In typical schooling fish including sardines, tuna and herrings, individuals show highly synchronised collective movement behaviours driven by a uniform search process, in which the environment is explored using a predominantly similar set of rules with little dependence on individual foraging experiences. By contrast, larval zebrafish exhibit an individually responsive and autonomous exploratory strategy, where individual larvae explore unknown regions using their own visual and mechanosensory systems rather than fixed group formations or coordinated leadership structures. Additionally, movement magnitude is adaptively controlled in accordance with individual foraging performance, such that a low-performing individual is afforded greater exploratory freedom through a fairness-weighted behavioural adjustment. This unique biological mechanism forms the foundation of the individual foraging-success-based exploration strategy in the proposed Zebrafish Optimization Algorithm (ZFO) and consequently contributes to enhancing population diversity, balancing the exploration-exploitation process and improving resistance to premature convergence.

In ZFO, exploration behaviour refers to the manner in which zebrafish conduct their random foraging movements when they are finding food sources. During this stage of learning, an agent will move in a random or arbitrary fashion to discover new areas of the search space for food.

ZFO’s method of regulating how much exploration takes place during each learning iteration involves the implementation of a time-dependent or adaptive step size. In the ZFO algorithm, the initial iterations use a larger-than-normal step size and gradually reduce the amount of exploration as optimization continues.

The second aspect of regulating exploration intensity is its incorporation of a fitness-dependent fairness factor. As a result of the incorporation of the fitness-dependent fairness factor, agents with the worst fitness (i.e. those with the least success) can take larger exploratory actions, thereby allowing them to escape from areas where they are unsuccessful and re-enter the competition for food. On the other hand, well-performing agents will engage in more conservative actions to avoid dominating the exploratory actions of the remaining agents.

The significant distinction between shoaling zebrafish and other standard forms is that spatial regrouping adjusts dynamically to local conditions. The dynamic spatial organization of zebrafish shoal’s results in constant local rearrangement within the group as each individual reacts to neighbour attraction and communal chemical cue detection. This process constitutes environmentally induced, information based dynamic modulation of spatial positioning rather than predetermined locomotion and random drift is the underlying basis for the neighbourhood centroid shoaling mechanic of ZFO and guarantees that the population does not become uniform or otherwise cease to be effective throughout the course of the search.

The shoaling behaviour of zebrafish is the coordinated swimming patterns with keeping safe distances between shoal members through the collective behaviour of zebrafish. In ZFO, every agent interacts with its local neighbourhood based only on the spatial proximity to other agents, not on any global information. When determining how to move, an agent will move towards the centroid of its neighbourhood to facilitate information sharing among nearby agents.

A separation mechanism is incorporated to prevent agents from colliding with one another when they become close together. An agent will be repelled when it comes too close to a neighbouring agent. Furthermore, perturbations are added to allow for continued diversity while not disrupting the overall trend of convergence. The shoaling behaviour acts as an intermediary stage in stabilizing population dynamics and allows for the gradual improvement of the most promising areas.

This ZFO attack technique is derived from the leaderless, synchronous predatory behaviour of larval zebrafish and distinct from many hierarchical predator models and conceptually dissimilar to hierarchically based search attack strategies of existing metaheuristic algorithms. Other predators typically coordinate attacks in a predetermined social structure or by a defined leader whose task assignments are known beforehand but larval zebrafish do not utilize a hierarchy and instead coordinate attack on the prey based on a dynamic interaction between their current distance to prey and individual search potential at that particular instant. This biologically uncommon, fitness-based, interaction-driven coordination serves as the foundation of the fitness-modulated with multi-agent exploitation method implemented in ZFO, where the population concentrates its search on potentially high-fitness regions while ensuring distribution and resistance to premature convergence.

Zebrafish exhibit a shift in their hunting behaviour once they detect a potential prey item. In this stage, they exhibit rapid and directed motion toward it. During this step, zebrafish are able to adapt their swimming paths to enhance their chance for a successful capture, often through the combination of sensory input from themselves and members of their immediate shoal.

Candidate solutions actively search prospective regions during the exploitation phase of the proposed optimiser, which is equivalent to the hunting activity mentioned in promising candidate solutions intensively and concurrently while pursuing both global and local optima.

The ZFO stagnation recovery mechanism draws inspiration from a biologically unusual mode of adaptation observed in larval zebrafish. While many predator-inspired models do not explicitly model after a bout of searching failures, larval zebrafish make an individual decision and, once they accumulate a specific number of consecutive search failures, immediately leave that poor patch and redisperse back into unexplored space. The intelligent adaptive reset forms the basis of ZFO stagnation recovery, the stagnant agents are triggered for reactivation in order to restore diversity, get out of the local optimum and improve the exploration of several potential basins.

Zebrafish show an important trait of fairness-driven adaptation in zebrafish shoals. In a shoal, less-successful individuals have more space for the possible recovery from unfavourable areas, while successful individuals adopt a more conservative strategy. At the same time, zebrafish will leave areas that do not yield any successful hunting opportunities and will become exploratory and continue searching for better alternatives.

These biological mechanisms motivate the integration of two key components to create ZFO, which includes fitness-based adaptation and the ability to switch between behaviours when they become stagnant, thereby improving the robustness of the system and improving long-term search efficiency.

The earlier stages of optimization for zebrafish are primarily characterized by exploratory responses. In the absence of known food sources, the exploratory response is analogous to a random search for potential locations of food. During this phase, an agent’s movements will be random throughout the search space to attempt to find areas to discover unexplored regions.

The update rule for position on exploration, for the ith zebrafish at time ( t ), is represented by: (3) Z i t + 1 = Z i t + α ( t ) ( 1 + ϕ i ) R , where R follows a Gaussian distribution N ( 0 , 1 ) in (3) and (4) α ( t ) is representation of time-varying exploration factor, defined by: (4) α ( t ) = α max ( 1 − τ ) + α min τ , τ = t T , where τ ∈ [ 0 , 1 ] is the normalized step of iteration progress (value from 0 to 1), t is the current iteration, T is the maximum number of iterations, α max and α min are the upper and lower bounds of the exploration step size respectively.

In (5) represents term ϕ i called the fairness factor, which encourages greater activity in poorly performing candidate solutions, is defined as: (5) ϕ i = max ( 0 , F median − F best + ε F i − F best ) . This exploration method reduces the risk of premature convergence, as well as increasing the area coverage of the search space. Where F median is the median fitness value of the population, F best + ε is the best fitness value encountered so far and is a tiny positive value 10 − 10 to avoid division by zero. The operator max ( 0 , · ) ensures that is ϕ i non-negative.

Zebrafish exhibit shoaling as a mechanism for social regrouping. Shoaling allows individuals to swim together while keeping safe distances apart from one another. Through shoaling behaviour, information can be shared between neighbouring zebrafish. It also allows the zebrafish to continue to move together as they migrate from one location to the next.

The local centre associated with the ith zebrafish is calculated using (6): (6) C i = 1 k ∑ j ∈ N i X j . Zebrafish update their position according to: (7) X i t + 1 = X i t + β ( 1 + 0.2 ϕ i ) ( C i − X i t ) − γ S i + δ s R . As given in (7), S i = X i t − X near , i , where k is the size of neighbourhood, N i is the set of the k nearest neighbours of agent i and C i is the local shoal centroid of agent i and X is the separation vector from nearest neighbour of the agent i to avoid congestion, β is the weight of shoaling attraction, γ is the weight of repulsion (separation), δ s is the weight of random perturbation and R is a vector of random numbers sampled from N ( 0 , 1 ). S i prevents zebrafish from clumping on each other β, γ and δ s are parameters used to define attraction, repulsion and random perturbation weight, respectively. Shoaling behaviour reflects the shift from exploration or exploitation behaviour to cohesion behaviour of the swarm.

After identifying areas with significant prey abundance, zebrafish begin to hunt by moving towards those areas in a directed manner. During this hunting process, zebrafish can use both local and global information to help improve their potential prey location.

D g = Z best − Z i is the global direction vector in (8) toward population best position Z best and D l = Z lbest − Z i is the local direction vector toward local best position. (8) D g = Z best − Z i , D l = Z lbest − Z i . These two hunting directions will be combined to form the hunting direction: (9) D h = w g D g + ( 1 − w g ) D l . As given in (9), D h represents the holistic hunting direction and w is the adaptive global influence weight within range w g [ 0 , 1 ].

The global influence weight w g is defined in (10), where τ = t T is the normalised iteration progress. When τ = 0 (early iterations), w g = 0.5 global and local guidance contribute equally to D h , where D h = 0.5 D g + 0.5 D l , maintaining balance between local and global search. As τ = 1 (late iterations), w g = 1.0, means D h = w g D g + ( 1 − w g ) D l . D l = 1.0 D g + 0 D l = D g . And hence D g , the direction is then dictated totally by the global best. If τ = 0 (early iterations), then w g = 0.5 and so the global and local directions equally contribute to the initial exploration: (10) w g = 0.5 + 0.5 t T .

The new hunting direction of zebrafish will be updated according to (11), where η is the hunting step scale factor, ξ is the hunting perturbation weight and R is a random vector drawn from N ( 0 , 1 ). The term ( 1 − τ ) makes the hunting step get smaller with each iteration, enabling fine-grained exploitation at later iterations: (11) Z i t + 1 = Z i t + η ( 1 + ϕ i ) ( 1 − τ ) D h + ξ R .

This feature enables zebrafish to exploit promising regions more thoroughly in later iterations while retaining mobility for lower-fitness individuals.

The natural behaviour of zebrafish also exhibits a periodicity in their hunting patterns. ZFO implements a mechanism that forces a zebra to reperform line searches on areas that are deemed to have experienced more than a fixed number of unsuccessful hunting attempts. (12) Z i t + 1 = L B + R ⊙ ( U B − L B ) , where L B and U B are the lower and upper bound vectors of the search space in (2), R ∼ U ( 0 , 1 ) is a vector of uniformly distributed random numbers and ⊙ denotes element-wise (Hadamard) multiplication. In (12) presents the vectorized form of initialization. This type of adaptive reset allows the algorithm the opportunity to recover from getting stuck in a local solution (e.g. local search) and continue its effective long-term search. An adaptive stagnation threshold S t h = max ( 3 , round ( 0.05 × T ) ), where T is the number of iterations. It refers to the number of unsuccessful search steps allowed consecutively before reinitialization. It is scaled dynamically according to optimization horizon and bounded from below by 3. This condition for reset is designed such that it triggers early enough for detecting premature stagnation in short time periods and sustained enough for detection of stagnation in long time periods before reset.

Exploration, shoaling and hunting are selected according to a probability-related method which relates to the interdependence of an individual agent’s fitness at a specific time and the agent’s progress through the optimization process: (13) P explore = 1 − τ + 0.5 ϕ i , (14) P hunt = τ + 0.3 ϕ i , (15) P shoal = 1 − ( P explore + P hunt ) , where τ = t T is the normalised iteration progress and φ i is the individual fairness value defined by (13), (14) and (15). P explore are the respective probabilities for entry into exploration, P hunt is the hunting and remaining shoaling state P shoal , which is normalized to unity before making state selection. The value of P explore is monotonically decreasing with progress, which implies that later iterations favour the exploration state. On the other hand, P hunt increases with both τ and φ i , so the high-fitness agent in later iteration has the biggest probability of directed hunting. During each iteration, the above three states probabilities are separately sampled by every agent to make the behaviour choices and generate a heterogeneous population that exploration, shoaling and hunting exist at the same time among the swarm. This individual-level adaptation strategy of phase-switching provides ZFO to maintain an everlasting compromise between global search and local exploitation during the process.

The performance of the Zebrafish Optimization Algorithm (ZFO) was compared against four of the most recently developed metaheuristic algorithms (CUTTLE, JS, POA and KH) across seven unimodal benchmark functions (F1–F7). Table 2 shows that the algorithms were tested under exactly the same conditions (50 separate trials with 500 iterations) and Table 3 shows the algorithms were tested under exactly the same conditions (50 separate trials with 1000 iterations) and the best, average and standard deviation (STD) of each algorithm’s fitness score were captured. Figures 6 and 7 show the convergence behaviour of the algorithms (50 separate trials with 500 iterations and 50 separate trials with 1000 iterations) compared on the unimodal benchmark functions, which is shown along with their optimization dynamics across the iteration process.

ZFO consistently outperforms the other tested algorithms on all unimodal test functions. Specifically, ZFO has lower mean fitness values and lower standard deviation values, indicating excellent exploitation and convergence behaviour. ZFO’s low variability (i.e. low variance) in independent experimental runs also suggests that it can be relied upon to produce similar results regardless of the run.

For simpler unimodal functions like F1, F2 and F7, ZFO quickly approaches the global optimum and generates more accurate solutions than those created by CUTTLE, JS, POA and KH. The ZFO Algorithm’s strength lies in its ability to perform intensive exploration of the more promising areas of the search space to find these points. The competing algorithms show higher mean errors and a greater spread, indicating they will take longer to converge and are more likely to find fewer optimum paths during their search process.

Additionally, for higher dimensional unimodal problems, ZFO continues to outperform the alternatives through its ability to honour the trade-off between encouraging movement toward the “optimal” solution while avoiding premature stagnation. The results for POA and CUTTLE were significantly poorer than ZFO, demonstrating relatively large mean fitness values and high standard deviations, indicating that they are less effective at exploiting the available resources within unimodal landscapes.

The extensive experiments with unimodal benchmark functions, the performance gained from using ZFO has been validated and is extremely effective, accurate and robust. ZFO is an extremely effective optimizer in terms of utilising its extreme capability for finding a singular, global optimum and is especially beneficial for solving optimization issues, including those with a singular optimizer.

A comparison between the ZFO algorithm and four other stochastic, multi-dimensional optimization methods is presented in Table 4 and 5. All methods were applied in three levels of testing, consisting of the recommended number of trials (50 with 500 iterations and 50 with 1000 iterations). The trial results were compiled into tables including the best trial results, average results and standard deviations for all trial results obtained for each level of testing. Additionally, Figs. 8 and 9 show the convergence behaviour of the algorithms (50 separate trials with 500 iterations and 50 separate trials with 1000 iterations) compared on the multimodal benchmark functions, which is shown along with their optimization dynamics across the iteration process.

Data was collected across all run sizes and iteration limits to develop the ZFO algorithm’s ability to do better than, or at least as well as the benchmark algorithms. The results indicate that ZFO has superior exploration capability in the early stages of the search process and gradually increases its ability to exploit as the number of iterations increases. Through this process, ZFO is able to traverse multi-modal optimally and prevents a premature fall back into local optima.

As the number of runs increasing from 25 to 50, the performance of ZFO is more consistent with lower values of standard deviation over all multimodal functions. These results demonstrate the robustness and reliability of the proposed algorithm when subjected to diverse stochasticity. On the other hand, other proposed algorithms, such as POA and KH demonstrate a high degree of performance variability, indicating that they are more susceptible to differences in both initialization and search randomness.

Additionally, by increasing the number of iterations from 500 to 1000 iterations, a greater number of iterations will allow ZFO to refine the quality of its solutions beyond their current mean fitness value without compromising the stability of their convergence rates. The ability of the algorithm to scale up in terms of both the number of runs and the number of iterations demonstrates that the algorithm can effectively utilize more computer resources to improve the accuracy of its optimization results.

The results of the experiments conducted in multiple runs, iterations and other configurations validate the capability, dependability and capacity of ZFO to solve multimodal optimization problems. These experimental configurations provide a consistent performance improvement across all trial types and substantiate the validity of ZFO as an appropriate tool for dealing with complex, high dimensional optimization problems that are characterized by multiple local optima.

The performance of ZFO is compared with CUTTLE, JS, POA and KH by evaluating each algorithm when solving benchmark functions that have several variables and are multimodal. These problems were carefully designed so that each has a fixed number of decision variables and represent very complex optimization problems because many of the variables have many local minima and they require very different exploitation strategies than traditional optimization problems. They provide an excellent means for testing an algorithm’s robustness, exploration capability and convergence properties.

To allow for a fair and statistically significant comparison between the algorithms through multiple experiments, the following factors were taken into account: the number of independent runs per algorithm. Table 6 and 7 show the algorithms were tested under exactly the same conditions (50 separate trials with 500 iterations and 50 separate trials with 1000 iterations). After completing the experiments for each of these variations, the algorithms’ real-time fitness values were analysed based on: best fitness value, mean fitness value and standard deviation (STD) for each configuration. Additionally, Figs. 10 and 11 show the convergence behaviour of the algorithms (50 separate trials with 500 iterations and 50 separate trials with 1000 iterations) compared on the fixed dimensional multimodal benchmark functions, which is shown along with their optimization dynamics across the iteration process of 50. Each independent run was completed with 500 iterations or 1000 iterations.

ZFO performed better than any other algorithm for the entire fixed-dimensional multimodal function suite. More importantly, ZFO is consistently able to avoid the pitfalls of local optima while maintaining a diverse population. This is evidenced by ZFO’s lower and more consistent average fitness over repeated runs. Therefore, ZFO has effectively found a good balance between exploration and exploitation when operating in a multi-modal environment.

An increased run count of 50 demonstrated improved stability of ZFO as indicated by consistently lower standard deviation and it shows greater stability in the presence of stochastic variation. ZFO can further enhance the quality of the solution when run over 500–1000 iterations, improving convergence accuracy without negatively impacting stability.

In the search phase, POA and KH often substitute a restricted number of search patterns for a variety of fixed-dimensional multimodal functions, which causes premature convergence. On the other hand, compared to the other competing algorithms, the overall performance of CUTTLE and JS is more variable. In addition, evaluation shows that the new algorithm ZFO is better able to maintain diversities in the search process and provide steady convergence to high-quality solutions.

Overall, the experimentation has demonstrated that ZFO is a high-performance method for Fixed-Dimension Multimodal Optimization Problems due to its superior robustness, scalability and convergence reliability and that the increases in performance seen across various run sizes and iteration limits support ZFO’s efficacy with respect to the solution of difficult complex optimization problems characterized by rugged and deceptive search spaces.

The Friedman test was conducted independently for each of the benchmark functions to test the algorithms’ abilities. The Friedman test was performed on three sets of data – the results of 25, 30 and 50 independent trials with 500 and 1000 iterations and a significance level of α = 0.05. The Friedman test provided mean rankings for the algorithms and the p-value associated with the mean. The results of the Friedman test showed ZFO had the lowest (and best) ranking among most unimodal and multimodal functions and fixed-dimensional multimodal functions. Additionally, the p-values generated by the Friedman test for many of the test functions indicated that the differences between the algorithms’ performance were statistically significant, thus indicating no random variability would account for the observed differences in performance.

The results of the Wilcoxon signed-rank test demonstrate that there are statistically significant differences in performance between the ZFO algorithm and four other competing algorithms for each of the twenty three benchmark functions when using the best fitness values across independent runs of the ZFO and each of the competing algorithms. To complete the statistical test, the results obtained from each of the four competing algorithms were treated as a “reference” algorithm against which to compare ZFO. Before performing the Wilcoxon signed-rank test, all comparisons with missing values and identical fitness values were excluded to ensure the validity of the comparisons performed. The results obtained from conducting the Wilcoxon signed-rank test resulted in the conclusion that ZFO outperformed the competing algorithms in a statistically significant manner for most benchmark functions. In two specific cases, multimodal and fixed-dimension multimodal problems, the results of the Wilcoxon signed-rank test confirm that the statistically significant performance improvements exhibited by the ZFO algorithm are due to its consistently superior performance rather than due to any form of stochastic variation of performance. Table 8 summarizes the p-value result for the Wilcoxon signed-rank test for statistically significant differences of performance between the algorithms.Across all the benchmark functions a trend was observed ( p < 0.05 ) on statistically significance of the ZFO performance.

This overall ranking considers all of the functions possible within a benchmark configuration, using averaged Friedman rankings to see how the different configurations stack up against one another based on the results of the global performance evaluation. All benchmarks, as expected, performed extremely well, with ZFO finishing at the top of the list each time. The results from the experimental configurations provided conclusive evidence that the overall performance of ZFO shows high levels of robustness in varying sizes of evaluations and budgets of evaluations (or iteration budgets) while allowing for generalising across other types of optimization landscapes. Averages of the Friedman rankings and overall ranks for all algorithms with different runs and iterations show that ZFO is consistently achieving the lowest average rank and demonstrating that it has superior and robust performance overall in Table 9.

For unimodal F1–F7 (with single global and no local optimum) performance is driven by Hunting Phase (Section 2.3.3). Update (11) includes global (to population best) and local (to neighbours best) directional vector, modulated by time-dependent factor τ = t T . When small local exploration dominates, to keep population coverage wide, whilst as grows global direction dominates and it increasingly targets the optimum. This temporal dependency fits well with convex, gradient-consistent landscapes such as F1 (Sphere), F2 (Schwefel 2.22) and F4 (Rosenbrock).

This Biological Inspiration directly relates to leaderless group predation observed in zebrafish; the intensity of the attack increases with the group as it closes in on the prey target. In contrast, the cuttlefish (CFO) use their camouflage-based search operator with a static, iteration independent, pull toward the best and pufferfish (POA) with their puffing mechanism use a radius search that increases equally regardless of distance to optimum. CFO and POA are unable to adapt directional search influence and therefore present higher mean values and a larger spread on F1–F7 due to the difference mechanism described. Fairness factor ϕ i (5) is also directly inspired by leaderless, individual-centric, foraging behaviour: we make a direct connection to zebrafish schooling where less capable individuals are allowed greater exploratory ranges and this is achieved at each hunting step, where weak agents (lower fitness) receive disproportionately larger hunting steps than strong (higher fitness) agents due to the fairness factor ϕ i .

ZFO is able to outperform on F8–F13 where, due to the multitude of local optima, the two working principles are responsible for the gain: the Fairness-Weighted Exploration Phase (Section 2.3.1) and the Stagnation-Aware Reinitialization (Section 2.3.4). The adaptable step t (4) is controlled by max and min and it remains sufficiently large to maintain spread of population in initial iterations, where the risk of being attracted by local optima is the highest, whilst the fairness coefficient ϕ i enables individuals trapped around a suboptimal local optimum to have larger exploratory movements in proportion to the severity of their trapping and therefore be capable of being kicked out of the basins of attraction. When they fail over iterations of escape, the stagnation count up forces a complete reinitialization to another random point (12), thus forcibly ensuring diversity at the individual level. It is the combination of this two-tiered system of step amplification, which accounts for the unfairest of being trapped around poor local optima and reinitialization after a defined threshold, that is key to ZFO being able to find high-quality solutions in difficult functions such as F8 (Schwefel 2.26), F9 (Rastrigin) and F12 (Penalised).

The fairness coefficient ϕ i serves the biologically inspired element, directly drawn from the concept of equal access foraging observed in zebrafish where the worse-off individuals exhibit a larger relative movement range than fitter individuals within the shoal, a component not present in jellyfish (JSO) which passively moves with currents and employs a dynamic, population-based state-transition between drifting and foraging or krill (KHO), which uses diffusive, random walks and has no agent-specific state of stagnation. The reinitialization based on the stagnation count up is inspired by the empirical observation of zebrafish abandoning unfavourable foraging sites and redistributing itself and this is again an agent-specific change in state, unlike the two algorithms. It is the lack of these two components which is the cause for the higher mean fitness values and variance produced by JSO and KHO for F8–F13.

For highly non-linear, deceptive, multi-modal fixed-dimension benchmark functions F14–F23, it is the Shoaling Phase (Section 2.3.2) which predominantly drives performance. The shoaling update (7) drives each individual towards the centroid of its k closest neighbours, as opposed to towards global best, facilitating rich neighbourhood-level information exchange and allowing the population to simultaneously explore and converge on distinct promising sub-regions. Minimum distance between individuals is encouraged through the separation parameter γ ( S i in 7), preventing population collapse and keeping distinct basins of attraction explored simultaneously. The stochastic perturbation parameter δ s (R in (7)) helps maintain local diversity in each neighbourhood and allows individuals to move through the sharp ridges found in F14 (Langermann), F16 (Six-Hump Camelback) and F18 (Goldstein-Price).

The neighbourhood-centroid based shoaling update is biologically inspired by the very real active spatial regrouping based on proximity in shoals of zebrafish, which continually reform their position based on local cues and shared gradient information. It is distinctly different from the passive, drift-driven positioning of jellyfish (JSO) and diffusion-based herding of krill (KHO). In the case of the zebrafish shoaling mechanism, γ and δ s contribute to its ability to efficiently navigate sharp, tricky terrain which is strongly linked to its unique neighbourhood-centroid style update combined with intelligent control of both separation between individuals and internal diversity. CFO’s performed inconsistent behaviour across these specific F14–F23 problems, as compared to JSO, is that neither these specific kinds of spatial search mechanics nor active local grouping behaviours are present in CFO and hence CFO’s performance has the largest variance of all.

The most unique parameter among the 4 comparison algorithms and ZFO is the adaptive behavioural transition strategy (13)–(15). The adaptive behavioural transition strategy assigned each agent with their own, fitness rank-dependent probability of moving in the direction of exploration, shoaling or hunting at each iteration. This strategy of assigning individual, phase-switching probabilities depended on the interplay of the normalized iteration progress τ = t T , which skewed the population toward hunting over exploration as optimization progressed, and the individual fairness factor ϕ i , which modified the individual agent’s personal phase probabilities based on their individual fitness rank. The higher is an agent’s fitness rank, the greater its P hunt, the greater its probability of exploiting the environment intensely, whereas low rank promoted greater P explore. This effectively caused each iteration to maintain a heterogeneous population stratified by fitness level and simultaneous exploration, shoaling and hunting occur within the swarm at varying times. This corresponds to the simultaneous occurrence of these three states at all times in a school of zebrafish.

The probabilities of phase transition are predicated on an empirically proven fitness-dependent zebrafish phase switching behaviour; individuals of low fitness will behave in an explorative or dispersive manner while those with high fitness will maintain the directed hunting behaviour. ZFO is the only model that incorporates an individual, fitness-dependent phase-switching mechanism. CFO has one visibility function-based operator, JSO has a population wide, time-dependent probability, POA utilizes a threshold-based mode switching between attraction and puffing and KHO is designed around static, pre-set weights on three distinct movement components without an individual assigned state.

α m a x , α m i n which govern the step range of the exploratory movement, β (shoaling attraction parameter), δ h (hunting step size) and the stagnation parameter S t h (controls the rate of reinitialization) are the parameter combination that characterizes the time and spatial profile of search by ZFO. When combined, these parameters determine the temporal/individual characteristic of search performed by ZFO and are confirmed to yield stable and effective search within all three categories of benchmarks without function-specific parameter adjustment, suggesting generality of the biologically-motivated design.

A summary of the phase-to-benchmark mapping as defined in this work, listing the core driver for each benchmark class, biological inspiration, underlying ZFO mechanism, dominant parameter and the relevant algorithmic shortcoming of comparison algorithms for each benchmark class.

Unimodal (F1-F7): Dominant driver Hunting Phase (11); Biological basis zebrafish leaderless time-intensified predation; Dominant parameter time-adaptive and fairness i and the reason for competitors fail is that CFO and POA utilize iteration-independent static attraction that is incapable of increasing directionality over time. Multimodal (F8–F13): Dominant driver Exploration Phase (4), Stagnation Reinitialization (12) zebrafish fairness-weighted dispersal and patch-abandonment and the Dominant parameters – α max , α min , ϕ i , stagnation S t h = max ( 3 , round ( 0.05 × T ) ) where T is the number of iterations. It refers to the number of unsuccessful search steps allowed consecutively before reinitialization. It is scaled dynamically according to the optimization horizon and bounded from below by 3. This condition for reset is designed such that it triggers early enough for detecting premature stagnation in short time periods and sustained enough for detection of stagnation in long time periods before reset and the reason of competitor fails is that JSO and KHO are devoid of individual level stagnation detection and threshold based reinitialization. Fixed-dim Multimodal (F14–F23): Dominant driver Shoaling Phase (7); zebrafish proximity-driven fluid spatial regrouping; Dominant parameters β (attraction), γ (separation), δ s (perturbation), k (neighbourhood size) and the reason for competitors fail is that CFO and JSO are devoid of structured neighbourhood-centroid movement including separation and perturbation.

In conclusion, the superior results demonstrated by ZFO cannot be attributed merely to favourable average benchmark statistics but are mechanistically explained by three distinct behavioural phases, each designed to address specific structural characteristics of optimization landscapes. The fairness factor, inspired by the egalitarian foraging behaviour of zebrafish and embedded across all behavioural phases and transition mechanisms acts as an overarching adaptive control mechanism at the individual-agent level. This mechanism helps overcome the limitations of CFO, JSO, POA and KHO, particularly in diversity preservation, stagnation avoidance and adaptive search regulation.