgrow Lantana camara. In Regime I( r1r2<1), where the growth rate of Lantana camara is lower than that of control plants, control costs are relatively low. In contrast, Regime II (( r1r2>1)), where Lantana camara outcompetes control plants with higher growth rates, incurs significantly higher management costs, as evident in Figures 3a, c. These trends highlight the crucial role of competition dynamics, as higher costs emerge when Lantana camara dominates resource consumption and growth. Figures 3b, d further illustrate that the behavior of the mean (averaged over various values of r2) control costs for scenarios of high and moderate Lantana camara abundances. The results clearly show that as r1 increases, the management costs consistently rise. While the system shows limited sensitivity to the small variations within each regime, the absolute cost strongly depends on the relative growth rates and the initial abundance of the Lantana camara. These findings significantly highlight that the effective management of invasive species, such as Lantana camara, requires addressing growth dynamics and competition, as higher growth rates and competitive dominance significantly escalate management efforts to restore ecological balance.