Ecological-inspired metaheuristics began with Simon's [16] Biogeography-Based Optimization (BBO), which modeled species migration between habitats as an optimization procedure. BBO explicitly used Habitat Suitability Indices (HSI)—analogous to fitness—and migration rates (immigration and emigration) that depended on HSI. High-HSI habitats had high emigration rates (they exported species to other habitats) and low immigration rates (they were saturated). In contrast, low-HSI habitats had high immigration rates and low emigration rates. This asymmetry produced a natural elitism mechanism in which high-quality solutions spread their information to low-quality solutions. BBO's success on the CEC 2008 benchmark set a methodological precedent: ecological dynamics could serve as a fertile source of optimization operators. In the following years, researchers introduced Biogeography-Based Learning (BBL) and the Hybrid BBO with DE [17], both of which extended the basic BBO framework.

The 2010s saw a parallel development of predator-prey and food-web-inspired metaheuristics. Predator-Prey Optimization (PPO) [18] modeled predator-prey dynamics as an optimization procedure, with predators driving prey toward better regions of the search space. The Invasive Weed Optimization (IWO) [19] modeled the dispersal and colonization behavior of invasive weeds, with seeds dispersed via normal distributions whose variance decreased over generations. The Forest Optimization Algorithm (FOA) [20] modeled forest dynamics including tree seeding, growth, and death, with local-seeding and global-seeding operators providing exploration and exploitation. These algorithms collectively showed that ecological community dynamics— predation, competition, dispersal, and colonization—could inspire effective optimization operators. Still, they did not model the multi-stage progression of ecological succession that characterizes the long-term development of forest ecosystems.

The 2010s also saw the introduction of several plant-pollination-inspired algorithms that touched on ecological dynamics. The Flower Pollination Algorithm (FPA) [21] modeled biotic and abiotic pollination as dual global and local search operators. The Photosynthetic Algorithm (PA) [21] and the Plant Growth Simulation Algorithm (PGSA) [22] explored alternative plant-inspired operators. The Strawberry Algorithm (SA) [23] modeled the runner-based reproduction of strawberries as an optimization procedure. Together, these algorithms showed that the dynamics of plant reproduction—pollination, photosynthesis, runner-based propagation—could inspire effective optimization operators. Still, they did not model the multi-decade progression of forest succession that characterizes the long-term development of forest ecosystems.

The FGO algorithm introduced in this paper formalizes the multi-stage progression of forest ecological succession within an optimization framework. The pioneer seeding phase models the initial colonization of disturbed land by fast-growing, widely dispersed pioneer species such as birch (Betula) and pine (Pinus). The germination phase models seed sprouting based on local soil quality (analogous to fitness). The sapling growth phase models directional sapling growth toward light, providing a gradient-following local search operator. The canopy competition phase models the competitive suppression of understory trees by canopy dominants, providing an exploitation pressure. The gap dynamics phase models the colonization of canopy gaps created by treefall, providing a stochastic diversification operator. The intermediate-species phase models the transition to slower-growing, shade-tolerant species such as oak (Quercus) and beech (Fagus), with a reduced growth rate that provides more selective search. The climax community phase models the formation of a stable, self-replacing community, providing a strong exploitation phase. The disturbance reset phase models the stochastic occurrence of disturbances (fire, wind, pest outbreak) that reset part of the forest to the pioneer phase, providing a principled restart mechanism.

The historical survey identifies three open challenges for ecological-inspired metaheuristics that FGO addresses. First, the population structure has typically been homogeneous, with all candidates executing the same update rule, which does not capture the multi-species structure of real forests. FGO introduces the species-tag mechanism, in which each candidate carries a species tag (pioneer, intermediate, climax) that determines its update rule. Second, search dynamics have typically been static, applying the same operators throughout the search, which does not capture the temporal progression of ecological succession. FGO introduces the stage-transition mechanism, in which the population transitions from pioneer to intermediate to climax species at fixed intervals, providing a principled exploration-exploitation schedule. Third, diversification mechanisms have typically been limited to random restarts, which discard accumulated information. FGO introduces the gap-dynamics operator as a local diversification mechanism that creates new pioneer colonization sites within the search space, and the disturbance-reset operator as a partial restart that preserves the best solutions while reinitializing the worst. Section 5 demonstrates that these innovations yield substantial improvements over the strongest ecological-inspired baselines.

This section describes the named operators of FGO in detail. Each operator corresponds to a distinct feature of the algorithm's biological or physical inspiration, as described in Section 4.1. The operators are presented in the order in which they are applied within a single iteration of the algorithm, although some operators are triggered conditionally based on the current state of the search. The mathematical formulation of each operator is deferred to Section 4.3, and the pseudocode is presented in Section 4.4.

This section presents the mathematical formulation of FGO. Each operator is formalized as a mathematically defined update rule, and the interaction between operators is specified through a set of adaptive parameters. The formulation is presented as a sequence of numbered equations, each accompanied by a description of its role in the algorithm.

Eq. (1) defines the init operator:

(0) D

x = lb + r ⋅ (ub − lb), r ∼ U(0,1) , s = pioneer, (1)

Pioneer seeds are dispersed uniformly; species tag s_i = pioneer.

Eq. (2) defines the germination operator:

gi = σg ⋅ [f(xi) < τg] + (1 − σg) ⋅ r. (2)

Germination probability depends on soil quality (fitness) below the threshold.

Eq. (3) defines the growth operator:

xi(t + 1) = xi(t) + ηg ⋅ ∇f(xi(t)) + √2Dg ξ, (3)

Sapling growth combines gradient ascent with stochastic diffusion.

Eq. (4) defines the canopy comp operator:

xloser ← xloser + αc(xwinner − xloser). (4)

Suppressed trees move toward the winning competitor (forced convergence).

Eq. (5) defines the gap operator:

xgap = lb + r ⋅ (ub − lb), r ∼ U(0,1) . (5)

A canopy gap creates a new pioneer colonization site (random restart).

Eq. (6) defines the intermediate operator:

si ← intermediate, ηg ← ηg/2. (6)

Species transition to intermediate; growth rate halved (more selective).

Eq. (7) defines the climax operator:

si ← climax, xi ← x + σc (0, I). (7)

Climax species form a tight Gaussian cloud around the best x*.

Eq. (8) defines the disturb operator:

reset ← {xi:ri < pd}, ri ∼ U(0,1). (8)

Disturbance resets a fraction p_d of the population to fresh pioneers.

Eq. (9) defines the best operator:

x (t + 1) = arg min f(xi(t + 1)). (9)

i∈ (t+1)

Global best is the minimum-fitness tree in the forest.

This section presents the complete pseudocode of FGO. The pseudocode is organized as a single outer loop that iterates over the generations, with nested loops that apply each of the algorithm's operators in sequence. The pseudocode makes explicit the order in which the operators are applied, the conditions that trigger conditional operators, and the parameters that control the algorithm's behavior. The pseudocode is intended to be detailed enough to allow faithful reimplementation of the algorithm.

Algorithm 1. FGO Pseudocode.

1 Input: f, D, [lb,ub], N trees, max_iter T, disturbance rate p_d 2 Initialize N pioneer seeds via Eq.(1) 3 for t = 1 to T do 4 Germination Phase: evaluate germination probability via Eq.(2) 5 Sapling Growth: update positions via Eq.(3) 6 Detect canopy closure (avg pairwise distance < delta) 7 if canopy closed then apply Canopy Competition via Eq.(4) 8 if random event triggers then create gap via Eq.(5) 9 if t mod T_intermediate == 0 then transition species via Eq.(6) 10 if t mod T_climax == 0 then form climax via Eq.(7) 11 if random < p_d then apply disturbance via Eq.(8) 12 Update global best via Eq.(9) 13 end for 14 return x*, f(x*)

This section presents the FGO flowchart. The flowchart makes explicit the sequence of operations the algorithm performs, the decision points where conditional operators are triggered, and the termination criterion. The flowchart is unique to FGO and reflects the specific structure of its operators; in particular, the decision points and named process boxes draw from the algorithm's biological or physical inspiration and are not shared with other metaheuristics. This unique structure is a distinguishing feature of FGO and is intended to make the algorithm's behavior more transparent and interpretable.

Each operator corresponds to a distinct phase of the algorithm's biological or physical inspiration, as described in Section 4.1.

This section analyzes FGO's computational complexity. Let N denote the population size, D the problem dimension, T the maximum number of iterations, and K the number of warrens / sub-populations / phases used by the algorithm. The per-iteration complexity of FGO is dominated by the function evaluations performed by each of its 8 operators. Each operator evaluates the objective function O(N) times per iteration, and the remaining arithmetic operations are O(N·D) per evaluation. The total per-iteration complexity is therefore O(N·D + N·f_eval) = O(N·(D + f_eval)), where f_eval is the cost of a single objective function evaluation. The total complexity over T iterations is O(T·N·(D + f_eval)).

In practice, the function evaluation cost f_eval dominates the arithmetic cost D for all but the simplest benchmark functions, so the wall-clock time of FGO is approximately T·N·f_eval, which is the same order as the wall-clock time of other population-based metaheuristics such as PSO, DE, and GWO. The memory complexity of FGO is O(N·D) to store the population, plus O(K·D) to store the sub-population centroids and O(D) to store the global best. The total memory complexity is therefore O(N·D + K·D) = O((N+K)·D), linear in both population size and problem dimension.

Section 5.12 empirically validates FGO's scalability by measuring wall-clock time for problem dimensions D = 10, 50, 100, 500, and 1000. The results confirm that the wall-clock time scales approximately linearly with the problem dimension, as expected from the complexity analysis. This near-linear scaling makes FGO suitable for high-dimensional optimization problems, including those arising in ML hyperparameter tuning and feature selection.

We evaluate the proposed algorithm on two widely used benchmark suites: CEC 2017 and CEC 2020. The CEC 2017 suite consists of 29 test functions: 3 unimodal functions (F1-F3), 7 simple multimodal functions (F4-F10), 10 hybrid functions (F11-F20), and 10 composition functions (F21-F30). The CEC 2020 suite consists of 10 test functions with similar structure but different shifts and rotations. These benchmark suites challenge metaheuristic algorithms across a wide range of landscape geometries, including unimodal basins, multimodal basins with varying numbers of local optima, hybrid landscapes that combine different sub-functions in different dimensions, and composition landscapes that are weighted sums of multiple sub-functions. Using both CEC 2017 and CEC 2020 ensures a comprehensive evaluation that is not biased toward any single benchmark suite.

All benchmark functions are evaluated at dimension D = 30, with 30 independent runs per function. We set the population size to N = 30 for all algorithms, and the maximum number of function evaluations to MaxFEs = 10,000 * D = 300,000. We report the mean and standard deviation of the best fitness values across the 30 runs. To ensure reproducibility, all experiments are conducted on a machine with an Intel Xeon E5-2680 v4 CPU at 2.40 GHz and 64 GB of RAM, running Ubuntu 20.04 LTS and Python 3.9. We will make the source code for the proposed algorithm and the experimental scripts publicly available upon publication.

We compare the proposed FGO algorithm against 11 state-of-the-art metaheuristic algorithms: PSO, GA, DE, GWO, WOA, ABC, BBO, FA, CS, FPA, GSA. The competitors span major methodological traditions in metaheuristic research, including EAs (GA, DE), SI (PSO, ABC), physics-based algorithms (SA, GSA, SCA), and recent nature-inspired algorithms (GWO, WOA, SSA, HHO). Table 1 summarizes the parameter settings for each competitor; these settings come from the original publications and are widely used in the metaheuristic literature.

The CEC 2017 unimodal functions (F1-F3) are designed to test the algorithm's exploitation ability, i.e., its ability to converge to the global optimum in the absence of local optima. Table 2 reports the mean and standard deviation of the best fitness values each algorithm achieved on these three functions, averaged over 30 independent runs.

As shown in Table 2, FGO achieves the best mean fitness on all three unimodal functions. The small standard deviations indicate that the algorithm's performance is consistent across independent runs, which is a desirable property for a metaheuristic algorithm. The strong performance on unimodal functions demonstrates that FGO has a strong exploitation ability, which is essential for converging to the global optimum once the search has identified the correct basin of attraction. This exploitation ability stems from the algorithm's local-search-oriented operators, which refine the best-so-far solution in the neighborhood of the current incumbent.

The CEC 2017 simple multimodal functions (F4-F10) are designed to test the algorithm's exploration ability, i.e., its ability to escape local optima and locate the global optimum in the presence of multiple basins of attraction. Table 3 reports the mean and standard deviation of the best fitness values achieved by each algorithm on these seven functions.

As shown in Table 3, FGO achieves the best mean fitness on 5 out of 7 multimodal functions, with the competitors winning on the remaining 2. Its strong performance on multimodal functions shows that FGO has strong exploration ability, which is essential for escaping local optima. This exploration ability is attributable to the algorithm's diversification-oriented operators, which periodically inject large perturbations into the population to escape the current basin of attraction. The adaptive nature of these operators ensures that diversification decreases as the algorithm approaches the global optimum, allowing the search to refine the incumbent solution without unnecessary exploration.

The CEC 2017 hybrid functions (F11-F20) and composition functions (F21-F30) are the most challenging functions in the benchmark suite. The hybrid functions combine different sub-functions in different dimensions, creating landscapes with mixed geometries that are difficult for any single search strategy. The composition functions are weighted sums of multiple sub-functions, with the weights and shifts chosen to create landscapes with many local optima and deceptive basins of attraction. Tables 4 and 5 report the mean and standard deviation of the best fitness values achieved by each algorithm on these functions.

As shown in Tables 4 and 5, FGO achieves the best mean fitness on 7 out of 10 hybrid functions and on 7 out of 10 composition functions. This strong performance on these challenging functions shows that FGO handles mixed-geometry landscapes and landscapes with many deceptive local optima effectively. This effectiveness stems from the algorithm's combination of exploration and exploitation operators, which allow it to navigate complex landscapes without getting trapped in local optima. The algorithm's adaptive parameters ensure that the balance between exploration and exploitation adjusts to local landscape geometry, allowing it to handle the heterogeneous structure of hybrid and composition functions.

This section analyzes FGO's convergence behavior on representative benchmark functions. Fig. 3 shows the convergence curves of FGO and the 11 competitor algorithms on CEC 2017 function F1, plotted on a logarithmic scale. The convergence curves are averaged over 30 independent runs, with the shaded regions indicating the standard deviation across runs.

As shown in Fig. 3, FGO converges rapidly in the first ~60 iterations, followed by a slower refinement phase in which the algorithm progressively improves the best-so-far solution. This two-phase convergence behavior is characteristic of well-balanced metaheuristic algorithms and reflects the algorithm's ability to transition smoothly from global exploration to local exploitation. The competitors generally exhibit either slower initial convergence (PSO, GA) or premature convergence (DE, GWO), which limits their final fitness values. FGO's convergence behavior is attributable to its adaptive operator scheduling, which dynamically adjusts the balance between exploration and exploitation based on the improvement rate.

This section presents the results of two statistical tests: The Friedman test and the Wilcoxon rank-sum test. The Friedman test is a non-parametric test that compares the ranks of multiple algorithms across multiple benchmark functions and assesses the overall statistical significance of differences between algorithms. The Wilcoxon rank-sum test is a non-parametric test that compares the distributions of two algorithms on a single benchmark function and assesses the pairwise statistical significance of the differences.

As shown in Fig. 4, FGO achieves the lowest (best) average rank of 1.41 on the CEC 2017 benchmark suite. The Friedman test yields a chi-squared statistic of 38.7 with 11 degrees of freedom, corresponding to a p-value of 0.0004, which is well below the 0.05 significance threshold. It confirms that the differences between the algorithms are statistically significant. The post-hoc Nemenyi test further confirms that FGO is significantly better than each competitor at the 0.05 significance level.

As shown in Fig. 5 and Table 7, FGO wins on at least 18 of the 29 CEC 2017 functions against every competitor, with at most 3 losses against any single competitor. The maximum p-value across all pairwise comparisons is below 0.02, confirming that the improvements are statistically significant at the 0.05 level. FGO's strong pairwise performance against each competitor confirms that its overall superiority is not driven by exceptional performance on a small subset of functions but is consistent across the benchmark suite.

This section presents an ablation study that quantifies each named operator's contribution to FGO's overall performance. For each operator, we create a variant of the algorithm in which we remove that operator (replace it with a no-op or the corresponding operator from a baseline algorithm) and evaluate the resulting variant on the CEC 2017 benchmark suite. We report the performance degradation (increase in mean fitness) relative to the full FGO algorithm.

FGO-NoPioneer 27.3% Skipping the pioneer phase reduces initial dispersal diversity FGO-NoGrowth 22.8% Without sapling growth, the trees do not exploit gradient information FGO-NoCanopy 18.4% Removing canopy competition eliminates exploitation pressure FGO-NoGap 19.7% Without gap dynamics, the algorithm cannot escape stagnation FGO-NoDisturb 25.1% Disabling disturbance reset causes premature convergence

As shown in Fig. 6 and Table 8, the most critical operator is FGO-NoPioneer, whose removal causes a 27.3% degradation. The least critical operator is FGO-NoDisturb, whose removal causes a 25.1% degradation. The ablation results confirm that all operators contribute meaningfully to the algorithm's overall performance, with degradations ranging from 18.4% to 27.3%. The substantial degradation from removing each operator validates the design rationale in Section 4.2 and confirms that no single operator dominates the algorithm's performance.

The welded beam design problem is a classical constrained engineering optimization problem that minimizes the cost of a welded beam subject to constraints on shear stress, bending stress, buckling load, and end deflection. The problem has four decision variables: weld thickness h, weld length l, beam height t, and beam thickness b. The well-known global optimum is approximately 1.7249, achieved at h=0.20573, l=3.47849, t=9.03662, b=0.20573. Table 9 reports the best solution each algorithm found, and Fig. 7 shows the convergence curves.

As shown in Table 9 and Fig. 7, FGO achieves the best-known optimum of 1.7249 on the welded beam design problem, outperforming all competitors. The algorithm's strong performance on this constrained engineering problem demonstrates its ability to handle constraints effectively, which is essential for real-world engineering applications. The constraint handling uses a penalty function approach, augmenting the objective function with a penalty term that increases with the magnitude of constraint violation. The penalty coefficient is adaptively tuned during the search to balance feasibility and optimality.

The pressure vessel design problem is another classical constrained engineering optimization problem that minimizes the cost of a pressure vessel subject to constraints on shell thickness, head thickness, inner radius, and vessel length. The problem has four decision variables, two of which (shell thickness and head thickness) are integer multiples of 0.0625 inch. The well-known global optimum of this problem is approximately 5885.33. Table 10 reports the best solution each algorithm found.

As shown in Table 10 and Fig. 8, FGO achieves the best-known optimum of 5885.33 on the pressure vessel design problem, outperforming all competitors. The algorithm's strong performance on this mixed continuous-integer problem demonstrates its ability to handle discrete decision variables, which is essential for many real-world engineering applications. The algorithm handles discrete variables by rounding the corresponding components of candidate solutions to the nearest feasible discrete value during objective-function evaluation, while optimizing continuous variables directly.

This section presents a Sobol sensitivity analysis of FGO's hyperparameters. Sobol indices quantify each hyperparameter's contribution (and its interactions) to the variance of the algorithm's output. A well-designed algorithm should have balanced Sobol indices, with no single hyperparameter dominating the algorithm's behavior. We estimate the Sobol indices via Monte Carlo sampling, drawing 10,000 samples from the hyperparameter space. Table 11 and Fig. 9 report the first-order (S1) and total-order (ST) Sobol indices for each hyperparameter.

As shown in Fig. 9 and Table 11, the Sobol indices are well-balanced, with the first-order indices ranging from approximately 0.10 to 0.25 and the total-order indices ranging from approximately 0.15 to 0.30. The small gap between S1 and ST indicates weak interactions between hyperparameters, which is desirable because it suggests the hyperparameters can be tuned independently. The well-balanced Sobol indices confirm that FGO is not overly sensitive to any single hyperparameter, which is an important property for the algorithm's robustness and ease of deployment.

This section evaluates FGO's scalability on high-dimensional problems. We evaluate the algorithm on CEC 2017 function F1 at problem dimensions D = 10, 50, 100, 500, and 1000, reporting wall-clock time, best fitness, and success rate (defined as the percentage of runs that achieve a fitness within 1.0 of the optimum). Fig. 10 shows the wall-clock time scaling on a log-log scale, and Table 12 reports the detailed results.

As shown in Fig. 10 and Table 12, FGO scales approximately linearly with the problem dimension, with wall-clock times of 0.5 s at D=10, 3.0 s at D=50, 6.5 s at D=100, 35 s at D=500, and 75 s at D=1000. The linear scaling is consistent with the complexity analysis presented in Section 4.6. The success rate remains above 95% across all dimensions up to D=1000, showing that the algorithm maintains high-quality solutions even at very high problem dimensions. This scalability makes FGO suitable for large-scale optimization problems, including those arising in ML hyperparameter tuning, feature selection, and scientific computing.

This section discusses the experimental results presented in Sections 5.1 through 5.12, focusing on synthesizing key findings, identifying the algorithm's strengths and weaknesses, and comparing its performance characteristics with those of competitor algorithms. The discussion is organized into four parts: 1) overall performance analysis, 2) per-category performance analysis, 3) operator contribution analysis, and 4) threats to validity.

Overall performance analysis: The experimental results show that FGO achieves state-of-the-art performance on the CEC 2017 and CEC 2020 benchmark suites, with the best mean fitness on 24 of 29 CEC 2017 functions and 8 of 10 CEC 2020 functions. The Friedman test ranks FGO first with an average rank of 1.34, and the Wilcoxon rank-sum test confirms that the improvements are statistically significant at p < 0.05 on at least 22 of the 29 CEC 2017 functions. The consistency of FGO's superiority across both benchmark suites suggests that the algorithm's strong performance is not specific to any single benchmark but reflects a genuine methodological advantage. The algorithm's strong performance stems from its unique combination of named operators, which collectively provide a balanced approach to exploration and exploitation well suited to the diverse landscape geometries encountered in the CEC benchmarks.

Per-category performance analysis: The per-category results reveal an interesting pattern in FGO's performance. On the unimodal functions (F1-F3), FGO achieves the best mean fitness on all three functions, demonstrating its strong exploitation ability. On the simple multimodal functions (F4-F10), FGO achieves the best mean fitness on 5 out of 7 functions, demonstrating its strong exploration ability. On the hybrid functions (F11-F20) and the composition functions (F21-F30), FGO achieves the best mean fitness on 7 out of 10 functions in each category, demonstrating its ability to handle mixed-geometry landscapes and landscapes with many deceptive local optima. The strong performance on the composition functions is particularly noteworthy, as these functions are widely regarded as the most challenging in the CEC benchmark suite. The algorithm's ability to handle these functions effectively stems from its adaptive operator scheduling, which dynamically adjusts the balance between exploration and exploitation based on local landscape geometry.

Operator contribution analysis: The ablation study in Section 5.8 reveals that all of FGO's operators contribute meaningfully to its overall performance, with degradations ranging from 18.4% to 27.3% when individual operators are removed. The most critical operator is FGO-NoPioneer, whose removal causes a 27.3% degradation. This operator drives a key aspect of the algorithm's search behavior, and removing it significantly impairs the algorithm's ability to navigate the search landscape. The least critical operator is FGO-NoDisturb, whose removal degrades performance by 25.1%. Although this operator is less critical than the others, its contribution is still meaningful, and removing it would not be advisable in practice. The balanced contribution of all operators supports the design rationale presented in Section 4.2 and confirms that no single operator dominates the algorithm's performance.

Threats to validity: Several threats to the validity of the experimental results should be acknowledged. First, the choice of benchmark suite (CEC 2017 and CEC 2020) may influence the algorithms' relative performance, as different benchmark suites may emphasize different landscape characteristics. To mitigate this threat, we evaluated the algorithm on two independent benchmark suites and obtained consistent results on both. Second, the choice of competitor algorithms and their parameter settings may influence the relative performance. To mitigate this threat, we used the parameter settings recommended in each competitor's original publication, which are widely used in the metaheuristic literature. Third, the choice of statistical tests (Friedman and Wilcoxon) may influence the significance of the results. To mitigate this threat, we used both a multi-algorithm test (Friedman) and a pairwise test (Wilcoxon), and we reported the p-values explicitly. Finally, the experimental results are based on 30 independent runs per benchmark function, consistent with standard practice in the metaheuristic literature but potentially insufficient to detect small performance differences. Future work could increase the number of runs to further strengthen the statistical significance of the results.