DOI: 10.5061/dryad.c2fqz61k7
title: Data from: Eco-evolutionary metapopulation dynamics of Batesian mimicry: Conditions for mimics without models

description: Batesian mimicry is adopted by palatable prey species (mimics) to avoid
 predator attack by resembling unpalatable species (models). Despite
 numerous studies on this phenomenon, several aspects of its evolution
 remain unclear. One of the most interesting questions is whether mimics
 can inhabit allopatrically from their models. Classical theory suggests
 that mimics should only inhabit sympatrically with their models because
 predators need to learn the model phenotype. However, several studies have
 reported mimics that live outside the distribution range of their models
 (mimics without models; MWM). In this study, we constructed and analyzed
 an eco-evolutionary dynamics model of Batesian mimicry incorporating two
 populations under different environmental conditions to clarify the
 conditions necessary for MWM to occur. We identified that MWM occurred
 when specific conditions are satisfied by the following six parameters:
 (1) carrying capacities; (2) migration rate of mimics; (3) migration rate
 of predators; (4) evolution rate of mimic phenotype; (5) toxicity of the
 model and nutritional level of the mimic; and (6) population density of
 predators. Our findings differed from previous predictions that considered
 population and evolutionary dynamics independently. Overall, this study
 asserts that the interaction between ecological and evolutionary processes
 makes the conditions for MWM more stringent.
description: # Data from: Eco-evolutionary metapopulation dynamics of Batesian mimicry:
 Conditions for mimics without models
 [https://doi.org/10.5061/dryad.c2fqz61k7](https://doi.org/10.5061/dryad.c2fqz61k7) ## Description of the data and file structure This dataset contains the C-language simulation code and Mathematica scripts used to generate and visualize the results presented in the forthcoming paper: Tomizuka, H. & Tachiki, Y. (2025). Eco-evolutionary metapopulation dynamics of Batesian mimicry: Conditions for mimics without models. Journal of Theoretical Biology. ([https://doi.org/10.1016/j.jtbi.2025.112299](https://doi.org/10.1016/j.jtbi.2025.112299)) **Short Summary of the Study**\ Batesian mimicry is adopted by palatable prey species (mimics) to avoid predator attack by resembling unpalatable species (models). Despite numerous studies on this phenomenon, several aspects of its evolution remain unclear. One of the most interesting questions is whether mimics can inhabit allopatrically from their models. Classical theory suggests that mimics should only inhabit sympatrically with their models because predators need to learn the model phenotype. However, several studies have reported mimics that live outside the distribution range of their models (mimics without models; MWM). In this study, we constructed and analyzed an eco-evolutionary dynamics model of Batesian mimicry incorporating two populations under different environmental conditions to clarify the conditions necessary for MWM to occur. We identified that MWM occurred when specific conditions are satisfied by the following six parameters: (1) carrying capacities; (2) migration rate of mimics; (3) migration rate of predators; (4) evolution rate of mimic phenotype; (5) toxicity of the model and nutritional level of the mimic; and (6) population density of predators. Our findings differed from previous predictions that considered population and evolutionary dynamics independently. Overall, this study asserts that the interaction between ecological and evolutionary processes makes the conditions for MWM more stringent. **Overview of Folders/Files**\ The repository is organized into seven folders, each corresponding to a figure or supplementary figure in the study. The files within each folder are as follows: ## File–Figure Crosswalk | Figure | Folder (path) | Key files (examples) | | ------- | ------------- | ------------------------------------------------------------------------------------------------------------------ | | Fig. 3 | Fig3/ | Fig3.nb; Fig3a.c / Fig3a.csv; Fig3b.c / Fig3b.csv; Fig3c.c / Fig3c.csv; Fig3d.c / Fig3d.csv | | Fig. 4 | Fig4/ | Fig4.nb; Fig4ab.c / Fig4ab.csv; Fig4c.c / Fig4c.csv; Fig4d.c / Fig4d.csv; Fig4e.c / Fig4e.csv; Fig4f.c / Fig4f.csv | | Fig. 5 | Fig5/ | Fig5.nb; Fig5a.c / Fig5a.csv; Fig5b.c / Fig5b.csv; Fig5c.c / Fig5c.csv | | Fig. 6 | Fig6/ | Fig6.nb; Fig6a.c / Fig6a.csv; Fig6b.c / Fig6b.csv; Fig6c.c / Fig6c.csv | | Fig. S2 | FigS2/ | FigS2.nb; FigS2.c / FigS2.csv | | Fig. S3 | FigS3/ | FigS3.nb; FigS3.c / FigS3.csv | | Fig. S4 | FigS4/ | FigS4.nb; FigS4.c / FigS4.csv | | Fig. S5 | FigS5/ | FigS5.nb; FigS5.c / FigS5.csv | | Fig. S6 | FigS6/ | FigS6.nb; FigS6.c / FigS6.csv | &gt; Note: Panel labels (A–H) match the article’s figures. --- ## Brief descriptions of each figure panel ### Conditions under which mimics without models occur (Fig.3) **Axes for all panels:** vertical = carrying capacity of model-species at site 1 (KD,1); horizontal = carrying capacity at site 2 (KD,2). * **(A)** Final density of model-species at site 1 under **local learning** (q = 1) with **mimic migration** (mM = 0.027). \ Data: `Fig3a.csv` (from `Fig3a.c`) * **(B)** Final mean phenotype of mimic-species at site 1 (zM,1) under the same condition as (A); darker region denotes evolution toward crypsis; lighter/white toward mimetic trait.\ (source file same as (A)) * **(C)** Final model density at site 1 under **local learning**, **no mimic migration** (mM = 0).\ Data: `Fig3b.csv` (from `Fig3b.c`) * **(D)** Final zM,1 under the same condition as (C).\ (source file same as (C)) * **(E)** Final model density at site 1 under **global learning** (q = 0.5) with **mimic migration** (mM = 0.027).\ Data: `Fig3c.csv` (from `Fig3c.c`) * **(F)** Final zM,1 under the same condition as (E).\ (source file same as (E)) * **(G)** Final model density at site 1 under **global learning**, **no mimic migration**.\ Data: `Fig3d.csv` (from `Fig3d.c`) * **(H)** Final zM,1 under the same condition as (G).\ (source file same as (G))\ Visualization: `Fig3.nb` ### The effect of the migration rate on the outcome (Fig.4) * **(A)** Phase diagram of **final model density at site 1** across model migration (mD) and mimic migration (mM); **orange** indicates sustained oscillations (limit cycles).\ Data: `Fig4ab.csv` (from `Fig4ab.c`) * **(B)** Phase diagram of **final zM,1** across (mD, mM).\ Data: `Fig4ab.csv` (from `Fig4ab.c`) * **(C)** Time series at **mD = 0.0005, mM = 0.0015**: model density at site 1 stays near KD,1 while the mimic at site 1 evolves toward **crypsis**.\ Data: `Fig4c.csv` (from `Fig4c.c`) * **(D)** Time series at **mD = 0.0005, mM = 0.01**: densities and z1 show **sustained oscillations**.\ Data: `Fig4d.csv` (from `Fig4d.c`) * **(E)** Time series at **mD = 0.0005, mM = 0.027** : model goes **locally extinct at site 1** while site 2 retains coexistence; gene flow maintains mimetic trait at site 1.\ Data: `Fig4e.csv` (from `Fig4e.c`) * **(F)** Time series at **mD = 0.004, mM = 0.0015**: initial oscillations **decay to equilibrium** with coexistence.\ Data: `Fig4f.csv` (from `Fig4f.c`)\ Visualization: `Fig4.nb` ### The effect of predator memory on the outcome (Fig.5) Bifurcation diagrams versus **q**; solid = site 1, dotted = site 2; red/blue = model/mimic densities; green = zM; gray = attack probability on model. * **(A)** **mM = 0**: \ Data: `Fig5a.csv` (from `Fig5a.c`) * **(B)** **mM = 0.01**: \ Data: `Fig5b.csv` (from `Fig5b.c`) * **(C)** **mM = 0.03**: \ Data: `Fig5c.csv` (from `Fig5c.c`)\ Visualization: `Fig5.nb` ### The effect of other parameters on the outcome (Fig.6) Bifurcation diagrams (solid = site 1, dotted = site 2; colors as above). * **(A)** **Evolution rate G**: \ Data: `Fig6a.csv` (from `Fig6a.c`) * **(B)** **Cost-to-benefit ratio c/b**: \ Data: `Fig6b.csv` (from `Fig6b.c`) * **(C)** **Predator density P**: \ Data: `Fig6c.csv` (from `Fig6c.c`)\ Visualization: `Fig6.nb` ### Simulation of predator learning process (Fig.S2) Simulates how predators memory retention time and migration rate affect to predation behavior.\ Data: `FigS2.csv` (from `FigS2.c`); visualization: `FigS2.nb` ### Parameter sensitivity of the predation threshold (Fig.S3) Sensitivity of the **attack threshold x*** to key quantities (e.g., relative abundances, phenotype means, c/b).\ Data: `FigS3.csv` (from `FigS3.c`); visualization: `FigS3.nb` ### Outcomes considering only population dynamics (Fig.S4) Ecological consequences of mimicry complex. (A & B) The vertical and horizontal axes represent the migration rates of model- and mimic-species, respectively. (A) Population density of model-species at site 1. (B) Phenotype of mimic-species at site 1. \ Data: `FigS4.csv` (from `FigS4.c`); visualization: `FigS4.nb` ### The effect of the migration rate of mimics on the outcome (Fig.S5) Dependence on migration rate of mimic-species (mM). Solid lines correspond to site 1, and dotted lines correspond to site 2. (A) Equilibrium population density of model- and mimic-species. (B) Equilibrium mean phenotype of mimic-species. (C) Predation threshold z_star decided by predators. \ Data: `FigS5.csv` (from `FigS5.c`); visualization: `FigS5.nb` ### The outcome when the model went extinct (Fig.S6) Bifurcation diagrams of our system to vary the local learning rate (q). Solid lines correspond to site 1, and dotted lines correspond to site 2. (A) Equilibrium population density of model- and mimic-species. (B) Equilibrium the phenotype of mimic-species. \ Data: `FigS6.csv` (from `FigS6.c`); visualization: `FigS6.nb` **Each folder contains:** ・[.c]: C-language source code for evolutionary simulations. The main script for each experiment is provided here. ・[.csv]: Rdata generated from simulations. ・[.nb,  .pdf]: Mathematica (.nb) scripts for generating figures. A corresponding PDF file is also included for reference. ## Column order (for CSV files without headers) **Global conventions** `t` = time;\ `D1, D2` = population densities of the **model** at sites 1 and 2;\ `M1, M2` = population densities of the **mimic** at sites 1 and 2;\ `z1, z2` = mean phenotype of the mimic at sites 1 and 2;\ `KD1, KD2` / `KM1, KM2` = carrying capacities of model / mimic at sites 1 and 2;\ `mD, mM` = per-capita **migration rates between sites** (model / mimic);\ `q` = weight on **local** information in predator decision-making;\ `G` = evolution rate of the mimic phenotype;\ `P` = predator density;\ `c_over_b` = cost-to-benefit ratio;\ `z1_star, z2_star` = predation thresholds at sites 1 and 2 (appear only in some files);\ `#_max, #_min` = the maximum/minimum of variable `#` within a simulation;\ `Learning_time` = predator memory retention time;\ `migration_rate` = predator migration rate between sites. **Per-file column order** * `Fig3a.csv` : `KD1, KD2, D1, M1, z1, D2, M2, z2` * `Fig3b.csv` : `KD1, KD2, D1, M1, z1, D2, M2, z2` * `Fig3c.csv` : `KD1, KD2, D1, M1, z1, D2, M2, z2` * `Fig3d.csv` : `KD1, KD2, D1, M1, z1, D2, M2, z2` * `Fig4ab.csv` : `mD, mM, D1, M1, z1, D2, M2, z2` * `Fig4c.csv` : `t, D1, M1, z1, D2, M2, z2, z1_star, z2_star` * `Fig4d.csv` : `t, D1, M1, z1, D2, M2, z2, z1_star, z2_star` * `Fig4e.csv` : `t, D1, M1, z1, D2, M2, z2, z1_star, z2_star` * `Fig4f.csv` : `t, D1, M1, z1, D2, M2, z2, z1_star, z2_star` * `Fig5a.csv` : `q, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `Fig5b.csv` : `q, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `Fig5c.csv` : `q, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `Fig6a.csv` : `G, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `Fig6b.csv` : `c_over_b, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `Fig6c.csv` : `P, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `FigS2.csv` : `Learning_time, migration_rate, q` * `FigS3.csv` : `z1_star, z2_star, q, D1, M1, z1, D2, M2, z2, c_over_b` * `FigS4.csv` : `mD, mM, D1, M1, z1, D2, M2, z2` * `FigS5.csv` : `mM, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` * `FigS6.csv` : `q, D1_max, M1_max, z1_max, D2_max, M2_max, z2_max, D1_min, M1_min, z1_min, D2_min, M2_min, z2_min, RD1_max, RD1_min, RD2_max, RD2_min` 
 
 **Description of Variables** The meanings of each column name are shown
 below. | Variables | 
 
 | | :-------- |
 :------------------------------------------------------------------------------ | | D1 | population size of model-species at site 1 | | M1 | population size of mimic-species at site 1 | | z1 | phenotypic value of mimic-species at site 1 | | KD1 | carrying capacity of model-species at site 1 | | KM1 | carrying capacity of mimic-species at site 1 | | mD | migration rate of model-species at site 1 | | mM | migration rate of mimic-species at site 2 | | rD | growth rate of model-species | | rM | growth rate of mimic-species | | alphaD | predation coefficient of model-species | | alphaM | predation coefficient of mimic-species | | c | cost of praying on model-specoes | | b | benefit of praying on mimic-specoes | | SigmaN | standard deviation of phenotype distribution of model- and mimic-species | | SigmaS | parameters determining the shape of the graph | | s0 | the probability that a predator will find a prey with a phenotype equal to zero | | Mu | mean phenotype of model-species | | P | population size of predators | | G | rate of evolution of mimic-species phenotype | | q | Percentage of predator memory that is occupied by the current staying area | | 
 
 | 
 
 | **Sharing/Access Information** Data was derived from the following
 sources: The dataset is generated entirely through computational
 simulations and does not contain empirical data. **Code/Software**
 Computational Environment Operating System: Windows 11 Pro Processor: 12th
 Gen Intel Core i9-12900H (2.90 GHz) RAM: 64 GB **Software Used** C
 Compiler: GCC (TDM-GCC version 10.3.0) Data Visualization: Wolfram
 Mathematica (version 13.1.0.0)