Abstract
Mechanistic trait models are increasingly considered to estimate persistence-related suitability for invasive species, although translating reproduction metrics into operational indicators for spatial decision-making remains challenging. Branching-process transformations are widely applied to convert mechanistic reproduction numbers into probabilities of establishment; however, such transformations may compress spatial gradients when persistence is broadly feasible. Using tomato red spider mite (Tetranychus evansi) as a case study across Africa, this study developed an uncertainty-aware, physiologically-based model and generated spatially explicit estimates of expected reproduction (R0). We compared three representations: raw mechanistic suitability, branching-process persistence probability, and quantile-normalised suitability. Although all representations preserved rank ordering, branching probability showed strong spatial saturation (>93% of pixels have close to 1), reducing effective resolution for prioritisation. In contrast, quantile normalisation retained uniform spatial contrast and stable hotspot delineation under fixed-effort targeting. Our findings demonstrate that indicator choice fundamentally affects interpretability, and that rank-based transformations may provide more robust ecological indicators for spatial and spatiotemporal decision support under uncertainty.
1 Introduction
Understanding whether a species, above all pests or pathogens, can persist under given environmental conditions is a central objective in ecological risk assessment and management (). Mechanistic models grounded in life-history traits, demographic rates, and environmental responses are increasingly used to characterise persistence potential, particularly in data-limited systems where correlative approaches may lack biological interpretability or extrapolative power (; ; ; ). These models often yield continuous outputs related to population growth or reproduction, such as reproduction numbers or suitability indices, which need then to be transformed into quantities that can inform decision-making ().
A widely adopted framework for translating mechanistic reproduction metrics into a probability of establishment is classical branching process theory, originally developed by to study the extinction of family lineages and later formalised mathematically by . In this framework, population growth during the early invasion phase is approximated as a stochastic birth-death process in which individuals reproduce independently (; ). When offspring production follows a Poisson distribution with expected value (the average number of offspring per individual), the probability of ultimate establishment is given by the complement of the extinction probability, which reduces to , for (). This connection between reproduction number and invasion probability was subsequently adopted in epidemic theory (e.g., ; ) and later integrated into modern -based frameworks in ecology and epidemiology (). In applied contexts, branching processes therefore provide a theoretically grounded and intuitive means of answering a feasibility question: given local demographic and environmental conditions, is persistence or establishment likely to occur at all? (). Because the transformation places different mechanistic outputs onto a common probabilistic scale bounded between 0 and 1, branching-derived probabilities have been recently used for invasion screening, early-warning systems, and comparative risk assessment in the case of Phlebotomus papatasi ().
In many applied cases, such as disease vectors and pest control, however, the primary decision problem is not whether persistence is possible, but where and when limited resources should be prioritised (; ). Surveillance, control, and mitigation actions are implemented spatially and often dynamically, requiring tools that support spatial and spatiotemporal prioritisation rather than binary feasibility judgments. Effective decision support, therefore, depends on representations that preserve relative ordering across landscapes, enable fixed-effort or fixed-area targeting (e.g., prioritising the top decile of risk), and remain interpretable under uncertainty in biological parameters (). Crucially, the operational relevance of persistence-related metrics only becomes evident when they are evaluated in a spatial or spatiotemporal context (; ). Conversely, rank-based representations preserve spatial gradients and temporal coherence, supporting consistent prioritisation across space and time even when absolute values are uncertain (; ). Among these approaches, quantile-oriented analysis offers a particularly effective alternative to persistence-related metrics in spatial and spatiotemporal contexts, as it maintains ecological ranking while enhancing interpretability for decision-making.
Quantile-oriented methods offer a practical alternative to probability-based persistence metrics for decision support. recently showed that quantile-based sensitivity analysis can preserve the same parameter-ranking information as failure-probability approaches, while avoiding the strong saturation that often arises near extreme probabilities. Quantile-oriented indices can be formulated using simple decision-relevant contrasts (e.g., the distance between a chosen output quantile and the mean response), yielding interpretable estimations of uncertainty that remain discriminative across the output range. , instead, showed that equi-quantile values can be used to construct decision alternatives even when uncertainty and dependence among inputs are both present. Taken together, these insights motivate comparing quantile-normalised suitability against persistence probability and underscore that the usefulness of persistence-related information depends on how it behaves across space and time in realistic mapping applications.
This background motivates the present study, as we aim to evaluate quantile normalisation as a decision-oriented transformation of uncertainty-aware mechanistic suitability derived from a persistence-related framework. Quantile normalisation is more suitable in terms of relative rank rather than absolute magnitude and preserves spatial and temporal gradients while avoiding saturation effects. For the sake of this study, we consider the case of tomato red spider mite (Tetranychus evansi), an invasive agricultural pest with well-characterised thermal biology (; ; ; ), across Africa.
Tropical African ecosystems are valuable bench tests to explore these methodologies, as they encompass diverse environmental conditions that can strongly influence the establishment and persistence of invasive species. Existing suitability assessments commonly rely on species distribution models and ecological databases, which provide valuable large-scale insights but may have limited capacity to capture mechanistic population processes and variation across spatial scales. By integrating multiple reproductive metrics, spatial indicators, and suitability representations within a mechanistic, trait-based framework, our approach provides a broader basis for assessing invasion risk and supporting actionable pest-management decisions. Notably, we compare quantile-normalised mechanistic suitability with persistence probability surfaces derived from a branching-process interpretation. Our objective is to evaluate how different representations of persistence-related information support spatial and spatiotemporal decision-making under uncertainty, focusing on interpretability, ranking stability, and hotspot delineation. Notably, we would not like to reassess persistence itself, nor to replace branching processes as tools for feasibility assessment. By explicitly contrasting feasibility-oriented and prioritisation-oriented representations, this study wants to point out that aligning ecological metrics with their intended operational use demonstrates how uncertainty-aware transformations can enhance the practical relevance of mechanistic models for ecological informatics and applied risk management.
2 Methods
2.1 Physiologically-based model of Tetranychus evansi
The starting point to develop the theory behind the present study is the general physiologically-based model of , ), which describes the stage-development of terrestrial arthropods over time, accounting for the effect of the environmental parameters, as well. It is worth pointing out that the methodology hereafter developed is general and fully transferable to any other terrestrial arthropod species. However, the above-mentioned general model, based on Ordinary Differential Equations (ODEs), requires a case study to be used in practice. For this reason, we will refer to the case of T. evansi, although the steps can be carried out for any other terrestrial arthropod.
The biological life cycle of T. evansi is composed of six stages: eggs , larvae , protonymphs , deutonymphs , and adult males and females . As the sex ratio of this species is strongly unbalanced towards females, for the sake of this study we collapse adults into a single stage, reducing the complexity of the model. Accordingly, under standard stage-structured assumptions with exponentially distributed development and mortality times (i.e., competing hazards), the low-density population dynamics are described by the following ODE system. All parameters depend on environmental variables, which, because of the literature available, are only represented by temperature. We hereafter denote the development and mortality of the preimaginal stages as and (with ), respectively, while adult survival is denoted by ; egg production rate, instead, is denoted by . For the sake of notation, we avoid to explicitly reporting temperature.
The system is:
We can use the system (1) to build a physiologically-based index that does not depend anymore on the population abundance, but that at the same time describes the population growth/decline. The methodology provides for the application of the next generation matrix approach to the system (1), an alternative to the method described in and .
As the first step, let us define the New Infection Vector and the Net Transfer Matrix as:
As the Next Generation Matrix is defined as , we now proceed to compute by using the Gauss elimination algorithm, recalling that , with the identity matrix. The computations lead to the following result:
where , , , , .
The Next-Generation Matrix of the system (1) is then defined as:
From the Equations 1–5, we can define the Basic Reproduction Number . Since only the first row is non-zero, the spectral radius is the entry, namely:
As for other risk indices based on similar approaches (; ), population growth occurs if , while extinction is expected when , and does not contain any information about the population abundance, being completely independent on the estimation of initial conditions. In general, the rate functions in Equation 6 generally depend on many factors, including the population density of specific stages; however, the quantitative knowledge that allows the identification of the specific equations is often limited to temperature. Thermal response, in fact, is usually explored through life tables studies (, ), experiments that consist of rearing cohorts of insects at different constant temperatures and that provide an estimation of the stage development time, the stage mortality rate, and of the egg production rate.
A complete list of the temperature-dependent rate functions involved in the computation of , together with their best fit parameters, was retrieved from the literature and summarised in Table 1. Notably, development rates for egg, larva, protonymph, and deutonymph stages were described using stage-specific Lactin-2 functions, parameterised with an upper thermal limit fixed at 45 °C () and fitted coefficients derived from published experimental studies. Egg mortality was treated differently because egg survival was available as a temperature-dependent function rather than as a single overall value. First, egg hatch probability was estimated using the fitted quadratic relationship with temperature. This stage-level survival was then combined with the temperature-dependent egg development time (i.e., egg duration) to be further converted into a daily mortality rate. In other words, egg mortality was derived from the observed probability of surviving the full egg stage and scaled according to how long the egg stage lasts at a given temperature, ensuring biological consistency between survival and development. Adult mortality was represented by a linear temperature response constrained to non-negative values. Direct empirical estimates of stage-specific post-embryonic mortality were not available; therefore, total larva-to-adult mortality was inferred from reported survival rates and partitioned proportionally across larval, protonymph, and deutonymph stages based on their relative mean developmental durations (). Specifically, because only overall survival from larva to adult was available (mean survival = 0.9093), total post-embryonic mortality was first calculated as 1 − 0.9093, giving 0.0907 (9.07%). Since stage-specific mortality was not reported, this total mortality was distributed across the larval, protonymph, and deutonymph stages in proportion to their mean developmental durations. Using average stage durations (2.435, 2.73, and 4.27 days, respectively), the resulting mean stage-level mortalities were 0.0234 for larvae, 0.0262 for protonymphs, and 0.0410 for deutonymphs, which sum to the observed total post-embryonic mortality (Table 1). This approximation is the results of the fragmented quantitative literature often available on the thermal biology of many species and does not limit the generality of the methodology hereby presented. In fact, every rate function can be developed or replaced independently, without further modifications to the other model components (until the rate functions do not depend on the states). Fecundity was modelled as a quadratic function of temperature with an explicit upper cutoff at 42 °C based on the data provided by , and effective fecundity was computed by combining temperature-dependent daily fecundity with a constant female sex ratio estimated as the mean of seven reported strain-specific values.
Table 1
| Biological process | Mathematical expression | Parameter values | Units | Source |
|---|---|---|---|---|
| Development eggs () | q=0.008257022; D = 1.861163; k=−1.000; Tmax=45 | day⁻¹ | () | |
| Development larva () | q=0.026209408; D = 0.500; k=−1.000; Tmax=45 | day⁻¹ | () | |
| Development protonymph () | q=0.029286797; D = 0.500; k=−0.9501334; Tmax=45 | day⁻¹ | () | |
| Development deutonymph () | q=0.027496083; D = 0.500; k=−1.000; Tmax=45 | day⁻¹ | () | |
| Mortality egg () | - | q=0.008257022; D = 1.861163; k=−1.000; Tmax=45 | day⁻¹ | (; ) |
| Mortality larva () | 0.023 ± 0.006 | – | Proportion day⁻¹ | () |
| Mortality protonymph () | 0.026 ± 0.007 | – | Proportion day⁻¹ | ) |
| Mortality deutonymph ( | 0.04 ± 0.01 | – | Proportion day⁻¹ | ) |
| Mortality adult () | -0.0313 + 0.0035T | – | day⁻¹ | () |
| Fecundity (Adult female) | -46.192 + 3.608T-0.056 | – | eggs day⁻¹ | () |
| Sex ratio (Adult) | 0.868 | – | proportion female | () |
Temperature-dependent life-history functions and parameterization used in the stage-structured model.
2.2 Mechanistic suitability model and uncertainty propagation
The allows the computation of the mechanistic suitability of the species. As explained in the previous section, T. evansi was modelled as a temperature-dependent, stage-structured persistence proxy including development, mortality, and reproduction across the life cycle. It is worth noticing, at this stage of the process, that the parameters listed in Table 1 come from experimental datasets and, by definition, are affected by uncertainty. In many similar studies, the common practice is to neglect the uncertainty associated with the parameters, but this might lead to a loss of information or reliability of the estimation (e.g. ; , , ).
For the sake of this study, parameter uncertainty was propagated using a non-Bayesian Monte Carlo simulation () with 500 iterations. The probability distributions specified for the simulation represent uncertainty distributions for model inputs rather than Bayesian priors; no likelihood-based updating or posterior estimation was performed. Uncertainty was introduced in larva-to-adult survival and sex ratio, while the remaining temperature-dependent functions and parameter estimates were treated deterministically at their fitted or literature-derived values. Larva-to-adult survival was sampled using normally distributed draws centred on the empirical mean (mean = 0.9093, SD = 0.02265) and bounded to 0.70–0.99, while the female sex ratio was sampled using normally distributed draws centred on the observed mean across strains (mean = 0.8683, SD = 0.02) and bounded to 0.75–0.95. For each Monte Carlo draw and temperature, mechanistic suitability was computed as a composite function of stage-specific development, mortality, and effective fecundity, and marginalised expected suitability was obtained by averaging across iterations. Thermal response curves with uncertainty were generated over a continuous temperature range from 10 to 50 °C, with uncertainty envelopes summarised using the 2.5th and 97.5th percentiles of the resulting Risk Index values across Monte Carlo draws at each temperature.
This framework provides a biologically grounded and reproducible representation of persistence-related suitability while explicitly accounting for uncertainty arising from incomplete life-history data. The analyses were carried out in R (), using the terra (), sf (; ), and ggplot2 () packages.
2.3 Spatial prediction and raster generation
Let denote the long-term mean temperature at location x, derived post hoc by averaging WorldClim monthly minimum and maximum temperature climatologies (). These climatological variables were selected because temperature is a major determinant of T. evansi development, survival, reproduction, and population growth, and therefore provides a biologically relevant basis for assessing persistence-related suitability. Monthly minimum and maximum temperatures capture the seasonal thermal conditions experienced by populations and allow the model to represent both favourable conditions and potential thermal constraints across Africa. The expected persistence-related mechanistic suitability at location x is expressed as Equation 7:
where the expectation is taken over Monte Carlo draws of uncertain life-history parameters. Here, represents a mechanistically derived persistence proxy rather than a realised epidemiological reproduction number.
An interpretable spatial prioritisation was carried out by transforming the expected suitability surface into a quantile-normalised index (Equation 8):
where ranks are computed across all valid spatial pixels. This transformation preserves relative ordering while enforcing uniform contrast across space.
For comparison with probability-based representations commonly used in persistence and invasion modelling, a branching-process approximation was applied to obtain a persistence probability (Equation 9),
Spatial predictions of persistence-related suitability were generated across Africa using long-term climate surfaces and raster-based evaluation of the mechanistic model. The climatological variables were harmonised to a common spatial and temporal resolution before analysis at 2.5-arcminute. Monthly temperature data were aggregated and matched to the spatial grid used for the persistence-related suitability assessment, ensuring that all variables were directly comparable across locations and time periods. All spatial analyses were implemented in R using the terra, sf, and ggplot2 packages (). Monthly temperature rasters were obtained from WorldClim v2.1 () at 2.5-arcminute resolution, based on long-term climatological averages of minimum and maximum temperature. These layers were post-processed to derive mean temperature by averaging minimum and maximum values prior to model evaluation. All rasters were cropped and masked to the African continent using a country-level shapefile to ensure spatial consistency. Expected mechanistic suitability was computed for each pixel by applying the fast expected suitability function to the mean temperature raster, yielding a continuous surface representing marginalised persistence-related suitability.
Suitability values were then transformed into a quantile-normalised index by ranking pixels and rescaling ranks to the unit interval, thereby preserving relative ordering while enforcing uniform contrast across space while providing an interpretable spatial prioritisation under fixed-effort decision rules. For comparison with probability-based representations commonly used in invasion and epidemic modelling, expected suitability was additionally transformed into a persistence probability using a branching-process approximation. The three representations (i.e., raw expected suitability, quantile-normalised suitability, and branching probability) were exported as georeferenced raster layers. To capture seasonal dynamics, the same procedures were applied to individual monthly temperature layers, producing monthly maps of raw suitability, quantile-normalised suitability, and branching probability. Together, these raster products provide a spatially explicit, uncertainty-aware representation of persistence-related suitability suitable for comparative decision-support analyses.
2.4 Uncertainty attribution and robustness analysis
Uncertainty attribution was evaluated using a Monte Carlo framework in which mechanistic suitability was recalculated across repeated stochastic draws of uncertain life-history parameters. Analyses were conducted at five fixed temperatures (20, 25, 30, 35, and 40 °C), spanning sub-optimal to near-upper thermal conditions relevant for T. evansi. For each temperature, suitability values obtained from individual Monte Carlo draws were correlated with sampled parameter values using Spearman rank correlations, providing a non-parametric measure of each parameter’s contribution to variability in suitability. Spearman’s rank correlation was selected because it provides a robust, non-parametric measure of monotonic associations without requiring normally distributed variables, while the Jaccard index was used to quantify spatial overlap between suitability classes in a straightforward and scale-independent manner. These methods were therefore appropriate for the distributional characteristics and spatial nature of our data, while avoiding stronger parametric assumptions. Both analyses were performed using the original (untransformed) values, as no transformation was required for these non-parametric and overlap-based measures.
Only parameters treated as stochastic in the model, immature-stage mortality rates, and sex ratio (effective fecundity) were included in the attribution analysis, while temperature-dependent egg and adult mortality were excluded to avoid conflating deterministic thermal responses with parametric uncertainty. To assess robustness to sample adequacy and missing or invalid values, the attribution analysis was repeated across a range of minimum valid Monte Carlo sample sizes (10, 20, 30, 50, 75, and 100 draws). This iterative filtering ensured that inferred dominance patterns were not driven by small-sample effects or numerical instability. Correlation results were synthesised across robustness thresholds and visualised as temperature-dependent summaries, providing an assessment of both consistency and thermal stability of uncertainty structure. This approach provides a transparent and reproducible quantification of how biological uncertainty propagates into mechanistic suitability estimates.
2.5 Decision-relevant comparison of persistence-related metrics
To evaluate the decision relevance of persistence-related spatial metrics, we compared quantile-normalised mechanistic suitability, persistence probability derived from a branching-process interpretation, and raw mechanistic suitability using a spatially explicit, decision-oriented framework. All analyses were conducted on georeferenced raster outputs aligned to a common spatial grid using the terra package in R. Pixel-wise comparability was ensured by resampling rasters where necessary.
First, we assessed whether the different representations preserved the same spatial ordering by computing Spearman rank correlations between raw mechanistic suitability, quantile-normalised suitability, and persistence probability across all pixels. Because both transformations are monotonic, this step served to confirm that differences among metrics arise from representation rather than reordering of ecological information. To evaluate effective spatial resolution, we quantified saturation in persistence probability by calculating the proportion of pixels exceeding high probability thresholds (0.90, 0.95, and 0.99), with high saturation indicating compression of values and reduced discriminative capacity. We then assessed performance under explicit prioritisation rules reflecting fixed-effort decision contexts. For each metric, priority areas were defined as the top 10%, 20%, and 30% of locations based on area-matched thresholds, rather than absolute values. Spatial accordance between priority areas derived from quantile-normalised suitability and persistence probability was quantified using the Jaccard overlap index, and the fraction of the landscape classified as priority was recorded to verify consistency with the intended targeting proportion.
3 Results and discussion
3.1 Temperature-dependent persistence potential and uncertainty
The persistence potential of the tomato red spider mite, quantified using the basic reproduction number (R0), showed a strongly temperature-dependent and unimodal response (Figure 1). At low temperatures, R0 remained close to zero, below approximately 17 °C, highlighting that populations cannot persist under cold conditions due to severely constrained development and reproduction. Moreover, low temperatures are also characterised by high mortalities which, towards the limits, are close to 1. This low-temperature inhibition aligns with previous observations that spider mite development rates are reduced at temperatures below 15 °C, with minimal reproductive output (). Moreover, these values are also compatible with other close spider mite species, such as Tetranychus urticae ().
Figure 1
Above this lower thermal threshold, R0 increased sharply with temperature, reflecting rapid gains in population growth potential as developmental rates accelerated and survival improved, in line with . R0 reached a clear maximum at intermediate temperatures, peaking around 27–30 °C, which corresponds to optimal conditions for mite development, fecundity, and overall life-history performance. Beyond this optimum, R0 declined progressively with increasing temperature, driven primarily by reductions in fecundity and increasing adult mortality under thermal stress. At high temperatures (approximately ≥ 42 °C), R0 dropped back to zero, indicating complete loss of persistence potential due to reproductive failure, despite continued development at sublethal levels. Similar unimodal patterns have been reported in life table studies of Tetranychus spp., where reproductive potential and intrinsic growth peaked at intermediate temperatures (27–33 °C) before declining at hotter conditions ().
Uncertainty propagated from constant life-history parameters produced a moderate uncertainty envelope around the mean R0 curve, which was narrow near the thermal optimum and widened towards the thermal extremes. Importantly, uncertainty did not alter the qualitative interpretation of population persistence: R0 remained consistently above the persistence threshold (R0 = 1) across the intermediate temperature range, while remaining below this threshold at both low and high thermal extremes. This pattern of narrow optimal windows flanked by strong thermal limitation is typical in temperature-dependent arthropod population models, underscoring the importance of temperature as a fundamental driver of pest population dynamics ().
3.2 Contribution of life-history parameters to uncertainty in persistence potential
Uncertainty in persistence potential was primarily driven by variation in effective fecundity (sex ratio) across the evaluated temperature range (Table 2). This aligns with empirical demography of the tomato red spider mite, where both fecundity and progeny sex ratio vary with temperature and jointly determine realised daughter production, the key reproductive component underlying R0 ().
Table 2
| Parameter | 20 °C | 25 °C | 30 °C | 35 °C | 40 °C |
|---|---|---|---|---|---|
| Larval mortality | −0.72 | −0.61 | −0.50 | −0.41 | −0.33 |
| Protonymph mortality | −0.72 | −0.61 | −0.50 | −0.41 | −0.33 |
| Deutonymph mortality | −0.72 | −0.61 | −0.50 | −0.41 | −0.33 |
| Effective fecundity (sex ratio) | 0.59 | 0.70 | 0.79 | 0.85 | 0.8 |
Contribution of stochastic life-history parameters to uncertainty in risk index (RI).
Spearman rank correlation (ρ) between RI and parameter values across Monte Carlo draws. Only parameters treated as stochastic in the model are shown.
Spearman rank correlations between R0 and effective fecundity increased monotonically with temperature, rising from moderate values at cooler temperatures (ρ ≈ 0.59 at 20 °C) to strong positive associations at warmer temperatures (ρ ≈ 0.85–0.89 at 35–40 °C). A comparable biological pattern is reported in laboratory studies () where egg-to-adult survival remains high across warm conditions (e.g., >88% through 37.5 °C across multiple strains), which reduces the variance contribution of survival and increases the relative importance of reproduction-related terms such as fecundity and sex ratio ().
This pattern indicates that uncertainty in reproductive output becomes increasingly dominant as thermal conditions approach the upper range of suitability. Consistent with this interpretation, constant-temperature life tables for tomato red spider mite show that rm and R0 can reach very high values under warm conditions (e.g., rm ≈ 0.432 and net reproductive rate ≈ 111.77 at 35 °C), indicating that small proportional changes in reproductive output can translate into large changes in R0 ().
Conversely, uncertainty associated with stage-specific post-embryonic mortality showed consistently negative correlations with R0. Larval, protonymph, and deutonymph mortality showed strong negative associations at lower temperatures (ρ ≈ −0.72 at 20 °C), which gradually weakened with increasing temperature (|ρ| ≈ 0.33 at 40 °C). This is consistent with experimental observations that immature survivorship is generally reduced under cooler temperatures, making survival constraints more influential on R0 near the lower thermal range ().
The identical correlation structure across immature stages reflects their shared origin from uncertainty in overall post-embryonic survival. Notably, the relative contribution of each uncertain parameter to R0 variability remained invariant across alternative minimum inclusion thresholds (n = 10–100), demonstrating that uncertainty attribution was robust to data-filtering choices. From a modelling perspective, this robustness is consistent with demographic sensitivity/elasticity studies in spider mites showing stable identification of key life stages/traits affecting population growth across temperatures, although the dominant lever can differ depending on which traits are allowed to vary (e.g., adult survival often ranks highest when treated as variable) ().
Given that egg and adult mortality were modelled deterministically as temperature-dependent functions, they did not contribute to Monte Carlo uncertainty and were therefore excluded from this analysis. Collectively, these results indicate that uncertainty in persistence potential is mainly driven by reproductive traits, with survival-related uncertainty exerting a secondary, temperature-modulated influence.
3.3 Spatial distribution of raw mechanistic suitability (R0)
The raw mechanistic reproduction metric R0 highlighted extensive spatial heterogeneity across Africa, with most of the continent showing values well above the persistence threshold (R0 > 1) (Figure 2). This portion of the results indicates that, under long-term mean thermal conditions, large areas are climatically suitable for sustained population growth of tomato red spider mite. Consistent with this broad mechanistic index, previous climate suitability modelling has similarly suggested that most of sub-Saharan Africa is environmentally suitable for tomato red spider mite, with moisture-related stresses acting as principal constraints at the margins (; ). Future extensions of the model could incorporate relative humidity and vapour pressure deficit as additional drivers of development, survival, reproduction, and dispersal through direct temperature–moisture transition rates. This would allow the model to account for potentially important interactions between temperature and atmospheric moisture, thereby improving its capacity to describe population dynamics under a wider range of environmental conditions and to assess how moisture-related stress may limit the distribution and establishment of tomato red spider mite. Such extensions would be particularly relevant under future climate scenarios, where changes in temperature may be accompanied by changes in atmospheric moisture demand.
Figure 2
High suitability zones (R0 > 90, and notably R0 > 120) were concentrated across the Sahelian belt, extending from West Africa through Niger and Chad towards Sudan, as well as across parts of central Africa and lowland East Africa. This Sahel-adjacent corridor is also reflected in earlier CLIMEX parameterisation, where dry-stress limits were calibrated using Sahelian occurrence records including Senegal and Niger, supporting the plausibility of persistence potential across warm semi-arid belts (; ). These regions correspond to warm semi-arid and sub-humid climates that align closely with the thermal optimum derived from the life-history model.
Similarly, substantial portions of southern Africa, including Botswana, Zimbabwe, Mozambique, and interior South Africa, showed very high reproduction numbers, reflecting prolonged exposure to temperatures near optimal developmental ranges. This area was also identified by as highly suitable for tomato red spider mite, matching the high reproduction potential predicted for the region. Moderately high values (60–90) were widespread across tropical West Africa, Central Africa, and some area of East Africa, indicating broad climatic permissiveness for rapid population increase and confirming the results of . Conversely, reduced suitability was primarily associated with cooler high-elevation systems and more temperate southern latitudes. The southern African escarpment and the northern coastal margins and portions of North Africa showed markedly lower values, with some localised areas approaching or falling below the persistence threshold (R0 1). Because the species does not have diapause, and reproduction is continuous, cooler high-elevation climates are expected to curtail persistence, consistent with stress-based modelling frameworks that identify cold and cold–wet constraints as limiting for non-diapausing pests (; ).
This pattern underscores the dominant role of temperature-dependent development, mortality, and fecundity in shaping continental-scale persistence potential. Importantly, the magnitude of R0 varies substantially across space, with extreme values in optimal thermal corridors. While this preserves the full mechanistic signal of population growth potential, the wide dynamic range (including very large R0 values) complicates direct interpretation for operational decision-making. Nevertheless, the raw surface provides the fundamental ecological baseline against which transformed representations, such as branching probability and quantile-normalised suitability, can be evaluated.
3.4 Comparison of quantile-normalised suitability and branching-process probability
Although both the quantile-normalised suitability surface and the branching-process probability transformation were derived from the same underlying mechanistic reproduction metric, their spatial behaviour differed markedly in terms of decision relevance and interpretability (Figure 3). The quantile-normalised surface (Figure 3a) preserved the full spatial ranking of suitability across the continent while redistributing values uniformly over the unit interval. This transformation retained fine-scale gradients between regions, clearly distinguishing low-suitability highland systems (e.g., Ethiopian Highlands, East African Rift high elevations, southern African escarpments) from intermediate semi-arid zones and from the most climatically favourable Sahelian and lowland tropical corridors.
Figure 3
The resulting surface shows balanced contrast, providing relative prioritisation across space without compressing large areas into a narrow range of values. Conversely, the branching-process probability surface (Figure 3b) showed a more pronounced saturation. Because the majority of Africa showed raw R0 values substantially greater than 1, the transformation mapped most pixels to probabilities close to 1. Accordingly, large contiguous areas across West, Central, and East Africa appeared uniformly in the highest probability class (>0.8), with limited internal differentiation. Only marginal thermal regions, such as high-elevation systems and cooler southern latitudes, remained distinguishable as low-probability areas.
The spatial ordering of pixels remained monotonic between the two representations; however, the branching transformation substantially reduced effective spatial resolution. While it preserved feasibility information (i.e., whether persistence is possible), it diminished the capacity to discriminate among highly suitable regions. In contrast, the quantile-normalised surface maintained relative ranking structure by design, allowing explicit area-based targeting (e.g., top 10% of suitability) under fixed-resource decision rules.
Moreover, while both representations maintain monotonic ordering, the branching transformation shows strong saturation across much of sub-Saharan Africa, with most pixels clustered in high-probability classes. In contrast, the quantile-normalised surface retains balanced spatial contrast and clearer differentiation among highly suitable regions. This corroborates , who showed that quantile-based sensitivity analysis can preserve the same parameter-ranking information as failure-probability approaches, while avoiding the strong saturation often observed near extreme probabilities. Another study by , instead, showed that quantile-based values can be used to formulate decision alternatives even when uncertainty and dependence among inputs are both present. Taken together, although both representations originate from the same mechanistic framework, they differ fundamentally in their utility for spatial prioritisation. The branching-process probability is informative for binary persistence assessment, whereas quantile-normalised suitability provides clearer contrast and greater operational interpretability for spatiotemporal decision support. Although this study focuses on climatological variables, persistence-related suitability may also be influenced by biotic interactions, time-lagged environmental conditions, data quality and quantity, and the ability of models to capture underlying ecological mechanisms. These factors should therefore be considered in future assessments, as they may affect both model transferability and the spatial prediction of tomato red spider mite persistence suitability (; ; , ).
3.5 Expanded interpretation of ranking, saturation, and hotspot overlap results
Branching probability, raw mechanistic suitability, and quantile-normalised suitability showed perfect rank accordance (Spearman ρ = 1), indicating that all three representations preserve the same global ordering of locations. This outcome reflects the strictly monotonic relationship between raw mechanistic suitability and its subsequent transformations into persistence probability and quantile-normalised values. In other words, none of the representations reorders spatial risk; rather, they differ in risk expression and interpretation.
Despite this good rank accordance, branching probability showed pronounced saturation across the study domain. More than 94% of pixels exceeded a probability threshold of 0.90, more than 93% exceeded 0.95, and approximately 74% exceeded 0.99. This compression of values towards the upper bound indicates that, once persistence is broadly feasible, branching probability collapses effective spatial resolution, assigning near-uniform high probabilities to large areas. Consequently, small but potentially important differences in underlying mechanistic suitability are no longer distinguishable in probability space.
Area-matched hotspot analyses further illustrate this distinction. When hotspots were defined as the top 10%, 20%, or 30% of locations by each metric, quantile-normalised suitability and branching probability selected almost identical spatial extents, as reflected by near-unity Jaccard overlap across all thresholds. However, the interpretability of these hotspots differs fundamentally between metrics. For quantile-normalised suitability, hotspot thresholds are explicitly defined by decision rules (e.g. prioritising the top decile of risk). In contrast, for branching probability, equivalent prioritisation requires imposing arbitrary probability cut-offs that lack a consistent operational meaning under saturation.
4 Conclusion
This study provides an analytical protocol to derive a physiologically-based index based on the Next Generation Matrix and demonstrates that the usefulness of mechanistic ecological indicators depends not only on their biological realism but also on how they are represented for decision-making. While branching-process transformations provide a theoretically sound indicator of persistence feasibility, their tendency to saturate under broadly suitable environmental conditions limits their effectiveness for spatial prioritisation. Conversely, quantile-normalised mechanistic suitability preserves ecological ranking while enhancing spatial contrast, enabling transparent and consistent identification of priority areas under fixed-effort management scenarios.
Practitioners can select between branching probability and quantile normalisation approaches according to the specific characteristics of their management context and available data. Branching probability is preferable when the objective is to explicitly represent alternative developmental or demographic pathways and when sufficient information is available to estimate the probability of each branch. In contrast, quantile normalisation is more appropriate when the main objective is to harmonise predictions across different datasets, models, or environmental conditions, particularly when the underlying distributions differ in scale or variability. As a general decision rule, branching probability should therefore be selected when mechanistic interpretation and pathway-specific dynamics are the priority, whereas quantile normalisation should be favoured when comparability and calibration across heterogeneous data sources are more important. This framework provides a straightforward basis for selecting the most appropriate approach while retaining flexibility for different management applications.
For ecological indicator development, these findings highlight the importance of aligning metric formulation with operational objectives. Indicators intended for feasibility screening and those designed for spatial targeting serve fundamentally different purposes and should be evaluated accordingly. Rank-based transformations of uncertainty-aware mechanistic models offer a scalable and interpretable framework for translating life-history theory into actionable ecological indicators, particularly in data-limited or climatically permissive systems.
Statements
Data availability statement
Publicly available datasets were analyzed in this study. This data can be found here: https://doi.org/10.5281/zenodo.18791914.
Ethics statement
The manuscript presents research on animals that do not require ethical approval for their study.
Author contributions
KA: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. LR: Validation, Writing – original draft, Writing – review & editing. QM: Writing – original draft, Writing – review & editing. BS: Writing – review & editing. AA: Writing – original draft, Writing – review & editing. TD: Writing – review & editing. EA-R: Writing – review & editing.
Funding
The author(s) declared that financial support was received for this work and/or its publication. The authors sincerely appreciate the financial support provided by the Government of Norway for the project “Malawi Digital Plant Health Service; MaDiPHS, Project/Grant No. 988 983 837) Additional funding was received the Swedish International Development Cooperation Agency (Sida); the Swiss Agency for Development and Cooperation (SDC); the Australian Centre for International Agricultural Research (ACIAR); the Government of Norway; the German Federal Ministry for Economic Cooperation and Development (BMZ); and the Government of the Republic of Kenya. L.R. was funded by the European Commission, HORIZON-JU-CBE-2024 project “ZELDA”, Grant n. 101214981.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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Summary
Keywords
branching-process persistence probability, invasive pest risk assessment, Monte Carlo uncertainty propagation, quantile-normalised suitability, spatial prioritisation, trait-based R0 modelling
Citation
Agboka KM, Rossini L, Meltus QJ, Sokame BM, Azrag AGA, Dubois T and Abdel-Rahman EM (2026) Quantile normalisation enables interpretable spatial prioritisation of persistence-related mechanistic suitability under uncertainty for tomato red spider mite (Tetranychus evansi) in Africa. Front. Agron. 8:1840571. doi: 10.3389/fagro.2026.1840571
Received
27 March 2026
Revised
19 August 2026
Accepted
24 August 2026
Published
08 September 2026
Volume
8 - 2026
Edited by
Fahad Al Basir, Asansol Girls’ College, India
Reviewed by
Lifei Wang, University of Toronto, Canada
Khalid Aldawsari, Prince Sattam Bin Abdulaziz University, Saudi Arabia
Updates
Copyright
© 2026 Agboka, Rossini, Meltus, Sokame, Azrag, Dubois and Abdel-Rahman.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Komi Mensah Agboka, kagboka@icipe.org
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.