Ecology Invasion Olog: A Category-Theoretic Ontology Log for the Geng et al. (2016) Invasion Model

Methodology

This olog follows the framework of Giesa, Spivak, and Buehler (2011) (buehler2011-reoccurring-patterns): a category C where objects are sets representing entities in the ecological invasion domain, arrows are unique functions between them, and commutative diagrams enforce consistency constraints. A hierarchical subcategory H forms a forest organizing objects by structural level.

Domain source: Geng et al. (2016) — Alternanthera philoxeroides (alligator weed) invasion across native range (Argentina) and two introduced ranges (USA, China). The paper’s core finding: phenotypic plasticity, not genetic diversity, drives invasion success. Chinese populations had extremely low genetic diversity (94% identical multi-locus genotypes) yet achieved full predicted bioclimatic niche occupancy through plastic responses to water availability.

Notation conventions:

Objects are labeled as “an X is a Y” following Spivak’s olog convention, with Y the type
Arrows are labeled with their meaning in gerund form (e.g., “has a”)
Composition is read left-to-right: path P = arrow1 followed by arrow2
Commutativity: given two paths P1, P2 with the same domain and codomain, P1 = P2 as functions (for all x in the domain, the images under P1 and P2 are identical)

1. Objects (Sets)

Objects are organized into five hierarchical levels. Each level corresponds to one rank in the forest subcategory H.

1.1 Level 1 — Molecular/Genetic Level

Object	Label	Interpretation
Genotype	a specific multi-locus genotype	A particular ISSR marker profile (band presence/absence pattern across 8 primers producing 60 bands). In Geng’s data: 61 unique multi-locus genotypes among 179 individuals. (geng2016-genetic-diversity-phenotypic-plasticity, Results)
MarkerLocus	an ISSR marker position	A specific PCR band position scored as present (1) or absent (0). Geng used 8 primers generating 60 bands total. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)
GeneticDiversity	a triple of diversity measures	The set of values (P, He, I) where P = percentage of polymorphic loci, He = Nei’s genic diversity index, I = Shannon diversity index. E.g., China: (2.22%, 0.0043, 0.0071); USA: (69.33%, 0.2293, 0.3445); Argentina: (60.00%, 0.1821, 0.2759). (geng2016-genetic-diversity-phenotypic-plasticity, Table 1A)

1.2 Level 2 — Organismal Level

Object	Label	Interpretation
Individual	a single plant ramet	A collected specimen of A. philoxeroides from field sampling. Geng sampled 179 individuals: Argentina n=21, USA n=32, China n=126. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)
Phenotype	a vector of trait measurements	An 8-element vector (leaf length, stem diameter, stem pith cavity diameter, internode length, specific leaf area, relative chlorophyll content, root/shoot ratio, storage root/fine root ratio) measured in a specific treatment. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)
PhenotypicPlasticity	a plasticity index value	A quantitative measure Ip = (Max − Min)/Mean for a given trait across terrestrial and aquatic treatments. The PCA showed 67.61% of phenotypic variation was plastic response to habitat. (geng2016-genetic-diversity-phenotypic-plasticity, Results)
ReactionNorm	a pattern of phenotypic expression across environments	The function f: EnvironmentPatch → Phenotype for a fixed genotype. Visualized as a line plot of trait value vs. environment (Figure 4 in Geng). Geng found clone-level reaction norm slopes varied significantly. (geng2016-genetic-diversity-phenotypic-plasticity, Results)

1.3 Level 3 — Population Level

Object	Label	Interpretation
PlantPopulation	a set of individuals at a geographic site	A sampling site population. Geng sampled 7 Argentine sites, 9 USA sites, 9 Chinese sites. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)
Clone	a genet — all ramets with the same multi-locus genotype	A group of genetically identical individuals produced through clonal propagation. In China: 119/126 ramets shared one multi-locus genotype (C-Dominant). In USA and Argentina, each ramet had a unique genotype. (geng2016-genetic-diversity-phenotypic-plasticity, Results)
CloneFrequency	a frequency distribution of genotypes in a population	A function mapping each genotype to its relative abundance. China: C-Dominant at 94.4% frequency, 7 minor genotypes at low frequency. USA and Argentina: each genotype at ~uniform frequency (each unique). (geng2016-genetic-diversity-phenotypic-plasticity, Results)
FounderEvent	the establishment of a new population by a small number of individuals	A specific introduction event. China: single introduction (the 8 Chinese genotypes form a single clade). USA: multiple introductions (USA genotypes intermingled with Argentine genotypes in the NJ tree). (geng2016-genetic-diversity-phenotypic-plasticity, Results)

1.4 Level 4 — Environmental Level

Object	Label	Interpretation
EnvironmentPatch	a specific habitat type	The set {terrestrial, aquatic}. Geng’s common garden experiment manipulated water availability at these two levels. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)
WaterAvailability	a water regime level	The set {high (aquatic), low (terrestrial)}. Aquatic: 1 m deep ponds. Terrestrial: raised garden beds. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)
BioclimaticNiche	the set of climate conditions where the species can persist	The CLIMEX model output: Environmental Index (EI, scaled 0–100) based on temperature, soil moisture, cold stress, and heat stress parameters fitted against native-range (Argentina) distribution data. (geng2016-genetic-diversity-phenotypic-plasticity, Methods)

1.5 Level 5 — Geographic Level

Object	Label	Interpretation
NativeRange	the native geographic range of A. philoxeroides	A singleton set {Argentina}. Specifically, populations along the Parana, Uruguay, San Borombon, and Salado rivers. (geng2016-genetic-diversity-phenotypic-plasticity, Methods, Figure 2)
IntroducedRange	a non-native geographic range	The set {USA, China}. USA: southern coastal plains Virginia to Texas and California. China: most provinces south of the Yellow River. (geng2016-genetic-diversity-phenotypic-plasticity, Methods, Figure 2)
PredictedNiche	a CLIMEX-predicted bioclimatic distribution	The set of areas within an IntroducedRange where the Environmental Index exceeds the threshold for persistence, as projected from the BioclimaticNiche model. (geng2016-genetic-diversity-phenotypic-plasticity, Results)
NicheOccupancy	a boolean occupancy outcome	The set {occupied, unoccupied} for each geographic region. Geng found: full potential distribution invaded in both China and USA. (geng2016-genetic-diversity-phenotypic-plasticity, Results)

2. Arrows (Functions)

Arrows are labeled in gerund form: arrow name: Domain → Codomain.

2.1 Genetic Arrows

hasDiversity: PlantPopulation → GeneticDiversity

Maps each plant population to its diversity measures (P, He, I).
Empirical basis: Table 1A — Argentina (60.00%, 0.1821, 0.2759), USA (69.33%, 0.2293, 0.3445), China (2.22%, 0.0043, 0.0071) after resampling correction for sample size.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Table 1A.

hasGenotype: Individual → Genotype

Maps each individual ramet to its ISSR multi-locus genotype.
Empirical basis: 61 unique multi-locus genotypes from 179 individuals across 8 ISSR primers.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

constitutes: Genotype → PlantPopulation

Maps each genotype to the population(s) in which it is found. Injects genotype into its population context.
Empirical basis: Chinese genotypes form a single clade closest to USA sites N4 and N8; USA and Argentine genotypes intermingle.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Figure 3.

2.2 Phenotypic Arrows

propagates: Individual → Clone

Maps an individual to its clonal genet (asexual reproduction). In A. philoxeroides, propagation occurs through vegetative structures (roots, broken stems); viable seeds are rare in introduced ranges.
Empirical basis: 94% of Chinese ramets belong to one clone (C-Dominant).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

expresses: Genotype × EnvironmentPatch → Phenotype

Maps a genotype-environment pair to the resulting phenotype (reaction norm evaluation). Takes both a specific genotype and a specific environment, returns the trait vector measured.
Empirical basis: two-way ANOVA showed significant treatment effect on all 8 traits (P < 0.01), confirming that genotypes express different phenotypes in aquatic vs. terrestrial conditions.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Table 2.

hasPlasticity: Genotype → PhenotypicPlasticity

Maps each genotype to its plasticity index Ip for each trait. Measures the magnitude of plastic response.
Empirical basis: plasticity indices calculated as (Max − Min)/Mean. No significant differences among regions for most traits (Figure 5). Some Argentine clones showed higher plasticity than USA and Chinese clones.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Figure 5, Table S3.

hasReactionNorm: Genotype × [EnvironmentPatch] → ReactionNorm

Maps a genotype and the set of environment patches to its complete reaction norm — the pattern of phenotypic expression across all environments.
Empirical basis: clone-level reaction norm slopes varied significantly (Table 2, clonal effect P < 0.01). The slope variation was greatest among Argentine clones (Figure S1).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Figure S1.

hasGeneticPlasticity: PlantPopulation → PhenotypicPlasticity

Maps a population to the plasticity of its dominant or representative genotype(s). This is the population-level aggregate of hasPlasticity on individual genotypes.
Empirical basis: Chinese population has C-Dominant as its representative genotype; the plasticity of C-Dominant drives population-level invasion success.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

2.3 Population Arrows

founds: PlantPopulation × EnvironmentPatch → FounderEvent

Maps a source population and the environment patch where establishment occurs to a founder event. A small subset of the source population establishes a new population.
Empirical basis: China experienced a single founder event from an unknown source (most closely related to USA sites N4 and N8). USA experienced multiple founder events from Argentine sources (multiple introductions).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

bottlenecks: FounderEvent → PlantPopulation

Maps a founder event to the resulting bottlenecked population. The arrow encodes that the new population is a subset of the source, with reduced genetic diversity.
Empirical basis: China bottleneck produced 94% genetic uniformity (P = 2.22%). USA bottleneck was mild (multiple introductions, P = 69.33%).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

countsClones: PlantPopulation → CloneFrequency

Maps a population to its clone frequency distribution.
Empirical basis: China: C-Dominant at 119/126 with 7 minor genotypes. USA and Argentina: each individual a unique genotype (n = 21 and 32 unique genotypes respectively).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

invades: PlantPopulation → NicheOccupancy

Maps a population to whether it occupies its potential niche. Encodes the invasion outcome at the population level.
Empirical basis: both China and USA populations achieved full predicted bioclimatic distribution.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

2.4 Environmental Arrows

hasWaterRegime: EnvironmentPatch → WaterAvailability

Maps a habitat type to its water availability level.
Empirical basis: aquatic treatment = 1 m deep ponds (high water availability); terrestrial treatment = raised garden beds (low water availability).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Methods.

shapes: WaterAvailability → Phenotype

Maps water availability to the phenotypic values it induces across individuals. This is the environmental main effect on phenotype.
Empirical basis: “plants in aquatic plots had longer leaves, longer internodes, thicker stems and larger stem pith cavity, larger specific leaf area, lower root/shoot ratio, lower relative chlorophyll content, and lower storage root/fine root ratio” than terrestrial plots (Results).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

2.5 Geographic Arrows

models: NativeRange → BioclimaticNiche

Maps the native range to the CLIMEX-predicted bioclimatic niche (calibrated on native distribution data). The model is fitted against Argentine distribution of A. philoxeroides.
Empirical basis: CLIMEX temperature and moisture parameters fitted to maximize match with Argentine occurrence data. The model overestimated native-range distribution, suggesting other factors (topography, competition) also limit native range.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Methods, Discussion.

projects: BioclimaticNiche × IntroducedRange → PredictedNiche

Maps the bioclimatic niche model applied to an introduced range to the predicted suitable area. This is the CLIMEX projection step.
Empirical basis: predictions mapped for both USA and China using CliMond world 10-minute climate dataset.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Methods, Figure 2.

checksOccupancy: PredictedNiche × PlantPopulation → NicheOccupancy

Maps a predicted niche and a plant population to whether the population actually occupies that niche.
Empirical basis: “bioclimatic modeling suggested that the full potential distribution of the species in the introduced ranges were invaded in China and the USA” (Results).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Results.

2.6 Cross-Level Arrows for Commutativity

enables: PhenotypicPlasticity → NicheOccupancy

Maps a plasticity level to niche occupancy. The core causal arrow: high plasticity enables full niche occupancy independent of genetic diversity.
Empirical basis: Chinese populations had low genetic diversity but high plasticity and full occupancy; USA populations had high genetic diversity, high plasticity, and full occupancy.
Source: geng2016-genetic-diversity-phenotypic-plasticity, Abstract, Discussion.

predicts: GeneticDiversity → NicheOccupancy

Maps a genetic diversity level to a PREDICTED (not actual) occupancy. Exists as the naive hypothesis: “low genetic diversity limits niche breadth.”
Empirical basis: this arrow FAILS to predict actual occupancy for China. Diversity range: P=2.22% (China) to 69.33% (USA). Predicted: China should have restricted occupancy. Actual: China has full occupancy. This arrow gives the wrong answer for bottlenecked populations (see Condition 4).
Source: geng2016-genetic-diversity-phenotypic-plasticity, Abstract, Discussion.

3. Commutativity Conditions

Each condition is stated formally as a path equality (or inequality) in the category. Empirical notes indicate where Geng’s data supports or qualifies the condition.

3.1 Condition C1: Plasticity-Diversity Trade-off

Formal statement: For any genotype g (in Genotype) belonging to population p (in PlantPopulation):

Path A: g →hasPlasticity→ PhenotypicPlasticity →enables→ NicheOccupancy Path B: g →constitutes→ PlantPopulation →hasDiversity→ GeneticDiversity →predicts→ NicheOccupancy

Paths A and B jointly determine NicheOccupancy. The diagram forms a cone: NicheOccupancy is the pullback of the two arrows enables and predicts along the two routes from Genotype. The trade-off principle states that either sufficient plasticity (Path A) OR sufficient diversity (Path B) is adequate for full occupancy.

Empirical support: There exists at least one population (USA) where Path B predicts full occupancy (P=69.33% → high→ full) AND Path A produces full occupancy (plasticity comparable to native range). There exists another population (China) where Path B predicts LOW occupancy (P=2.22% → low → restricted) but Path A still produces full occupancy. This confirms the joint-determination model: Path A dominates in bottlenecked populations.

Formally: let f = enables ∘ hasPlasticity ∘ constitutes⁻¹ and let g = predicts ∘ hasDiversity. For any population p:

If hasDiversity(p) is high (USA, Argentina), then f(p) = g(p) (both predict full occupancy)
If hasDiversity(p) is low (China), then f(p) != g(p) (plasticity path dominates; diversity gives wrong prediction)

Source: geng2016-genetic-diversity-phenotypic-plasticity, Abstract, Results, Discussion.

flowchart LR
    subgraph C1["Condition C1: Plasticity-Diversity Trade-off"]
        G["Genotype"] -->|hasPlasticity| PP["PhenotypicPlasticity"]
        G -->|constitutes| PoP["PlantPopulation"]
        PoP -->|hasDiversity| GD["GeneticDiversity"]
        PP -->|enables| NO["NicheOccupancy"]
        GD -->|predicts| NO
    end

    style NO fill:#f9f,stroke:#333,stroke-width:2px

C1: NicheOccupancy as the pullback of enables and predicts routes from Genotype. The arrow predicts gives the wrong answer for bottlenecked populations (China), establishing that enables is the dominant route.

3.2 Condition C2: Founder Bottleneck Condition

Formal statement: For any founder event f (in FounderEvent) that produces a bottlenecked population p = bottlenecks(f) (in PlantPopulation), and letting h = hasDiversity ∘ bottlenecks:

The composite FounderEvent →bottlenecks→ PlantPopulation →hasDiversity→ GeneticDiversity produces a residual diversity measure that depends on bottleneck severity:

If severity(f) = “severe” (single introduction, as in China), then residual diversity → low (P ≈ 2.22%)
If severity(f) = “mild” (multiple introductions, as in USA), then residual diversity → high (P ≈ 69.33%)

The condition states that bottleneck severity determines residual diversity, with two discrete regimes.

Empirical support: China: single introduction → 94% of ramets identical (C-Dominant), only 8 genotypes total, P=2.22%. USA: multiple introductions from Argentine sources → 32 unique genotypes, P=69.33%, comparable to native Argentina (P=60.00%). The NJ tree confirms: Chinese genotypes form a single, well-supported clade; USA genotypes are intermingled with Argentine genotypes across multiple clades (Figure 3).

Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Figure 3, Table 1A.

flowchart LR
    subgraph C2["Condition C2: Founder Bottleneck"]
        FE["FounderEvent"] -->|bottlenecks| PP2["PlantPopulation"]
        PP2 -->|hasDiversity| GD2["GeneticDiversity"]
        PP2 -->|hasGeneticPlasticity| PP2b["PhenotypicPlasticity"]
        FE -->|determines| Sev["BottleneckSeverity<br/>{severe, mild}"]
        Sev -->|controls| GD2
        Sev -->|does NOT control| PP2b
    end

C2: Bottleneck severity determines residual genetic diversity but does NOT erode phenotypic plasticity (dashed line from Sev to PP2b is absent — plasticity is conserved).

3.3 Condition C3: Plasticity Conservation

Formal statement: Let g_N be a genotype in the native range and g_I a genotype in an introduced range. Then:

hasPlasticity(g_N) = hasPlasticity(g_I)

for any pair of genotypes. That is, the plasticity index is invariant under geographic range: plasticity is a species-level trait.

Empirical support: Two-way ANOVA found “no significant differences among regions for most examined traits when comparing plasticity indices” (Results). The treatment-by-region interaction was non-significant (Table 2), meaning the magnitude of plastic response was comparable across Argentina, USA, and China. The Mantel test found no significant correlation between marker distance and plasticity dissimilarity (r = 0.15, p = 0.29), confirming that plastic response norm is an inherent (species-level) acclimation, not under local genetic control.

Qualification: Some Argentine clones showed higher plasticity than USA and Chinese clones for certain traits (Figure 5, Figure S1). The equality is approximate at the regional mean level, not exact at the individual clone level.

Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Table 2, Figure 5, Figure S1.

flowchart LR
    subgraph C3["Condition C3: Plasticity Conservation"]
        NR["NativeRange"] -->|has representative| G_N["Genotype<br/>(Argentina)"]
        IR["IntroducedRange"] -->|has representative| G_I["Genotype<br/>(China)"]
        G_N -->|hasPlasticity| PP_N["PhenotypicPlasticity<br/>Ip = 0.3-0.7"]
        G_I -->|hasPlasticity| PP_I["PhenotypicPlasticity<br/>Ip = 0.3-0.7"]
        PP_N ---|equal| PP_I
    end

C3: Plasticity is conserved across ranges. The hasPlasticity arrow produces the same output regardless of whether the genotype originated in the native or introduced range.

3.4 Condition C4: Niche Occupancy Independence

Formal statement: For any population p (in PlantPopulation):

Path A: p →hasDiversity→ GeneticDiversity →predicts→ NicheOccupancy

WRONG outcome for bottlenecked populations: predicts(p) = “restricted” when hasDiversity(p) = low.

Path B: p →hasGeneticPlasticity→ PhenotypicPlasticity →enables→ NicheOccupancy

CORRECT outcome for all populations: enables(p) = “full” regardless of hasDiversity(p) when plasticity is high.

The condition states that Path A is not an arrow in the correct model of the system — it is a false expectation. The only valid functional arrow from PlantPopulation to NicheOccupancy factors through PhenotypicPlasticity, not GeneticDiversity. Formally: there is NO arrow predicts: GeneticDiversity → NicheOccupancy that makes the diagram commute for all populations. Such an arrow exists only for high-diversity populations (USA, Argentina).

Empirical support: Chinese populations had P = 2.22%, He = 0.0043, I = 0.0071 — among the lowest genetic diversity values reported for any clonal plant. Under the naive diversity-dependence model, this should have restricted them to a narrow range of habitats. Yet “the full potential distribution of the species in the introduced ranges were invaded in China and the USA” (Results). The PCA showed that 67.61% of phenotypic variation was a plastic response to habitat, not genetic variation.

Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Discussion.

flowchart LR
    subgraph C4["Condition C4: Niche Occupancy Independence"]
        PP3["PlantPopulation"] -->|hasDiversity| GD3["GeneticDiversity<br/>P = 2.22% (China)"]
        PP3 -->|hasGeneticPlasticity| PP3b["PhenotypicPlasticity<br/>Ip high (China)"]
        GD3 -->|predicts ✗| NO3["NicheOccupancy<br/>(restricted? No!)"]
        PP3b -->|enables ✓| NO3
    end

C4: The path through GeneticDiversity (solid arrow marked with ✗) gives the WRONG prediction for bottlenecked populations. The correct path (through PhenotypicPlasticity) is marked ✓.

3.5 Condition C5: Common Garden Commutativity

Formal statement: For any environment patch e (in EnvironmentPatch), and any individual i_N from the native range (Argentina) and i_I from an introduced range (USA or China) that came through the same common garden protocol:

expresses(hasGenotype(i_N), e) = expresses(hasGenotype(i_I), e)

That is, the phenotype produced by a native-range genotype in environment e equals the phenotype produced by an introduced-range genotype in the same environment e. The diagram commutes: the phenotype depends on the environment and the species-level developmental program, not on the genotype’s geographic origin.

Empirical support: Two-way ANOVA showed “significant effects of treatment on all the traits” but “treatment-by-region interaction was not significant for most examined traits” (Table 2). This means the phenotypic response to water treatment was consistent across regions: genotypes from Argentina, USA, and China all responded similarly to the aquatic/terrestrial contrast. The NJ tree grouping patterns (Figure 3) support this: Chinese genotypes form a single clade but express the same phenotypes as the interspersed USA-Argentine genotypes in both treatments.

Formal qualification: The equality holds at the level of the species-wide reaction norm but not at the level of individual clone-level trait values (clonal effect P < 0.01 in Table 2). Individual genotypes within each region varied in their exact trait values. The commutativity condition holds for the REGIONAL MEAN phenotype, not for every individual.

Source: geng2016-genetic-diversity-phenotypic-plasticity, Results, Table 2, Figure 4.

flowchart LR
    subgraph C5["Condition C5: Common Garden Commutativity"]
        subgraph Native["Native Range"]
            IN["Individual (Argentina)<br/>n = 21"]
        end
        subgraph Introduced["Introduced Ranges"]
            II["Individual (USA/China)<br/>n = 32/126"]
        end
        EP["EnvironmentPatch<br/>{terrestrial, aquatic}"] -->|hasWaterRegime| WA["WaterAvailability"]
        IN -->|hasGenotype| G_N2["Genotype (Argentina)"]
        II -->|hasGenotype| G_I2["Genotype (USA/China)"]
        G_N2 -->|expresses| PH_N["Phenotype<br/>(aquatic: long leaves,<br/>thick stems,<br/>low root/shoot ratio)"]
        G_I2 -->|expresses| PH_I["Phenotype<br/>(same pattern)"]
        EP -->|environment factor| PH_N
        EP -->|environment factor| PH_I
        PH_N ===|equal| PH_I
    end

C5: The phenotype expressed by native-range and introduced-range genotypes in the same environment is equal at the regional mean level. Arrows into Phenotype from both Genotype and EnvironmentPatch show the joint determination. The equal sign (=) marks commutativity.

4. Summary Diagram: The Complete Olog

The following diagram shows the full olog with all objects organized by hierarchical level and the key arrows connecting them. Objects are grouped by level (G = genetic, O = organismal, P = population, E = environmental, Gg = geographic). Not all arrows are shown; only the structurally essential ones for the commutativity conditions are included.

flowchart TD
    subgraph L1["Level 1: Genetic"]
        G["Genotype"]
        M["MarkerLocus"]
        GD["GeneticDiversity<br/>(P, He, I)"]
    end

    subgraph L2["Level 2: Organismal"]
        I["Individual"]
        PH["Phenotype"]
        PP["PhenotypicPlasticity<br/>(Ip)"]
        RN["ReactionNorm"]
    end

    subgraph L3["Level 3: Population"]
        POP["PlantPopulation"]
        CL["Clone"]
        CF["CloneFrequency"]
        FE["FounderEvent"]
    end

    subgraph L4["Level 4: Environmental"]
        EP["EnvironmentPatch"]
        WA["WaterAvailability"]
        BN["BioclimaticNiche"]
    end

    subgraph L5["Level 5: Geographic"]
        NR["NativeRange<br/>{Argentina}"]
        IR["IntroducedRange<br/>{USA, China}"]
        PN["PredictedNiche"]
        NO["NicheOccupancy"]
    end

    %% Genetic arrows
    POP -->|hasDiversity| GD
    I -->|hasGenotype| G
    G -->|constitutes| POP

    %% Organismal arrows
    I -->|propagates| CL
    G -->|expresses, w/ EP| PH
    G -->|hasPlasticity| PP
    G -->|hasReactionNorm, w/ [EP]| RN
    EP -->|hasWaterRegime| WA
    WA -->|shapes| PH

    %% Population arrows
    POP -->|founds, w/ EP| FE
    FE -->|bottlenecks| POP
    POP -->|countsClones| CF
    POP -->|hasGeneticPlasticity| PP

    %% Geographic arrows
    POP -->|invades| NO
    NR -->|models| BN
    BN -->|projects, w/ IR| PN
    PN -->|checksOccupancy, w/ POP| NO

    %% Commutativity arrows
    PP -->|enables| NO
    GD -.->|predicts (wrong for China)| NO

    %% Level labels
    classDef level1 fill:#e1f5fe,stroke:#0288d1
    classDef level2 fill:#e8f5e9,stroke:#388e3c
    classDef level3 fill:#fff3e0,stroke:#f57c00
    classDef level4 fill:#fce4ec,stroke:#d32f2f
    classDef level5 fill:#f3e5f5,stroke:#7b1fa2

    class G,M,GD level1
    class I,PH,PP,RN level2
    class POP,CL,CF,FE level3
    class EP,WA,BN level4
    class NR,IR,PN,NO level5

Full ecology invasion olog. Solid arrows represent valid functional relationships. Dashed arrow (predicts: GeneticDiversity → NicheOccupancy) represents the naive expectation that empirically fails for bottlenecked populations (Condition C4). Color coding: blue = genetic level, green = organismal level, orange = population level, red = environmental level, purple = geographic level. Dependencies: expresses(Genotype, EnvironmentPatch), founds(PlantPopulation, EnvironmentPatch), projects(BioclimaticNiche, IntroducedRange), checksOccupancy(PredictedNiche, PlantPopulation).

5. Hierarchical Subcategory H (Forest)

The hierarchical subcategory H is a forest (disjoint union of trees) in which each object belongs to exactly one level, and the levels are partially ordered by composition. An object at level L is composed of objects at level L−1.

5.1 Forest Structure

Level 5 (Geographic):
  NativeRange ──{has distribution of}──> Level 4: BioclimaticNiche
  IntroducedRange
  NicheOccupancy

Level 4 (Environmental):
  EnvironmentPatch ──{provides conditions for}──> Level 3: PlantPopulation
  WaterAvailability
  BioclimaticNiche

Level 3 (Population):
  PlantPopulation ──{consists of}──> Level 2: Individual
  Clone                        ──> Level 2: Individual (identical genotypes)
  FounderEvent
  CloneFrequency

Level 2 (Organismal):
  Individual ──{has}──> Level 1: Genotype
  Phenotype
  PhenotypicPlasticity
  ReactionNorm

Level 1 (Genetic):
  Genotype            ──{scored at}──> MarkerLocus
  GeneticDiversity    ──{measured at}──> MarkerLocus
  MarkerLocus

5.2 Level Membership

Objects are assigned to levels by a function level: Ob(C) → {1, 2, 3, 4, 5}:

Level	Objects	Composition principle
1 (Genetic)	Genotype, MarkerLocus, GeneticDiversity	Molecular entities measurable by ISSR markers
2 (Organismal)	Individual, Phenotype, PhenotypicPlasticity, ReactionNorm	Entities and properties of single plant ramets
3 (Population)	PlantPopulation, Clone, CloneFrequency, FounderEvent	Collections of organisms and their dynamics
4 (Environmental)	EnvironmentPatch, WaterAvailability, BioclimaticNiche	Abiotic conditions that shape phenotype and distribution
5 (Geographic)	NativeRange, IntroducedRange, PredictedNiche, NicheOccupancy	Macro-scale spatial entities

The forest condition (each object has at most one parent at the next level, following Buehler/Spivak 2011) is satisfied: every object at level L has a unique compositional arrow to exactly one object at level L+1, forming a tree. Objects at the top level (5) have no outgoing compositional arrows.

6. Integration with Empirical Data

Every object and arrow in this olog is grounded in Geng et al. (2016). The following table summarizes the empirical support for the core functional relationships:

Arrow	Domain	Codomain	Primary evidence	P-value / value
hasDiversity	PlantPopulation	GeneticDiversity	Table 1A: P, He, I by region	China P=2.22%, USA P=69.33%, Argentina P=60.00%
hasGenotype	Individual	Genotype	All 179 individuals genotyped; 61 unique multi-locus genotypes	8 ISSR primers, 60 bands
expresses	Genotype × EnvironmentPatch	Phenotype	Two-way ANOVA: treatment effect on all 8 traits	Treatment P < 0.01 for all traits (Table 2)
hasPlasticity	Genotype	PhenotypicPlasticity	Plasticity indices per clone per trait	No sig. regional differences (Figure 5)
hasGeneticPlasticity	PlantPopulation	PhenotypicPlasticity	Regional mean plasticity indices	No treatment × region interaction (Table 2)
bottlenecks	FounderEvent	PlantPopulation	Clonal structure in NJ tree (Figure 3)	China: 94% identical; USA: unique each
invades	PlantPopulation	NicheOccupancy	CLIMEX projection vs. actual distribution	Full distribution invaded in both USA and China
projects	BioclimaticNiche × IntroducedRange	PredictedNiche	CLIMEX EI maps for USA and China	EI scaled 0–100 (Figure 2)
models	NativeRange	BioclimaticNiche	CLIMEX fitted on Argentine data	Parameters in Table S2

6.1 Key Numerical Values for Arrow Codomains

GeneticDiversity codomain values (Table 1A):

Population	P (%)	He	I
Argentina	60.00	0.1821	0.2759
USA	69.33 (71.67 resampled)	0.2293 (0.2323)	0.3445 (0.3495)
China	2.22 (11.67 resampled)	0.0043 (0.0144)	0.0071 (0.0260)

PhenotypicPlasticity codomain (plasticity indices, from Figure 5 and Table S3):

Ip values vary by trait. For leaf length: consistent across regions (no significant pairwise differences).
PCA axis: “67.61% of the phenotypic variation within the common garden experiment was a plastic response to habitat treatment” (Results).

NicheOccupancy codomain:

China: full potential distribution occupied
USA: full potential distribution occupied
Argentina: actual distribution smaller than predicted (overestimated by CLIMEX, suggesting topography/competition constraints)

6.2 Commutativity Condition Status

Condition	Formal status	Empirical status	Key statistic
C1 (Plasticity-Diversity)	Pullback cone (not strict commutativity)	Supported: plasticity dominates when diversity is low	PCA: 67.61% of variation is plastic response
C2 (Founder Bottleneck)	Commutes for each severity regime	Supported: single vs. multiple introductions produce distinct outcomes	C-Dominant at 94% (China) vs. unique each (USA)
C3 (Plasticity Conservation)	Commutes at regional mean level	Qualified: some Argentine clones more plastic than introduced	Treatment × region: non-significant (Table 2); Mantel: r=0.15, p=0.29
C4 (Niche Occupancy Independence)	Non-commutativity (identifies invalid arrow)	Supported: diversity path fails for bottlenecked populations	China P=2.22% but full occupancy
C5 (Common Garden)	Commutes at regional mean level	Qualified: clones differ within regions	Clonal effect P<0.01; treatment × region non-significant

7. Cross-Domain Mapping Notes

This olog is the source category for a functor to the cancer evolution domain olog. The mapping F: EcologyCat → CancerCat preserves the hierarchical structure and commutativity conditions. Key correspondences (from dual-regime-evolution and population-bottleneck):

Ecology object	Cancer analogue	Mapping rationale
PlantPopulation	TumorCellPopulation	Both are populations of clonally propagating cells with heritable variation
FounderEvent	TherapyBottleneck	Both collapse genetic diversity: single introduction (ecology) = deep treatment response (cancer)
GeneticDiversity (P, He, I)	SubclonalMutationalHeterogeneity	Both are measures of genetic variation in the population
PhenotypicPlasticity (Ip)	EpigeneticPlasticity	Both enable phenotypic variation without genetic change
WaterAvailability	TherapeuticPressure	Both are environmental variables that shape the adaptive landscape
NicheOccupancy	MetastaticColonization / TherapyResistance	Both are measures of adaptive success in the target environment
NativeRange (Argentina)	PrimaryTumor	Source population with full genetic diversity
IntroducedRange (USA/China)	MetastaticSite	Target of colonization with reduced or maintained diversity

Functorial preservation of commutativity:

C2 (Founder Bottleneck): F maps bottlenecks: FounderEvent → PlantPopulation to therapyBottleneck: TherapyBottleneck → TumorCellPopulation. The condition that bottleneck severity determines residual diversity holds in both domains: deep therapy response (cancer) = single introduction (ecology); mild response = multiple introductions.
C3 (Plasticity Conservation): F maps hasPlasticity: Genotype → PhenotypicPlasticity to hasEpigeneticPlasticity: EpigeneticState → EpigeneticPlasticity. The condition that plasticity is a species-level (cancer: disease-level) trait preserved across environments holds in both domains.
C4 (Niche Occupancy Independence): F maps the non-commutativity: just as low genetic diversity does not restrict niche occupancy in ecology, low subclonal heterogeneity does not restrict adaptive success in cancer.

Strict vs. non-strict functor: The functor F is non-strict in two ways:

The object Genotype in ecology maps to a combination of EpigeneticState and GeneticSubclone in cancer (the dual-regime model dual-regime-evolution)
The arrow expresses: Genotype × EnvironmentPatch → Phenotype maps to a more complex relationship in cancer (epigenetic regulation has bidirectional coupling to environment)

The non-strictness is informative: it identifies where the analogy is exact (bottleneck dynamics, plasticity compensation) and where it breaks (cancer’s dual-regime coupling has no ecological analogue in the single-species invasion model). See dual-regime-evolution and population-bottleneck for the full cross-domain synthesis.

8. Limitations and Caveats

Oversimplification of phenotypic plasticity. The olog treats PhenotypicPlasticity as a single numerical value (Ip), but Geng’s data show it varies across traits, clones, and treatments. The reaction norm is multi-dimensional (8 traits × 2 treatments × 25 clones). The scalar representation is a projection.
Binary niche occupancy. The olog uses {occupied, unoccupied} for NicheOccupancy, but the CLIMEX model produces continuous Environmental Index values. The binary threshold is an abstraction.
Static representation. As with Buehler/Spivak’s ologs, this representation is static. It does not capture temporal dynamics (invasion as a process unfolding over time, clonal expansion rates, or the sequence of multiple introductions).
Deterministic assumption. The olog treats arrows as deterministic functions. Genetic drift, sampling variation, and stochasticity in founder events are not represented. A Markov-categorical extension would be needed for probabilistic extension.
Single species. The olog is specific to A. philoxeroides. Generalization to other clonal invaders requires verifying that the commutativity conditions hold for other species, which Geng explicitly cautions against: “the role of genetic diversity in invasion success might be variable for plant species with different reproductive modes” (Discussion).
Bioclimatic model overestimation. The CLIMEX model “greatly overestimated the native-range distribution of A. philoxeroides” (Discussion), suggesting that the arrow models: NativeRange → BioclimaticNiche is only a partial function — it systematically over-predicts native occupancy, which may carry over to introduced-range projections.
Experimental design constraints. The two-level water treatment (terrestrial/aquatic) may not capture the full environmental heterogeneity of natural populations. The arrow expresses: Genotype × EnvironmentPatch → Phenotype is defined over only two patches.

9. Formal Olog Summary

Category C (EcologyInvasionOlog):

Objects (18): Genotype, MarkerLocus, GeneticDiversity, Individual, Phenotype, PhenotypicPlasticity, ReactionNorm, PlantPopulation, Clone, CloneFrequency, FounderEvent, EnvironmentPatch, WaterAvailability, BioclimaticNiche, NativeRange, IntroducedRange, PredictedNiche, NicheOccupancy

Arrows (19 excluding identity): hasDiversity, hasGenotype, constitutes, propagates, expresses, hasPlasticity, hasReactionNorm, hasGeneticPlasticity, founds, bottlenecks, countsClones, invades, hasWaterRegime, shapes, models, projects, checksOccupancy, enables, predicts

Commutativity conditions (5): C1–C5 as specified in Section 3

Hierarchical subcategory H: A forest {Genetic, Organismal, Population, Environmental, Geographic} with level function assigning each object to exactly one rank

Source: geng2016-genetic-diversity-phenotypic-plasticity for all empirical grounding buehler2011-reoccurring-patterns for olog methodology

Revision history

2026-07-05 — Olog created. Constructed from Geng et al. (2016) with category-theoretic methodology from Giesa, Spivak, & Buehler (2011). Five commutativity conditions formalize the plasticity-diversity trade-off, founder bottleneck dynamics, plasticity conservation, niche occupancy independence, and common garden invariance. 18 objects across 5 hierarchical levels. (geng2016-genetic-diversity-phenotypic-plasticity, buehler2011-reoccurring-patterns)

Clonal Evolution

Explorer

Ecology Invasion Olog

Ecology Invasion Olog: A Category-Theoretic Ontology Log for the Geng et al. (2016) Invasion Model

Methodology

1. Objects (Sets)

1.1 Level 1 — Molecular/Genetic Level

1.2 Level 2 — Organismal Level

1.3 Level 3 — Population Level

1.4 Level 4 — Environmental Level

1.5 Level 5 — Geographic Level

2. Arrows (Functions)

2.1 Genetic Arrows

2.2 Phenotypic Arrows

2.3 Population Arrows

2.4 Environmental Arrows

2.5 Geographic Arrows

2.6 Cross-Level Arrows for Commutativity

3. Commutativity Conditions

3.1 Condition C1: Plasticity-Diversity Trade-off

3.2 Condition C2: Founder Bottleneck Condition

3.3 Condition C3: Plasticity Conservation

3.4 Condition C4: Niche Occupancy Independence

3.5 Condition C5: Common Garden Commutativity

4. Summary Diagram: The Complete Olog

5. Hierarchical Subcategory H (Forest)

5.1 Forest Structure

5.2 Level Membership

6. Integration with Empirical Data

6.1 Key Numerical Values for Arrow Codomains

6.2 Commutativity Condition Status

7. Cross-Domain Mapping Notes

8. Limitations and Caveats

9. Formal Olog Summary

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