Modelling the effect of motivation on mental health components with fuzzy logic among elite athletes

Abstract

Elite athletes operate in high-stress, feedback-rich environments where mental health risk is shaped by motivation and training climate. This study models the joint effects of intrinsic motivation, psychological safety, and mental well-being on anxiety, depression, athlete-specific strain, and burnout using an interpretable fuzzy-logic framework alongside conventional regression. A sample of 247 athletes completed validated measures (SMS-6 Intrinsic Motivation, Psychological Safety, SWEMWBS, GAD-7, PHQ-9, APSQ, and BMS). After pre-processing and standard analyses, we constructed a Mamdani-type Fuzzy Inference System. To accurately represent the non-linear transitions between psychological states, the model specifies trapezoidal membership functions for boundary linguistic variables and triangular membership functions for intermediate categories. The system utilises a transparent rule base (primary risk from low IM; protection from PS/MWB; synergistic protection when both are high), min–max aggregation, and centroid defuzzification. Rule weights and breakpoints were calibrated against observed score distributions to minimise mean absolute error (MAE). Multiple regressions indicated that PS and MWB independently predicted lower risk across all outcomes; however, IM emerged as a significant positive predictor of depression and anxiety (p < 0.05). Further diagnostics confirmed this as a suppression effect, with all VIF values < 1.5. Visual and quantitative analyses confirmed three regularities: (i) a primary risk gradient when IM, PS, and MWB are low; (ii) buffering as PS or MWB increase; and (iii) a low-risk “basin” when PS and MWB are jointly high. Comparative metrics (MAE/RMSE) showed that the FIS model offers superior predictive accuracy and interpretability compared to standard linear approaches.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-39718-7.

Introduction

Elite sport is a demanding laboratory for human performance, where physiological limits¹, decision-making under pressure² and complex social dynamics³ converge. It is also a domain where mental health can be both a foundation and a fault line⁷. Athletes negotiate selection pressures⁴, injuries⁵, role ambiguity⁶, and public scrutiny⁸, encountering periods of anxiety⁹, low mood⁴, strain⁷ and burnout¹⁰ alongside moments of thriving. Recent evidence suggests that these psychological states are closely linked to the physical training environment, where optimising variables such as player numbers and training bout durations can directly impact mood balance and technical performance¹¹. Capturing this dual reality requires concepts from sport psychology, such as motivation and well-being^12–14, as well as tools from applied mathematics, including formal representations of uncertainty, rule-based reasoning, and predictive modelling^15–17. The present study deliberately positions itself at this intersection. It poses a psychological question with mathematical discipline: how do motivation and the training climate combine to shape mental health risk, and how can we model that process in a way that is both accurate and interpretable?

The psychological core of our argument is simple: the quality of motivation matters. Intrinsic motivation—training and competing for inherent interest, enjoyment or mastery—tends to support emotion regulation, persistence and a sense of purpose. Motivation is a fundamental psychological construct as it dictates the direction, strength and persistence of human behaviour¹². While identified regulation may support the initial adoption of an activity, intrinsic motivation is uniquely predictive of long-term adherence and persistence in physical activity¹³. Furthermore, this internal drive has been shown to predict an athlete’s perceived performance and their subsequent intention to remain physically active¹⁸. Research indicates that such autonomous states are closely linked to emotional intelligence, specifically supporting a person’s ability to attend to, clarify and regulate their emotions¹⁹. Because it is rooted in personal goals and inherent interest, intrinsic motivation leads to a more profound and sustained impact on continuous improvement and skill mastery²⁰. These self-determined processes ensure a positive alignment between psychosocial determinants, which is essential for maintaining sustainable exercise practices and overall wellness²¹. High-quality motivation is also positively associated with dispositional flow and serves as a critical protective factor against the risk of athlete burnout²². Ultimately, supporting basic psychological needs for autonomy, competence, and relatedness facilitates the internalisation process, allowing individuals to engage in training for their own satisfaction rather than external pressure²³.

When motivation becomes contingent on external approval or on fragile self-worth, effort may persist but become more brittle, with greater susceptibility to exhaustion and dysphoria^10,24–26. However, motivation does not act alone. Psychological safety, introduced by Edmondson²⁷, refers to the shared belief that one can speak up, make mistakes, and seek help without ridicule. This condition determines whether athletes disclose difficulties, mobilise support, and take learning risks^28–30. Well-being reflects positive functioning and balance, enabling athletes to manage stress effectively, recover quickly, and maintain a healthy perspective^31,32. These three elements form a coherent psychological system: Morris et al.³³ review evidence that dysfunction/low intrinsic motivation is associated with greater vulnerability to psychopathology (e.g., anhedonia, depression), though causal mechanisms remain to be established; psychological safety and well-being buffer that vulnerability^4,34; together, psychological safety and well-being can offer synergistic protection⁷.

From an applied mathematics perspective, this system invites a particular kind of modelling. The constructs are continuous, noisy, and context-dependent³⁵. They resist crisp thresholds and binary classification³⁶. Traditional linear models are valuable for estimating average effects but can struggle with non-linear interactions and local context³⁷. Highly flexible machine-learning methods capture complexity but often do so at the cost of transparency, making it difficult for practitioners to see “why” a recommendation is produced^38,39. This is particularly evident in the recent proliferation of deep learning and complex AI architectures in sports analysis, which, despite their predictive power, often function as opaque ‘black boxes’ that limit clinical and practical implementation⁴⁰. As highlighted in recent systematic reviews, while wearable technology and motion analysis benefit from high-dimensional AI models, the transition toward injury prevention and athlete monitoring requires models that maintain high interpretability to be clinically actionable⁴¹. Fuzzy logic offers a principled middle ground. It operationalises linguistic knowledge—if motivation, safety, and well-being are low, then anxiety is high—within a rigorous calculus of partial set membership. Instead of forcing an athlete into a single category, the model represents graded belonging to terms such as low, medium and high via membership functions. Inference proceeds through a transparent rule base, and the resulting fuzzy states are defuzzified into actionable indices for anxiety, depression, strain and burnout.

Methodologically, this approach has several advantages that resonate with both audiences. For sport psychologists, the rules mirror professional reasoning and can be examined, debated and refined; they also make explicit the protective and synergistic roles of psychological safety and well-being. For applied mathematicians, the framework specifies the full mapping from inputs to outputs: membership function families (trapezoidal), normalisation to a shared metric, a Mamdani inference engine with min–max operators, and centroid defuzzification⁴². The system’s parameters (e.g. rule weights and membership/breakpoint settings) can be calibrated against empirical data [see 43, 44]. While these calibrations influence predictive performance, framing them explicitly as a bias–variance trade-off is a modelling choice. Importantly, interpretable, rule-based/fuzzy models are intended to be auditable and not ‘black boxes’ — an argument strongly advocated by Rudin³⁸.

This study provides a structured approach to translating clinical intuition into computational rules, enabling finer distinctions than categorical cut-offs allow. It offers a real-world application where linguistic uncertainty is addressed through fuzzy membership functions, measurements are imperfect yet meaningful, and model outputs must be explainable to non-technical stakeholders. While standard FIS primarily handles semantic vagueness, the model’s structure enables a more nuanced representation of athletes’ states than traditional binary systems. Furthermore, the study illustrates a reproducible pipeline—pre-processing, normalisation, membership design, rule specification, inference configuration, calibration, and validation—that others can adapt to different psychological domains or performance contexts.

Athletes and support teams require routine, interpretable monitoring to detect early signs of burnout — for example, rising anxiety despite stable performance or progressive declines in motivation — and to identify likely contributing factors (e.g., chronic stress, need-thwarting environments). This recommendation is supported by recent syntheses that show robust links between burnout and adverse mental health outcomes⁴⁵ and by calls for early, system-level onboarding and mental health surveillance during transitions into elite sport⁴. By embedding psychological theory in a mathematically explicit structure, we aim to produce indicators that are sensitive enough to guide early intervention and transparent enough to foster trust. The approach respects the lived complexity of athletes while satisfying the rigour expected in quantitative modelling.

The study weaves together two strands—sport psychology’s focus on the human experience of striving and coping, and applied mathematics’ capacity to formalise uncertain knowledge—into a single fabric. It proposes that low intrinsic motivation acts as a primary driver of risk, that psychological safety and well-being provide protective and synergistic counterweights, and that a fuzzy-logic system can capture these relations with both nuance and clarity.

Aims and hypotheses

This study aims to model how intrinsic motivation, psychological safety, and well-being jointly influence athletes’ mental health outcomes—namely, anxiety, depression, strain, and burnout—using an interpretable fuzzy-logic framework alongside conventional analyses. Specifically, we seek to quantify the primary risk associated with low intrinsic motivation, estimate the protective and synergistic buffering effects of psychological safety and well-being, and deliver calibrated, practitioner-ready risk indices.

Hypothesis 1:

Low levels of Intrinsic motivation in athletes lead to increased levels of anxiety, depression, strain, and burnout.

Hypothesis 2:

When psychological safety and well-being levels are high, the negative impact of low Intrinsic motivation on anxiety, depression, strain, and burnout is reduced (a protective buffering effect).

Hypothesis 3:

When both psychological safety and well-being are high, the protective effect is synergistically strengthened, providing a greater protective effect than the sum of their individual contributions.

Materials and methods

Participants

We recruited 247 athletes for the study (X̅_age ± SD = 22.25 ± 6.10 years). Of the participants, 37.2% were female (n = 92), 62.3% were male (n = 154), and 0.4% (n = 1) chose not to disclose their gender. The sample included athletes from various sports: 19.0% (n = 47) football, 14.6% (n = 36) American football, 13.8% (n = 34) judo, 8.1% (n = 20) volleyball, 8.1% (n = 20) hockey (including cricket), and 6.1% (n = 15) gymnastics. Regarding competition level, 47.0% (n = 116) were professional athletes, while 53.0% (n = 131) were amateurs. A total of 25.5% (n = 63) of the participants identified as national athletes, defined as individuals who have competed at least once for their national teams. The average sports experience was X̅ = 8.34 ± sd = 5.03 years. The mean training frequency was X̅_day ± sd = 4.35 ± 1.61 days per week, with an average daily training duration of X̅_duration ± sd = 2.55 ± 1.30 h. The average duration of working with the same coach was X̅_coach ± sd = 3.87 ± 3.81 years, and the average duration of being in the same team was X̅_team ± sd = 4.05 ± 3.77 years. Regarding mental health support utilisation, 8.5% (n = 21) of participants reported receiving mental health support within the past year, while 91.5% (n = 226) reported no such support during this period. When asked about lifetime mental health support utilisation, 18.2% (n = 45) indicated they had ever received mental health support, whereas 81.8% (n = 202) reported never having accessed such services.

Measurement tools

Following the recommendations of Schinke et al.⁴⁶, our mental health screening encompassed both symptoms of mental illness and mental well-being. Specifically, athletes reported on Intrinsic motivation using the Intrinsic Motivation subscale of the Sport Motivation Scale-6 (SMS-6), developed by⁴⁷ and adapted into Turkish by⁴⁸. Higher scores indicate higher motivation. Psychological safety was assessed using the scale developed by Edmondson²⁶. (Turkish adaptation by⁴⁹; generalized anxiety symptoms using the GAD-7 (Generalized Anxiety Disorder developed by^4,50 and Turkish adaptation by⁵¹ with standard cutoffs of 0–4 (minimal), 5–9 (mild), 10–14 (moderate), and 15–21 (severe); depressive symptoms using the PHQ-9 (Patient Health Questionnaire developed by⁵²; Turkish adaptation by⁵³, scored 0–3 per item and summed to a total between 0 and 27 with categories 0–4 (none/minimal), 5–9 (mild), 10–14 (moderate), 15–19 (moderately severe), and 20–27 (severe); mental well-being using the SWEMWBS (Short Warwick-Edinburgh Mental Well-being Scale developed by⁵⁴; Turkish adaptation by⁵⁵; athlete-specific distress using the APSQ (Athlete Psychological Strain Questionnaire developed by⁵⁶; Turkish adaptation by⁵⁷, including Self-Regulation, Performance, External Coping, and Total scores, with APSQ severity ranges of 15–16 (moderate), 17–19 (high), and 20+ (severe); and Burnout Measure-Short Version (BMS developed by⁵⁸; Turkish adaptation by⁵⁹. All instruments are widely used and have demonstrated validity and reliability in athlete populations^33,60,61 except for BMS. We ran CFA to validate the structure of the measurement in a sport context, and the results showed acceptable fit with the data [χ² (33) = 70.988, χ²/df = 2.15, CFI = 0.97, TLI = 0.97, RMSEA = 0.06 (95% C.I.: 0.04–0.09), SRMR = 0.03, factor loadings were between 0.49 and 0.90].

Data collection process

After obtaining institutional ethical approval (decision number: 250115/11, date: 24 October 2025), we recruited athletes through multi-sport clubs, university teams, and federations, inviting those aged 18 and over who had trained and competed within the previous six months and could complete the Turkish self-report measures. Participants received an information sheet outlining the aims, voluntariness, confidentiality, and data use, and then provided their electronic consent. The questionnaire (approximately 10–15 min) followed a fixed order, covering demographics/sport history, Intrinsic Motivation (SMS-4), Psychological Safety, GAD-7, PHQ-9, SWEMWBS, APSQ, and BMS, and was scored according to validated guidelines. Core items employed mandatory responses and range checks; attention checks and response-time flags supported data quality.

Analysis

All statistical analyses and preliminary data screening were conducted using SPSS. After missing-value checks, outlier diagnostics and the inclusion of relevant covariates (age, sex, sport, training frequency and duration), multiple linear regressions were run to quantify the direct contribution of intrinsic motivation and the two protective factors—psychological safety and mental well-being—on anxiety, depression, athlete-specific strain, and burnout. These classical models served a dual purpose: they provided effect-size estimates under standard assumptions and yielded empirical distributions against which the fuzzy model could later be calibrated. To capture the uncertainty and graduations that are inevitably lost in crisp cut-offs, a Mamdani-type Fuzzy Inference System (FIS) was developed in MATLAB (version R2021a) by using Fuzzy Logic Designer. Trapezoidal membership functions—labelled low, medium and high—were defined for each input: Intrinsic Motivation, Psychological Safety and Well-being. The four outputs—Anxiety, Depression, Strain and Burnout—were likewise expressed on suggested guidelines where high values denote poorer mental-health status.

During fuzzification, raw scale scores (e.g., a PHQ-9 of 11 or a GAD-7 of 8) are translated into graded memberships across low/medium/high categories, revealing the partial truths that crisp diagnostics overlook. Inference proceeds through the rule base, generating a set of fuzzy conclusions for each outcome. Defuzzification employs the centroid-of-area method, returning a single risk estimate between suggested thresholds that can be interpreted on the same footing as the underlying clinical scales.

By integrating conventional regression with a calibrated fuzzy rule base, the analysis respects both the quantitative rigour of statistical modelling and the qualitative nuance of human psychological variability. Such a hybrid approach offers practitioners a flexible decision aid: raw scores can still be interpreted in the usual categorical fashion, while fuzzy outputs provide a graded, holistic risk profile sensitive to small but clinically meaningful shifts in motivation, safety, and well-being.

The scientific programme follows a hybrid pathway. First, we characterise mental-health and well-being profiles in a multi-sport cohort of elite athletes. Secondly, we estimate conventional multiple regressions to quantify the unique and combined contributions of intrinsic motivation, psychological safety and well-being to each outcome. These analyses provide effect-size benchmarks and identify linear structure in the data. Thirdly, we construct a fuzzy inference system (FIS) that encodes the theory-led rule base: low intrinsic motivation functions as a primary risk signal; psychological safety and well-being serve as protective signals; their coincidence yields the strongest buffering. Membership functions are defined for low/medium/high categories using a hybrid trapezoidal configuration. The FIS was calibrated using an iterative heuristic optimisation process; rule weights and membership parameters were systematically adjusted to minimise the Mean Absolute Error (MAE) with respect to the observed distributions in the training dataset. This data-driven approach ensures that the model’s mathematical surface is objectively grounded in the empirical data while strictly retaining the theoretical semantics of the expert-defined rules.

This integration delivers three contributions. Conceptually, it reframes intrinsic motivation as a stability parameter in a dynamical environment: when this parameter is low, small perturbations—such as illness, selection disappointment, or online criticism—can push the system towards higher-risk states. Conversely, when psychological safety and well-being are high, the system exhibits robustness, absorbing perturbations without loss of function. Mathematically, it demonstrates how fuzzy sets can encode psychological constructs that are inherently vague without compromising granularity, and how rule-based inference can embody expert knowledge in a form that allows for calibration and validation. Practically, it yields risk profiles that practitioners can read at a glance, with the accompanying “why” traceable through specific rules and memberships. A coach can see that raising psychological safety or enhancing well-being would, under current memberships, reduce predicted anxiety by a quantifiable margin; a psychologist can simulate the impact of targeted interventions on the combined risk landscape.

Fuzzy logic approach

In Zadeh’s fuzzy set theory⁶², each observation belongs to a concept to a partial degree, formalised as: µ_A: X → [0,1]. Membership functions translate linguistic terms, such as “low,” “medium,” and “high,” into numerical degrees of membership. Unlike classical (crisp) sets that assign 0/1 membership, fuzzy sets allow graded membership, thereby preserving information that would otherwise be lost at sharp cut-offs. Crisp sets are, in this sense, a limiting case within the fuzzy framework.

Intrinsic Motivation, Psychological Safety, Well-being (SWEMWBS), and the mental-health symptom measures (GAD-7, PHQ-9), athlete-specific strain (APSQ subscales and total), and burnout (BMS) were modelled with trapezoidal membership functions for categories such as “low–medium–high” (and “very low/very high” when warranted by score distributions). A generic trapezoidal form is:

• 0, for x < α.

• (x − α)/(β − α), for α ≤ x ≤ β.

• (γ-x)/(γ-β), for β ≤ x ≤ γ.

• 0, for x > γ.

Parameter settings were aligned with established score ranges and widely used clinical cut-offs (e.g., GAD-7 and PHQ-9 category thresholds; APSQ severity bands), ensuring that fuzzy categories correspond meaningfully to recognised interpretations of the instruments.

Figure 1 displays the general form of the FIS system. Decisions in the FIS are derived via a standardised set of IF–THEN rules. The rule base reflects domain knowledge specific to elite athletes, formalising the premise that lower intrinsic motivation is a primary risk factor, while psychological safety and well-being act as protective buffers. To ensure transparency and reproducibility, the core inferential logic is organised into a formal rule base (Table 1), where input variables are represented by their conceptual domains.

Fig. 1.

FIS system general form.

Table 1.

Standardised fuzzy rule base.

Rule ID	Intrinsic motivation	Psychological Safety	Well-being	Psychological Strain	Predicted Risk
R1	High	High	High	Low	Low
R2	Moderate	Low	Moderate	High	High
R3	Low	Moderate	Low	Moderate-High	Moderate-to-High
R4	High	Moderate	High	Moderate	Low-to-Moderate
R5	Low	Low	Low	High	Very High

These principles were instantiated separately for each outcome, with logical operators following standard Mamdani configurations: the minimum operator for AND (intersection) and the maximum operator for OR (union). Initial rule weights were set to 1 and subsequently refined through an iterative heuristic calibration process to minimise error metrics. Fuzzification maps psychometric scores to graded memberships, accounting for measurement error-related uncertainty. The inference mechanism aggregates these activations, and centroid defuzzification is employed to return a single, interpretable index for decision-making.

Figure 2 shows the FIS system used in the model tested in this study. The data obtained from participants were utilised. The dataset included three independent variables measuring participants’ levels of intrinsic motivation (4–20), psychological safety (13–35) and mental well-being (9–35), along with anxiety (0–21), depression (0–27), athlete strain (5–50), and burnout (10–70). All scales were derived from multi-item measurement tools with established validity in the literature. A Mamdani-type fuzzy inference system (FIS)was developed to predict psychological outcomes. This model has three input variables (intrinsic motivation, psychological safety, mental well-being) and four output variables (anxiety, depression, strain, burnout). Each output variable is divided into different levels based on the relevant clinical and psychometric literature:

Fig. 2.

FIS system for the given model.

Anxiety: 4 membership functions – Very Low, Low, Medium, High.

Depression: 5 membership functions – None/Minimal, Mild, Moderate, Moderate-Severe, Severe (adapted to PHQ-9 thresholds).

Strain: 4-point scale – Low, Moderate, High, Very High.

Burnout: 3-point scale – Low, Moderate, High.

The scales are defined as trapezoidal, taking into account data distributions and clinical thresholds.

The fuzzy rules used in the model were established in line with domain experts’ opinions and existing theoretical frameworks. Ten natural language rules were defined for each psychological outcome variable, and these rules were converted into 27 FIS rules to cover all possible combinations of input variables.

The Mamdani-type fuzzy inference system developed in this study was analysed separately for each psychological outcome variable (anxiety, depression, strain, burnout). Below, the model’s operation is explained in detail using the images obtained for the anxiety output.

Results

Table 2 presents a summary of bivariate correlations and descriptive statistics for athletes. The reliability coefficients (Cronbach’s α ranging from 0.80 to 0.93) indicate high internal consistency across all scales, reflecting stable measurement quality within this athlete sample. Intrinsic motivation (IM) exhibited a strong reliability coefficient (α = 0.83). It was positively associated with both Psychological Safety (r = 0.29, p < 0.01) and Mental Well-being (r = 0.42, p < 0.01), suggesting that intrinsically motivated athletes tend to perceive their training environments as psychologically secure and report higher levels of mental functioning.

Table 2.

Descriptive statistics, internal consistency, and correlations among study variables.

	Average ± sd	Skewness	Kurtosis	α	2	3	4	5	6	7
1. Intrinsic motivation	16.90 ± 3.21	−0.95	0.44	0.83	0.29**	0.42**	− 0.20**	− 0.16**	− 0.09	− 0.19**
2. Psychological Safety (PS)	27.19 ± 5.38	−0.40	−0.78	0.80	-	0.41**	− 0.51**	− 0.37**	− 0.34**	− 0.45**
3. Mental Wellbeing (MWB)	28.63 ± 5.26	−0.77	0.30	0.88		-	− 0.50**	− 0.56**	− 0.51**	− 0.54**
4. Athlete Strain (APSQ)	20.90 ± 8.07	0.78	0.45	0.87			-	0.65**	0.63**	0.64**
5. Generalised Anxiety (GAQ)	6.08 ± 5.17	0.59	−0.58	0.90				-	0.76**	0.73**
6. Depression (PHQ-9)	8.62 ± 6.27	0.58	−0.37	0.86					-	0.77**
7. Burnout (BMS)	25.51 ± 13.70	1.17	0.90	0.93						-

Notes: ^**p < 0.01, α: Cronbach’s alpha.

The correlations are theoretically coherent: higher Psychological Safety and Mental Well-being were moderately and negatively correlated with adverse outcomes—namely, Athlete Strain, Anxiety, Depression, and Burnout (r values ranging from − 0.34 to −0.56, p < 0.01). This pattern reinforces the interpretation of PS and MWB as protective constructs within the motivational–mental health system. In contrast, the positive and significant associations among the outcome variables (APSQ, GAD-7, PHQ-9, and BMS; r = 0.63–0.77, p < 0.01) reflect the expected comorbidity and overlap among strain dimensions in elite sport contexts.

Interestingly, intrinsic motivation demonstrates small yet significant negative correlations with Strain, Anxiety, and Burnout, but no significant relationship with Depression (r = − 0.09, ns). This may suggest that while intrinsic motivation provides partial protection against perceived stress and exhaustion, it is not sufficient on its own to buffer depressive symptoms in the absence of psychological safety or subjective well-being. The skewness and kurtosis values indicate acceptable normality, and the moderate dispersion of scores implies adequate variability within the sample, making it suitable for regression and fuzzy-modelling analyses.

Table 3 presents a summary of the multiple-regression analyses, in which the three predictors—Intrinsic Motivation (IM), Psychological Safety (PS), and Mental Well-being (MWB)—were entered simultaneously to predict each mental health outcome (Athlete Strain, Generalised Anxiety, Depression, and Burnout).

Table 3.

The summary of multiple regression analyses.

Outcome (DV)	R²	Adj. R²	F	p(F)	Predictor	Standardised Beta	p	VIF	Tolerance
APSQ (Athlete Strain)	0.37	0.36	48.08	< 0.001	IM	0.06 (−0.06–0.18.06.18)	0.30	1.25 1.23 1.37	0.79 0.80 0.72
					PS	−0.38 (−0.53–0.27)	< 0.001
					MWB	−0.37 (−0.54-0.27.54.27)	< 0.001
GAQ (Anxiety)	0.34	0.34	43.28	< 0.001	IM	0.11 (0.001–0.21.001.21)	0.047
					PS	−0.18 (−0.28–0.06)	0.004
					MWB	−0.53 (−0.64–0.40)	< 0.001
PHQ-9 (Depression)	0.30	0.29	36.40	< 0.001	IM	0.18 (0.05–0.25.05.25)	0.003
					PS	−0.18 (−0.27–0.05)	0.002
					MWB	−0.51 (−0.59–0.36)	< 0.001
Burnout (BMS)	0.36	0.35	46.50	< 0.001	IM	0.09 (−0.03–0.35.03.35)	0.10
					PS	−0.28 (−0.71–0.31)	< 0.001
					MWB	−0.46 (−1.06–0.63)	< 0.001

Table 4 presents the comparative performance of the Multiple Linear Regression (MLR) and the calibrated Fuzzy Inference System (FIS). Across all mental health outcomes, the FIS demonstrated lower error metrics (MAE and RMSE) during the training phase, indicating that the rule-based framework effectively captures the nuanced patterns within the dataset. Crucially, the testing metrics for the FIS remained closely aligned with the training results, particularly for anxiety and athletes’ strain, where the FIS yielded higher correlation coefficients (r = 0.66 and 0.67, respectively) than the linear model. This stability between the training and testing phases confirms that the model is well-calibrated and shows no evidence of overfitting. These results justify the use of fuzzy logic over standard linear approaches, as the FIS more accurately maps the non-linear, synergistic relationships between motivation, psychological safety, and mental health outcomes.

Table 4.

Comparative performance metrics of MLR and FIS Models.

Outcome Variable	Model	Train MAE	Train RMSE	Test MAE	Test RMSE	r (Pred. vs. Obs.)
Anxiety (GAQ)	MLR	3.08	4.02	3.55	4.47	0.62
	FIS	2.41	3.21	3.48	4.29	0.66
Depression (PHQ-9)	MLR	4.05	5.20	4.35	5.36	0.64
	FIS	3.10	4.04	4.76	5.63	0.60
Athlete Strain (APSQ)	MLR	4.65	6.53	4.48	6.02	0.65
	FIS	3.60	4.99	4.72	5.96	0.67
Burnout (BMS)	MLR	7.73	10.74	8.45	11.60	0.66
	FIS	6.07	8.17	8.60	11.70	0.65

Figure 3 illustrates the regions where the membership functions are active in the system for a sample input set (Intrinsic Motivation = 18.2, Psychological Safety = 32.1, Mental Well-being = 29.7), along with the corresponding rules for these inputs. The red lines indicate the extent to which the input values contribute to the membership functions, while the rules highlighted in yellow represent the rules triggered by this input combination. This visual clearly shows that the model produces a low anxiety prediction in the ‘high intrinsic motivation + high psychological safety + high well-being’ case (Anxiety = 1.63).

Fig. 3.

Example input combination for anxiety exit and visualisation.

Figure 4 shows the model’s anxiety prediction as a function of two main input variables: intrinsic motivation and psychological safety. Low intrinsic motivation significantly increases anxiety. Particularly when motivation is in the 4–10 range, anxiety levels are generally high, confirming the relationship in the literature of “low motivation = high stress. ”Increased psychological safety has an anxiety-reducing effect. When the psychological safety value is 30+, the anxiety level remains low or moderate, especially if motivation is also high. Even if motivation is low, anxiety levels decrease somewhat when psychological safety is high (e.g., motivation = 8, psychological safety = 32, so anxiety = 6). This provides evidence that “psychological safety acts as a protective factor.” ”The sharp transitions on the response surface reflect the nonlinear behaviour of fuzzy logic according to the model’s rules. For example, when motivation increases from 12 to 14, a sudden drop in anxiety level is observed. This indicates that the “medium motivation” level is a critical threshold. This graph confirms that the model is consistent with psychological reality and clinically interpretable. Furthermore, it serves as a concrete visual guide for coaches and psychologists, enabling them to reduce anxiety risk while increasing athletes’ motivation and considering their psychological safety.

Fig. 4.

Anxiety output surface graph according to intrinsic motivation and psychological safety variables.

Figure 5 shows how the model’s anxiety prediction relates to two main input variables: intrinsic motivation and mental well-being. It shows that it changes from green to yellow. Low intrinsic motivation significantly increases anxiety. Especially when motivation is in the 4–10 range, anxiety levels are generally high, supporting the findings in the literature that “low motivation = high stress.” Increased mental well-being reduces anxiety. When the mental well-being value is 30+, the anxiety level remains low or moderate, especially if motivation is also high. Even if motivation is low, anxiety levels decrease somewhat when well-being is high. This provides evidence that well-being functions as a “protective factor.” The sharp transitions on the response surface reflect the nonlinear behaviour of fuzzy logic, as dictated by the model’s rules. For example, when motivation increases from 12 to 14, a sudden drop in anxiety level is observed, indicating that the “medium motivation” level is a critical threshold.

Fig. 5.

Anxiety output surface graph according to intrinsic motivation and mental well-being variables.

Figure 6 visualises the regions in the system where the membership functions are active for an example input set (Intrinsic Motivation = 17.2, Psychological Safety = 31.3, Mental Well-being = 28.7), along with the rules corresponding to these inputs. The red lines indicate the extent to which the input values contribute to the membership functions, while the rules highlighted in yellow represent the rules triggered by this input combination. This visual clearly demonstrates that the model produces a moderate burnout prediction in the ‘high intrinsic motivation, high psychological safety, moderate mental well-being’ scenario (Burnout = 28.1).

Fig. 6.

Example input combination for burnout exit and visualisation.

Figure 7 shows the relationship between the model’s burnout prediction and the two main input variables: intrinsic motivation and psychological safety. Low intrinsic motivation significantly increases burnout. Specifically, when motivation is in the 4–10 range, burnout is generally high, confirming the relationship described in the literature: “low motivation = high burnout.” Increased psychological safety reduces burnout. When psychological safety is 30+, burnout remains low or moderate, especially when motivation is also high. Even if motivation is low, burnout levels decrease somewhat when psychological safety is high. For example, if motivation = 8 and psychological safety = 32, burnout ≈ 35. This provides evidence that psychological safety functions as a “protective factor.” The sharp transitions on the response surface reflect the nonlinear behaviour of fuzzy logic, as dictated by the model’s rules. For example, when intrinsic motivation increases from 12 to 14, a sudden drop in burnout level is observed. This indicates that the “medium motivation” level is a critical threshold.

Fig. 7.

Burnout output surface graph according to intrinsic motivation and psychological safety variables.

Figure 8 shows the model’s burnout prediction for the two main input variables: intrinsic motivation and mental well-being. Low intrinsic motivation significantly increases burnout. Especially when motivation is in the 4–10 range, burnout is generally high, confirming the sports psychology relationship of “low motivation = high burnout.” Increased mental well-being reduces burnout. When the well-being value is 30+, the burnout level remains low or moderate, especially if motivation is also high. Even when motivation is low, when well-being is high, burnout levels are partially reduced. This provides evidence that well-being functions as a “protective factor.” The sharp transitions on the response surface reflect the nonlinear behaviour of fuzzy logic as defined by the model’s rules. For example, when motivation increases from 12 to 14, a sudden drop in burnout level is observed. This indicates that the “medium motivation” level is a critical threshold.

Fig. 8.

Burnout output surface graph according to intrinsic motivation and mental well-being variables.

Figure 9 illustrates the regions in the system where the membership functions are active for an example input set (Intrinsic Motivation = 12, Psychological Safety = 24, Mental Well-being = 22), along with the corresponding rules for these inputs. The red lines indicate the extent to which the input values contribute to the membership functions, while the rules highlighted in yellow represent the rules triggered by this input combination. This visual clearly demonstrates that the model produces a high depression prediction in the ‘medium intrinsic motivation + medium psychological safety + medium wellbeing’ scenario (Depression = 21).

Fig. 9.

Example input combination for depression exit and visualisation.

Figure 10 shows how the model’s depression prediction changes according to levels of intrinsic motivation and psychological safety. Low intrinsic motivation, especially in the 4–10 range, significantly increases depression. Depression scores are generally high in this region. High psychological safety has a protective effect on depression. When the psychological safety level is 30+, the depression level remains low or moderate, especially if intrinsic motivation is also high. The sharp transitions on the surface reflect the rule-based nonlinear structure of the fuzzy inference system. For example, when motivation rises from 12 to 14, a significant decrease in depression is observed, indicating that the “medium motivation” level is a critical threshold. This visual demonstrates that psychological safety can reduce the risk of depression even in individuals with low motivation.

Fig. 10.

Depression output surface graph according to intrinsic motivation and psychological safety variables.

Figure 11 shows the relationship between the model’s depression prediction and its two main input variables: intrinsic motivation and mental well-being. Low intrinsic motivation significantly increases depression. Especially when motivation is in the range of 4–10, the level of depression is generally high, confirming the relationship of “low motivation = high depression” found in the literature. Increased mental well-being has a depressant effect. When the well-being value is 30+, the level of depression remains low or moderate, especially if motivation is also high. Even if motivation is low, depression levels partially decrease when well-being is high (e.g., if motivation = 8, well-being = 32, then depression ≈ 10). This provides evidence that well-being functions as a “protective factor.” The sharp transitions on the response surface reflect the nonlinear behaviour of the fuzzy logic model, as defined by its rules. For example, when motivation increases from 12 to 14, a sudden drop in depression levels is observed. This indicates that the “medium motivation” level is a critical threshold.

Fig. 11.

Depression output surface graph according to intrinsic motivation and mental well-being variables.

Figure 12 visualises the regions in the system where the membership functions are active for an example input set (Intrinsic Motivation = 12, Psychological Safety = 24, Mental Well-being = 22), along with the rules corresponding to these inputs. The red lines indicate the extent to which the input values contribute to the membership functions, while the rules highlighted in yellow represent the rules triggered by this input combination. This visual clearly demonstrates that the model produces a prediction in the ‘medium intrinsic motivation + medium psychological safety + medium mental wellbeing’ scenario.

Fig. 12.

Example input combination for strain exit and visualisation.

Figure 13 shows the relationship between the model’s stress prediction and two main input variables: intrinsic motivation and psychological safety. Low intrinsic motivation significantly increases stress. Especially when motivation is in the 4–10 range, stress levels are generally high, confirming the relationship in the literature of “low motivation = high stress.” Increased psychological safety reduces stress. When the psychological safety value is 30+, the stress level remains low or moderate, especially if motivation is also high. Even when motivation is low, stress levels decrease somewhat when psychological safety is high (e.g., if intrinsic motivation = 8 and psychological safety = 32, then stress ≈ 25). This provides evidence that psychological safety functions as a “protective factor.” The sharp transitions on the response surface reflect the nonlinear behaviour of fuzzy logic as defined by the model’s rules. For example, when motivation increases from 12 to 14, a sudden drop in stress level is observed. This indicates that the “medium motivation” level is a critical threshold.

Fig. 13.

Strain output surface graph according to intrinsic motivation and psychological safety variables.

Figure 14 shows the relationship between the model’s stress prediction and two main input variables: intrinsic motivation and mental well-being. Low intrinsic motivation significantly increases stress. In particular, when intrinsic motivation is in the 4–10 range, stress levels are generally high, confirming the literature’s finding that “low motivation = high stress.” Increased mental well-being reduces stress. When the well-being value is 30+, the stress level remains low or moderate, especially if motivation is also high. Even if motivation is low, stress levels decrease somewhat when well-being is high (e.g., if motivation ≈ 8, well-being ≈ 32, then stress ≈ 25). This provides evidence that “well-being acts as a protective factor.” The sharp transitions on the response surface reflect the nonlinear behaviour of fuzzy logic as defined by the model’s rules. For example, when motivation increases from 12 to 14, a sudden drop in stress level is observed. This indicates that the “medium intrinsic motivation” level is a critical threshold.

Fig. 14.

Strain output surface graph according to intrinsic motivation and mental well-being variables.

The comprehensive analysis of the surface graphs and input combinations confirms that the fuzzy logic model remains consistently aligned with psychological reality and offers high clinical interpretability. The model clearly identifies low intrinsic motivation (within the 4–10 range) as the most potent predictor of negative outcomes, including anxiety, burnout, depression, and strain. Crucially, as psychological safety and mental well-being reach levels of 30+, they function as vital ‘protective factors’ that mitigate these adverse states, even in low-motivation situations. The sharp transitions observed in the findings—particularly the sudden decrease in psychological risk when motivation crosses the 12–14 threshold—highlight critical tipping points that linear models often fail to capture. This non-linear behaviour accurately reflects the complex, multi-dimensional nature of human psychology within a sporting context.

Discussion

This study aimed to examine the following: (a) the predictive role of intrinsic motivation and psychological safety, and mental well-being on mental health outcomes (strain, anxiety, depression, and burnout); (b) to model the role of psychological safety and mental well-being when intrinsic motivation is low and high by using the fuzzy logic method. Our study provides an interdisciplinary contribution to the literature by empirically testing the interplay between psychological safety, mental well-being, intrinsic motivation, anxiety, strain, burnout, and depression in elite athletes. We hypothesised fuzzy-logic models to test various conditions, including low, moderate, and high levels of motivation, psychological safety, mental well-being, and mental health.

We provided a multiple linear regression model testing the relationships between motivation, psychological well-being, and mental well-being as predictors (separately) of mental health outcomes. The results showed that each regression model was significant. Except for the intrinsic motivation, psychological safety, and mental well-being negatively predicted mental health outcomes, which indicated that psychologically safe environments and mental well-being might be effective in reducing anxiety, depression, burnout, and strain. These findings are consistent with recent literature suggesting that psychological safety and organisational/relational factors in sport are linked to improved mental health outcomes^28,63. Empirical and review evidence suggest that athletes in psychologically supportive settings report increased well-being, and that awareness/intervention programs can improve mental health outcomes and help-seeking behaviours^64,65. Our results demonstrate that mental well-being and psychological safety can reduce adverse mental health outcomes. Intrinsic (autonomous) motivation and psychological safety exert complementary yet distinct influences on athletes’ mental health outcomes, with overall mental well-being serving as a crucial protective factor. Studies show that autonomous forms of motivation (including intrinsic and integrated regulation) are generally associated with lower levels of depression, anxiety and burnout. In contrast, controlled motivation and amotivation relate to greater psychological distress and higher risk of exhaustion^66–68. Psychological safety within sporting environments — that is, climates that allow for authentic self-expression, reduce stigma, and encourage help-seeking — is empirically and theoretically linked to reduced strain and burnout, as well as improved help-seeking behaviour^28,63. Moreover, higher levels of athlete mental well‑being are associated with lower anxiety, depression, strain and subsequent burnout, and may mediate or amplify the protective effects of both autonomous motivation and psychologically safe team climates^28,63. Longitudinal evidence indicates that athletes’ motivational profiles (including autonomy and relatedness), perfectionistic tendencies (strivings vs. concerns), and social support processes predict trajectories of athlete burnout, highlighting the need for interventions that promote autonomous motivation, foster autonomy-supportive coaching, and strengthen well-being to mitigate competitive anxiety, depressive symptoms, sleep disruptions, and burnout⁶⁹. The multivariate analysis revealed a complex relationship between Intrinsic Motivation (IM) and mental health outcomes. While bivariate correlations suggested a protective role for IM, it emerged as a significant positive predictor of depression and anxiety in the regression models (p <.05). Diagnostic testing yielded low Variance Inflation Factor (VIF) values (all < 1.5) and high tolerance levels, confirming that this sign reversal was not a product of multicollinearity but rather a classical suppression effect. This indicates that IM accounts for residual variance in Psychological Safety (PS) and Mental Well-being (MWB) that is associated explicitly with psychological strain. In the elite sporting context, this suggests that the protective effect of motivation is conditional: When environmental supports such as PS and MWB are held constant, high internal drive may manifest as ‘obsessive’ passion, potentially increasing vulnerability to strain. These findings align with recent conceptualisations of athlete welfare and suggest that the link between motivation and mental health is indirect, potentially mediated by the athlete’s perceived safety and well-being. Future research should employ longitudinal designs to clarify these indirect pathways further.

The fuzzy visualisations presented in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14 extend and complement the statistical analyses by uncovering the complex, non-linear, and context-dependent dynamics among intrinsic motivation (IM), psychological safety (PS), and mental well-being (MWB). Whereas multiple regression models quantified additive and directional relations, the fuzzy inference system (FIS) provides a multidimensional portrayal of how these psychosocial variables interact continuously within a graded risk landscape. In doing so, the model translates psychological relationships into interpretable, rule-based geometries that preserve theoretical coherence while improving interpretability in practice.

As shown in Table 3, the FIS model consistently demonstrated superior or comparable predictive accuracy and higher correlation coefficients (e.g., r = 0.67 for Athlete Strain) than the MLR model. This confirms that the rule-based framework—which posits intrinsic motivation as a protective factor and identifies synergistic effects between psychological safety and well-being—captures the non-linear complexities of athlete psychology more effectively than standard linear approaches. Furthermore, the close alignment between training and testing error metrics (MAE/RMSE) demonstrates the robustness of the FIS. It confirms that the model is well-calibrated and shows no evidence of overfitting.

Across all outcome surfaces—anxiety, burnout, depression, and athlete-specific strain— a consistent pattern appears. Low intrinsic motivation yields an upward gradient of psychological risk; however, this gradient flattens markedly as psychological safety or mental well-being increases (Hypothesis 1). The fuzzy surfaces make this protective buffering effect visually evident: high motivation coupled with high psychological safety and well-being generates a low-risk zone characterised by blue or green regions in the output maps. Conversely, when psychological safety and well-being fall below moderate levels, the slopes steepen sharply, confirming that motivational vulnerability scales non-linearly with contextual fragility. These results align with the empirical data^70,71.

In the case of anxiety (Figs. 3–5), the lowest predicted risk emerged in high psychological safety/high mental well-being configurations regardless of intermediate fluctuations in motivation, lending strong support to Hypothesis 2. While the linear regression models indicated a positive association between intrinsic motivation and risk (a suppression effect), the FIS model reveals a non-linear ‘protection basin’ where robust environmental and well-being factors effectively neutralise motivational risks. Given that the FIS model demonstrated superior predictive accuracy, with lower MAE and RMSE values than the regression (Table 3), this non-linear interaction is interpreted as a more precise reflection of the complex psychological dynamics in elite sports.

The burnout surfaces (Figs. 6, 7 and 8) reveal that while motivation makes a modest contribution, the climate of psychological safety is more decisive in preventing exhaustion, aligning with the notion that relational factors mediate the perceived meaning of effort. Here, burnout acts as a slow variable in the psychological system—less sensitive to short-term motivational shifts but highly responsive to chronic deficits in safety and well-being. This also supports the theory in sport psychology^45,69.

The depression surfaces (Figs. 9, 10 and 11) further illustrate how medium levels of motivation do not necessarily confer protection when safety and well-being are diminished. In these mid-range zones, the fuzzy model predicts high depressive risk, even without overt motivational collapse. This finding aligns with the existing literature, which identifies conditional vulnerability among athletes who thrive in climates characterised by low interpersonal trust^45,63. In such cases, fuzzy logic captures the latent risk arising from discrepancy dynamics: the mismatch between internal drive and external climate leads to an erosion of affective stability that is not visible in mean-based models.

The strain surfaces (Figs. 12, 13 and 14) show the steepest risk gradients under low psychological safety and intrinsic motivation conditions. The model suggests that psychological strain increases disproportionately when relational safety deteriorates in tandem with a decline in intrinsic engagement. However, where mental well-being remains high, this risk surface flattens—the “green plateau” observed in these figures indicates global resistance to perturbation. Conceptually, this supports the view that well-being functions as a systemic moderator, dampening the translation of motivational disruptions into affective distress^13,29.

Taken together, the fuzzy system substantiates Hypothesis 2 and Hypothesis 3: Psychological safety and well-being act not only as individual protective factors but also as synergistic stabilisers. When both are elevated, the model produces a “low-risk basin” across all outcome surfaces. This basin may be viewed metaphorically as an attractor within a dynamical system, representing a state of psychological stability and resilience. Athletes operating within such zones exhibit reduced emotional volatility and a higher capacity for stress recovery. At the same time, those in “high-slope” regions remain vulnerable to rapid transitions towards anxiety, depression, and burnout following even minor perturbations^2,69. This interpretable “landscape of resilience” also aligns with self-determination theory²³, which posits that autonomy, relatedness, and competence create an equilibrium that sustains motivation over time.

In practice, these fuzzy outputs have several implications. First, they illustrate that preventive interventions should prioritise the interaction of factors rather than isolated constructs. Enhancing athlete motivation without fostering psychological safety risks producing effort without recuperation, ultimately hastening burnout. Conversely, even moderate motivation can yield healthy psychological outcomes in climates marked by trust, openness, and emotional balance. Second, the graphical interpretability of the FIS provides a potential decision-support interface: practitioners can visualise real-time changes in athlete well-being or motivation and assess how such changes reposition the individual within the fuzzy landscape. This capacity for graded sensitivity constitutes an early warning mechanism — small losses in MWB or PS result in visible increases in predicted risk before crossing clinical thresholds.

From a methodological standpoint, the study demonstrates the value of fuzzy modelling for bridging the classical “black box versus transparency” divide in mental health analytics. The Mamdani-type FIS preserves rule-based clarity while accommodating non-linearity and uncertainty, reducing the conceptual tension between empirical rigour and practitioner accessibility^43,44. Crucially, the heuristic calibration of rule weights was constrained to ensure that the system’s semantic integrity remained intact; the final weights (ranging from 0.7 to 1.0) consistently prioritised the established theoretical hierarchy, with psychological safety and well-being as the dominant protective logic. Thus, predictive accuracy was enhanced without compromising the logical transparency required for practitioner decision-support. Moreover, the empirical congruence between the model’s surfaces and observed psychological regularities enhances both its construct validity and its translational potential for sport settings. Beyond the technical achievement, the analysis signals a shift in how mental health processes might be conceptualised — from categorical events to continuously evolving states within a fuzzy, relational field.

In a broader theoretical sense, the convergence of these fuzzy outputs with established sport-psychology frameworks reaffirms that motivation, safety, and well-being function as a coherent psychological ecology. Within this ecology, intrinsic motivation is necessary but not sufficient; it requires a relational milieu conducive to authenticity and learning. In the absence of such safety and affective replenishment, motivation may become over-controlled or perfectionistic, resulting in the very risk patterns captured by higher surface slopes in the model. Hence, fuzzy logic provides a quantifiable mirror for complex human processes, allowing dynamic, partial truths to be rendered measurable without reductively dichotomising them.

The fuzzy-logic visualisations elucidate the geometry of psychological resilience and vulnerability among elite athletes. They transform abstract theoretical relations into tangible, data-driven surfaces where “risk valleys” and “resilience basins” can be inspected, compared, and utilised for applied practice. Future research may build on this by implementing longitudinal fuzzy models to trace how athletes move across these surfaces over seasons or interventions, and by assessing whether targeted safety or well-being training can alter the topology of risk itself. Such advances would mark a decisive step towards interpretable, adaptive, and ethically robust decision-support systems for athlete mental health management—bridging the science of motivation with the art of care. This interdisciplinary approach contributes to the broader goal of advancing sports science by integrating psychological insights into comprehensive strategies to enhance performance, mitigate injury risk, and optimise athlete rehabilitation⁷². Consequently, these visual outputs serve as concrete guides for coaches and sport psychologists, enabling them to anticipate mental health risks more effectively. The data underscores the necessity of intervention programmes that prioritise both motivation enhancement and the cultivation of a supportive, secure environmental climate. Ultimately, this model provides a robust framework for both testing theoretical hypotheses and conducting practical risk assessments in sports science. Such an approach reinforces the decisive role of holistic strategies in fostering athlete well-being and perceptions of performance.

Conclusion and practical implications

This study demonstrates that mental health outcomes among elite athletes cannot be adequately explained through linear or unidimensional approaches. By integrating fuzzy-logic modelling with psychological theory, we have shown that intrinsic motivation, psychological safety, and mental well-being operate as interdependent components within a dynamic system. The fuzzy inference framework successfully captured the graded, nonlinear character of athletes’ mental states, identifying zones of heightened risk and regions of resilience. These results provide a computationally traceable yet psychologically meaningful map of how internal and environmental factors combine to influence anxiety, depression, strain, and burnout.

From a theoretical standpoint, the study reinforces the notion that resilience in sport emerges not from isolated traits but from configural balance — the co-activation of internal drive and supportive social contexts. Motivation alone, though essential for performance, offers limited protection against mental strain when the surrounding climate lacks psychological safety or emotional replenishment. Conversely, the synergy between safety and well-being generates stability, allowing athletes to maintain high engagement without escalating distress. The fuzzy-logic approach, therefore, represents a methodological advance: it quantifies interactional processes that traditional models often overlook, highlighting the continuous interplay between individual and contextual determinants of mental health.

In applied terms, these insights are highly relevant to coaches, performance staff, and sport psychologists. Interventions should prioritise climates of trust, open communication, and genuine care rather than focusing solely on motivational enhancement. Routine assessments incorporating fuzzy-derived “risk maps” could serve as early warning systems, identifying subtle declines in well-being or safety before overt symptoms appear. Moreover, the use of transparent, interpretable models aligns with ethical goals in elite sport — promoting evidence-informed, non-stigmatising support frameworks that respect athlete autonomy. Future research should explore how such models can be embedded into long-term monitoring programmes and whether dynamic changes in fuzzy risk surfaces correspond to recovery, adaptation, or maladjustment across different phases of competition.

Limitations

Despite its interdisciplinary strengths, this study has limitations. The design was cross-sectional, precluding causal inference regarding the temporal direction between motivation, safety, well-being, and mental health outcomes. Although the fuzzy-logic model captured non-linear and synergistic relationships, its rule base was calibrated against a single data wave, meaning that longitudinal fluctuations in affective states or team climates were not observed. Future research should therefore employ repeated-measures or time-series approaches to determine whether the fuzzy surfaces remain stable across competitive cycles or fluctuate in response to contextual pressures, such as injury, selection stress, or seasonal transitions.

A second limitation concerns sampling and measurement considerations. The participants were drawn from multiple sports but primarily represented a young adult cohort, which may restrict generalisability to older or para-athlete populations. Additionally, the study relied exclusively on self-report instruments. Although all showed acceptable reliability, responses may have been influenced by social desirability or self-presentation biases common in elite environments. Hybrid data streams — combining psychometric, behavioural, and physiological indices — could enhance future fuzzy models by integrating both subjective and objective indicators of athlete well-being. Finally, while the fuzzy-logic approach provides transparent and interpretable outputs, membership functions and rule structures remain partly researcher-defined; therefore, ongoing empirical refinement is essential to ensure the stability and cross-cultural validity of the modelling framework.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

A.A.Ş. conceptualised and developed the fuzzy logic modelling framework, performed the mathematical analyses, and contributed to the interpretation of results.G.E.A. coordinated participant recruitment, data collection, and initial statistical processing.E.Ş. (corresponding author) designed the overall research protocol, supervised the study, and integrated the psychological and applied mathematics perspectives.A.A.Ş. and G.E.A. prepared the initial draft of the manuscript.E.Ş. critically revised and finalised the manuscript for submission.All authors reviewed, discussed, and approved the final version of the manuscript.

Data availability

We confirm that the data used in the research are available. The data can be obtained by emailing the corresponding author (or at 10.6084/m9.figshare.30580361).

Declarations

Competing interests

The authors declare no competing interests.

Ethical approval

The Institutional Ethical Board reviewed and approved all procedures prior to recruitment and data collection. The study was conducted in accordance with the ethical principles outlined in the Declaration of Helsinki. We obtained ethical approval from the Muğla Sıtkı Koçman University Social and Human Sciences Research Ethics Committee, as mandated by decision number 250115/11, officially ratified on October 24, 2025.

Informed consent

All participants were fully informed about the process and purpose of the study, and written consent/assent was obtained.