---
title: "Spatial variation in allometric growth of invasive lionfish has management implications"
author: "Juan Carlos Villaseñor-Derbez, Sean Fitzgerald"
institution: "Bren School of Environmental Sciences and Management, University of California Santa Barbara, Santa Barbara, California, U.S."
abstract: |
  Lionfish (\textit{Pterois volitans / miles}) are an invasive species in the Western Atlantic and the Caribbean. Improving management of invasive lionfish populations requires accurate total biomass estimates, which depend on accurate estimates of allometric growth. Sedentary species like lionfish often exhibit high levels of spatial variation in life history characteristics. We review 17 published length-weight relationships for lionfish taken throughout their invasive range and found substantial regional differences in allometric growth parameters. The spatial pattern we observed is consistent with findings from other studies focusing on genetics or age-at-length. We show that the use of \textit{ex situ} parameters can result in up to a threefold under- or overestimation of total weight, but using parameters from nearby regions reduces this error. These findings can have major implications for management in terms of predicting effects on local ecosystems, evaluating the effectiveness of removal programs, or estimating biomass available for harvest.
keywords:
  - Lionfish
  - invasive species
  - length-weight
  - allometric growth
  - regional variations
bibliography: references.bib
output:
  pdf_document:
    keep_tex: yes
    number_sections: no
    template: peerj_template.tex
header-includes:
   - \usepackage{float}
   - \floatplacement{figure}{H}
   - \restylefloat{table}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(echo = F, warning = F, message = F, fig.width = 5.5)

options(knitr.table.format = "latex")

windowsFonts(Times=windowsFont("TT Times New Roman"))
```


```{r load packages}
suppressPackageStartupMessages({
  library(stringr)
  library(agricolae)
  library(magrittr)
  library(stargazer)
  library(sandwich)
  library(lmtest)
  library(broom)
  library(tmap)
  library(rgdal)
  library(sf)
  library(cowplot)
  library(tidyverse)
})
```

\section*{Introduction}

Lionfish (*Pterois volitans/miles* complex) are an invasive species in the western Atlantic and Caribbean Sea, likely introduced through liberation of aquarium-kept organisms \citep{betancurr_2011}. They are the first invasive marine vertebrates established along the North Atlantic Caribbean coasts \citep{schofield_2009,schofield_2010,sabidoitza_2016} and their presence has been labeled as a major marine invasion because they threaten local biodiversity, spread rapidly, and are difficult to manage \citep{hixon_2016}. Lionfish have established invasive populations in coral reefs, estuaries, mangroves, hard-bottomed areas, and mesophotic reefs \citep{barbour_2010,jud_2011,muoz_2011,claydon_2012,andradibrown_2017,gress_2017}.

A substantial amount of research describes lionfish impacts throughout its invaded range. A meta-analysis by \citet{peake_2018} showed that invasive lionfish prey on at least 167 different species across the tropical and temperate North Atlantic. Their feeding behavior and high consumption rates can reduce recruitment and population sizes of native reef-fish species, and can further endanger reef fish (\citet{green_2012,rocha_2015}; but see \citet{hackerott_2017}). For example, field experiments by \citet{albins_2008} showed that lionfish establishment led to reduced recruitment of native fishes by nearly 80% over a five-week period in Florida. \citet{green_2012} reported that prey fish biomass declined by 65% over two years as lionfish biomass increased along Bahamian coral reefs. Their trophic impacts can be minimized if local lionfish biomass is controlled by culling \citep{ariasgonzalez_2011}.

Governments and non-profit organizations have sought to reduce lionfish densities through removal programs and incentivizing its consumption \citep{chin_2016}. In some cases, these have shown to significantly reduce --but not quite eliminate-- lionfish abundances at local scales \citep{deleon_2013,sandel_2015}. Complete eradication of lionfish through fishing is unlikely because of their rapid recovery rates and ongoing recruitment to shallow-water areas from persistent populations in mesophotic ecosystems \citep{barbour_2011,andradibrown_2017}. However, promoting lionfish consumption might create a level of demand capable of incentivizing a stable fishery while controlling shallow-water populations, thus creating alternative livelihoods and avoiding further impacts to local biota.

The feasibility of establishing fisheries through lionfish removal programs has been extensively evaluated through field observations and empirical modeling \citep{barbour_2011,morris_2011,deleon_2013,johnston_2015,sandel_2015,usseglio_2017}. Determining the feasibility of such initiatives requires modeling the change in biomass in response to changes in fishing mortality (*i.e.* culling). A common way to model this is via length-structured population models, where fish lengths are converted to  weight to calculate total biomass \citep{barbour_2011,cote_2014,andradibrown_2017}. The allometric length-weight relationship is thus an essential component of these models, but this relationship can vary across regions as a response to biotic and abiotic conditions \citep{johnson_2016}.

Outcomes of previous studies suggest lionfish are likely to exhibit spatial heterogeneity in the length-weight relationship, which we summarize in two main causes. First, culling programs are effective in reducing local adult populations largely because lionfish exhibit high levels of site fidelity and small home ranges \citep{Fishelson_1997,kochzius_2005,jud_2012,cote_2014}. It is know that fish with sedentary behavior are likely to exhibit high levels of spatial variation in important life history characteristics such as growth or natural mortality rates \citep{gunderson_2008,hutchinson_2008,wilson_2012,guan_2013}. Second, genetic analysis of lionfish suggests biological differences due to the existence of two genetically distinct invasive subpopulations between the northwest Atlantic and the Caribbean \citep{betancurr_2011}. Site-specific parameters are necessary to accurately estimate biomass when allometric relationships are spatially variable, and this variability is increasingly important when estimating the potential effectiveness of lionfish culling programs \citep{barbour_2011,morris_2011,cote_2014,johnston_2015}. However, the region-wide differences in allometric growth parameters has remained unexplored for lionfish, despite the large number of site-specific studies reporting the length-weight relationship.

Here, we compare previously published length-weight relationships for lionfish populations in North Carolina, Northern and Southern Gulf of Mexico, the Southern Mexican Caribbean, Bahamas, Little Cayman, Jamaica, Bonaire, Puerto Rico, and Costa Rica \citep{barbour_2011,darling_2011,deleon_2013,fogg_2013,dahl_2014,edwards_2014,toledohernndez_2014,sandel_2015,aguilarperera_2016,sabidoitza_2016,sabidoitz_2016,chin_2016}. We also collected lionfish length and weight data in the central Mexican Caribbean and report the first allometric growth equation for this region. The objective of this paper is to describe the spatial pattern of length-weight relationships of lionfish across the Caribbean and Western Atlantic and to discuss implications of these spatial differences.

\clearpage

\section*{Methods}

```{r load fish data, include = F}
fish <- read_csv(here::here("data", "lionfish_data.csv"), col_types = cols()) %>%
  group_by(Id, Site, Location, Latitude, Longitude, Depth_m) %>% 
  summarize(Length = 10*mean(TotalLength_cm),
            Weight = mean(TotalWeigth_gr),
            Stomach = mean(StomachLength_cm)) %>% 
  ungroup()

review <- read.csv(here::here("data", "biometric_review.csv"), stringsAsFactors = F, strip.white = T, blank.lines.skip = T) %>% 
  mutate(c = log10(a),
         Gender2 = case_when(Gender == "Male" ~ "M",
                             Gender == "Female" ~ "F",
                             TRUE ~"B"))
```

We reviewed 12 published studies and obtained 17 length-weight relationships for the North Atlantic (n = 1), Gulf of Mexico (n = 7), and Caribbean (n = 9, Table \ref{tab:all_params}, Fig \ref{fig:all_allo}). We collected information on sampling methods, sex differentiation, location, and depth ranges from each study when available. Only two studies reported parameters for each gender \citep{aguilarperera_2016,fogg_2013}, so we assumed both genders were included in a study if gender was unspecified. Reviewed studies presented information for organisms obtained at depths between `r min(review$minD, na.rm = T)` m and `r max(review$maxD, na.rm = T)` m. Three studies explicitly stated that their organisms were sampled with pole spears \citep{dahl_2014,aguilarperera_2016,chin_2016,sabidoitz_2016}, and five studies mentioned that some of their organisms were obtained with pole spears (or other type of harpoon) but also hand-held nets or fish traps \citep{barbour_2011,fogg_2013,edwards_2014,toledohernndez_2014,sandel_2015,sabidoitza_2016,sabidoitz_2016}, and two studies did not specify how organisms were sampled \citep{darling_2011,deleon_2013}. \citet{fogg_2013} use spineless weight in their calculations, so their parameters likely underestimated total weight. Since no spineless to total weight conversions were available, these parameters were taken as reported.

```{r}

proj <- CRS("+proj=longlat +datum=NAD27")

# Load coastline for Mexico and World

data(World)
W <- spTransform(World, proj) %>% 
  st_as_sf()

# Map with SF
fig_map <- ggplot() +
  geom_sf(data = W) +
  geom_point(data = review %>%
      mutate(N = ifelse(is.na(N), 1, N)),
    aes(x = lon, y = lat, fill = b, size = N), shape = 21) +
  theme_classic() +
  scale_fill_gradientn(colors = colorRamps::matlab.like(20)) +
  theme(legend.background = element_blank(),
        panel.grid.major = element_line(colour = "transparent")) +
  lims(x = c(-100, -60), y = c(9, 36)) +
  labs(x = "longitude", y = "latitude") +
  geom_text(aes(x = -70, y = 30), label = "North Atlantic\nOcean") +
  geom_text(aes(x = -90, y = 25), label = "Gulf of Mexico") +
  geom_text(aes(x = -75, y = 15), label = "Caribbean\nSea")
  
```

```{r map, fig.cap = "\\label{fig:map}Locations where allometric growth parameters of lionfish (*Pterois spp*) have been reported. Circle sizes indicate sample size from each study, colors indicate the $b$ coefficient from Eq. \\ref{eq:allometric}.", fig.height = 3.5}
fig_map
```

We also used data from \citet{villaseorderbez_2014}, who collected organisms from 10 sampling sites along the central Mexican Caribbean coast in 2010 (Supplementary Table 1). Sampling locations included wall and carpet reefs at depths between 5.7 m and 38.1 m. All observed lionfish (n = 109) were collected using hand nets and numbered collection bottles. The use of hand nets prevented any weight loss due to bleeding and allowed better representation of small sizes by eliminating gear selectivity. Organisms were euthanized via pithing and Total Length (TL; mm) and Total Weight (TW; g) were recorded.

The weight-at-length relationship for lionfish in the central Mexican Caribbean was calculated with the allometric growth function:

\begin{equation}
\label{eq:allometric}
TW = aTL^b
\end{equation}

Where $a$ is the ponderal index and $b$ is the scaling exponent or allometric parameter.

\clearpage

Transforming this equation via base-10 logarithms we obtain:

\begin{equation}
\label{eq:log-alo}
log_{10}(TW) = b\times log_{10}(TL) + log_{10}(a)
\end{equation}

This can be simplified and re-written as:

\begin{equation}
\label{eq:log-alo-trans}
Y = bX + c
\end{equation}

Where $Y = log_{10}(TW)$, $X = log_{10}(TL)$, and $c = log_{10}(a)$. The coefficients ($c$ and $b$) were estimated with  an Ordinary Least Squares Regression and heteroskedastic-robust standard error correction \citep{zeileis_2004}. When the $b = 3$, it is said that the organism exhibits a perfect isometric growth, so the $b$ coefficient was tested against the null hypothesis of isometric growth (*i.e.* $H_0: b = 3$). Coefficients were tested with a two-tailed Student's t, and the significance of the regression was corroborated with an F-test.

Some of the reviewed studies inconsistently defined $a$ as either the ponderal index from Eq. \ref{eq:allometric} or the y-intercept ($c$) from Eq. \ref{eq:log-alo-trans}. Other studies incorrectly reported parameters as mm-to-g conversions when they were in fact cm-to-g conversions. We standardized each study by converting coefficients and report all parameters as TL(mm) to TW (gr) conversions. Locations where allometric studies have been performed are shown in Figure \ref{fig:map} and summarized in Table \ref{tab:all_params}.

We obtained a total of 18 parameter pairs by combining length-weight parameters extracted from the literature and the additional pair calculated here. We used the central Mexican Caribbean as a case study of how the use of *ex situ* parameters influences the accuracy of weight estimates for lionfish. We estimated TW from the TL observations we collected in the central Mexican Caribbean (n = 109, with $TL \in (34, 310)$) using each of the 18 parameter pairs and divided predicted weights by known observed weights to obtain a simple measure of over- or underestimation. Difference in mean weight ratios across the different parameter pairs were tested with a one-way analysis of variance (ANOVA) and Tukey's test was used for *post-hoc* tests. All analyses were performed in R version `r paste(version$major, version$minor, sep = ".")` \citep{rcore_2018}. Raw data and code used in this work are available on github.

```{r model fitting}
model <- lm(log10(Weight) ~ log10(Length), data = fish)

TidyModel <- model%>%
  lmtest::coeftest(vcov = sandwich::vcovHC(., type = "HC1")) %>%
  broom::tidy()

robust_se <- TidyModel$std.error %>%
  set_names(TidyModel$term)
```

```{r testing coefficients}
test_intercept <- 2*pt(abs(coef(model)[[1]]/robust_se[[1]]), df = 107, lower.tail = F)

test_slope <- 2*pt(abs(coef(model)[[2]]/robust_se[[2]]), df = 107, lower.tail = F)

test_slope_3 <- 2*pt(abs((coef(model)[[2]]-3)/robust_se[[2]]), df = 107, lower.tail = F)

c <- coefficients(model)[[1]]
b <- coefficients(model)[[2]]
```

```{r}
Predicted_biomass <- review %>% 
  mutate(Length = list(fish$Length),
         Weight = list(fish$Weight)) %>% 
  unnest() %>% 
  mutate(pWeight = a*(Length^b),
    Biomass_ratio = pWeight / Weight)

anova <- aov(Biomass_ratio ~ Study_id, Predicted_biomass)
groups <- HSD.test(y = anova, trt = "Study_id")

Rsq <- formatC(summary(model)$r.sq, digits = 3, format = "f")
Fstring <- waldtest(model, vcov = vcovHC(model, type = "HC1")) %>% 
  tidy()

Fstring <- paste0("F(df = 1; ", nobs(model)-2, ") = ", formatC(Fstring$statistic[2], digits = 2, format = "f"))
```

\section*{Results}

The length-weight relationship for organisms from the central Mexican Caribbean resulted in the coefficient values $a = `r 10^c`$, $b = `r b`$ and $c = `r c`$ ($R^2 = `r Rsq`$, `r Fstring`, $p < 0.001$). The allometric factor ($b$) was significantly different from $b = 3$ ($t(107) = `r formatC((coef(model)[[2]]-3)/robust_se[[2]], digits = 2, format = "f")`; p<0.001$) indicating that lionfish present allometric growth. The length-weight coefficients estimated in this study were within the range identified by studies in other regions (Table \ref{tab:all_params}). Figure \ref{fig:l-w-carib} shows the relationship between TL and TW for this region, and model fit statistics are presented in Table \ref{tab:reg_table}.

```{r}
TidyModel <- model %>%
  lmtest::coeftest(vcov = sandwich::vcovHC(., type = "HC1")) %>%
  broom::tidy()

robust_se <- TidyModel$std.error %>%
  set_names(TidyModel$term)
wald_results <- waldtest(model, vcov = vcovHC(model, type = "HC1"))
f_string <- c("F Statistic", paste0(formatC(wald_results$F[2], digits = 2, format = "f"), "*** (df = 1; ", dim(fish)[1]-2,")"))
```

```{r}
get_eqn <- function(model){
  substitute(italic(TW) == a %*% "10"^"-6" ~ italic(TL)^b,
             list(a = "3.2056",
                  b = "3.2347")) %>% 
    as.expression() %>% 
    as.character()
}

get_eqn2 <- function(model){
  substitute(italic(log[10](TW)) == b %*% italic(log[10](TL)) ~ a,
             list(a = "-5.4940",
                  b = "3.2347")) %>% 
    as.expression() %>% 
    as.character()
}

get_rsq <- function(model){
  substitute(italic(R)^2~"="~r2,
             list(r2 = formatC(summary(model)$r.squared, digits = 3))) %>% 
    as.expression() %>% 
    as.character()
}
```

```{r}
fit <- ggplot(data = fish, mapping = aes(x = Length, y = Weight)) +
  geom_point(size = 3, shape = 21, fill = "steelblue", color = "black") +
  theme_cowplot() +
  labs(x = "Total length (mm)", y = "Total Weight (gr)") +
  theme(text = element_text(family = "Times")) +
  scale_color_brewer(palette = "Set3") +
  theme(text=element_text(family = "Times")) +
  geom_smooth(method = "lm", formula = y ~ I(c*x^b), se = F, color = "black", linetype = "dashed") +
  annotate(geom = "text",
            x = 125, y = 390,
            label = get_eqn(model),
            parse = T,
            size = 4,
            family="Times",
            face="italic") +
  annotate(geom = "text",
            x = 125, y = 350,
            label = get_eqn2(model),
            parse = T,
            size = 4,
            family="Times",
            face="italic") +
  annotate(geom = "text",
            x = 125, y = 310, 
            label = get_rsq(model),
            parse = T,
            size = 4,
            family="Times",
            face="italic")
            

fig_l_w_carib <- ggExtra::ggMarginal(p = fit,
                                     type = "density",
                                     margins = "both",
                                     fill = "steelblue",
                                     color = "black",
                                     alpha = 0.5)
```

```{r fit1,  fig.height = 3.5, fig.pos = "H", fig.cap = "\\label{fig:l-w-carib}Length-weight relationship for 109 lionfish sampled in the central Mexican Caribbean. Points indicate samples, dashed black line indicates curve of best fit, marginal plots represent the density distribution of each variable."}
fig_l_w_carib
```


```{r, results = "asis"}
stargazer(model,
          dep.var.caption = "$log_{10}(TW)$",
          dep.var.labels.include = F,
          header = F,
          se = list(robust_se),
          type = "latex",
          t.auto = T,
          p.auto = T,
          intercept.bottom = F,
          single.row = T,
          report = "vcs*",
          omit.stat = c("f", "rsq"),
          add.lines = list(f_string),
          covariate.labels = c("c", "b"),
          dep.var.labels = "log10(TW)", title = "\\label{tab:reg_table}Coefficients of the linear model fit to Eq \\ref{eq:log-alo-trans}. Numbers in parenthesees represent heteroskedastic-robust standard errors.")
```

There were significant differences in our predicted weights for the central Mexican Caribbean when using the different pairs of parameters ($F(df = 17; 1944) = 61.55; p < 0.001$). The lowest weight estimates resulted from using the allometric parameters from Banco Chinchorro in the Caribbean, with mean $\pm$ SD of `r filter(Predicted_biomass, Study_id == "Sabido2BC") %$% mean(pWeight) %>% formatC(digits = 2, format = "f")` $\pm$ `r filter(Predicted_biomass, Study_id == "Sabido2BC") %$% sd(pWeight) %>% formatC(digits = 2, format = "f")` gr \citep{sabidoitz_2016}, and the highest weight estimates came from the Northern Atlantic with `r filter(Predicted_biomass, Study_id == "Barbour") %$% mean(pWeight) %>% formatC(digits = 2, format = "f")` $\pm$ `r filter(Predicted_biomass, Study_id == "Barbour") %$% sd(pWeight) %>% formatC(digits = 2, format = "f")` gr \citep{barbour_2011}. To put this in context, true observed weights were `r filter(Predicted_biomass, Study_id == "Barbour") %$% mean(Weight) %>% formatC(digits = 2, format = "f")` $\pm$ `r filter(Predicted_biomass, Study_id == "Barbour") %$% sd(Weight) %>% formatC(digits = 2, format = "f")` gr. These correspond to predicted-to-observed weights ratios of `r filter(Predicted_biomass, Study_id == "Sabido2BC") %$% mean(Biomass_ratio) %>% formatC(digits = 2, format = "f")` $\pm$ `r filter(Predicted_biomass, Study_id == "Sabido2BC") %$% sd(Biomass_ratio) %>% formatC(digits = 2, format = "f")` and `r filter(Predicted_biomass, Study_id == "Barbour") %$% mean(Biomass_ratio) %>% formatC(digits = 2, format = "f")` $\pm$ `r filter(Predicted_biomass, Study_id == "Barbour") %$% sd(Biomass_ratio) %>% formatC(digits = 2, format = "f")` (mean $\pm$ SD), respectively.

The calculated ratio of predicted-to-observed weight ranged from `r min(Predicted_biomass$Biomass_ratio) %>% formatC(digits = 2, format = "f")` to `r max(Predicted_biomass$Biomass_ratio) %>% formatC(digits = 2, format = "f")`, indicating that *ex situ* parameters can result in major under- and overestimation. Tukey's *post-hoc* test suggests that weight ratios for the central Mexican Caribbean were not different from those obtained with parameters from Little Cayman, the Bahamas, and some sites in the Gulf of Mexico (Tukey's HSD $p > 0.05$). Weight estimates using parameters from the Gulf of Mexico and North-Western Atlantic were higher on average than those from the Caribbean (Fig \ref{fig:all_allo}). The average ($\pm$ SD) predicted-to-observed weight ratios from these three regions were 1.24 $\pm$ 0.309, 1.76 $\pm$ 0.496, and 1.17 $\pm$ 0.398, respectively. Predicted-to-observed weight ratios are presented in Figure \ref{fig:bio_ratio}. Spineless weight parameters from \citet{fogg_2013} still produced predicted-to-observed weight ratios > 1.

```{r}
review %>% 
  select(Region, Gender2, N, a, b, c, r2, Study) %>% 
  filter(!Study == "Fishbase") %>% 
  arrange(Region) %>% 
  mutate(r2 = ifelse(is.na(r2), "-", r2),
         N = ifelse(is.na(N), "-", N),
         a = formatC((a*1e5), digits = 2)) %>% 
  knitr::kable(row.names = F,
               digits =  c(0, 0, 0, 0, 2, 2),
               booktabs = T,
               col.names = c("Region", "Sex", "n", "a", "b", "c", "Fit", "Reference"),
               caption = "\\label{tab:all_params}Summary of 18 allometric growth parameters available for lionfish in the invaded range from peer-reviewed literature and this study. All parameters have been adjusted to convert from millimeters to grams. n = Sample size, Sex specifies whether data was presented for Females (F), Males (M), or both genders combined (B), a = scaling parameter for Eq. 1 (presented in $\\times 10^{-5}$), c = y-intercept for Eq. 3, b = exponent or slope for Eq. 1 or Eq. 3, respectively. The Fit column contains the reported $R^2$ of the model fit.", escape = T)
```

```{r}
fig_all_allo <- review %>% 
  filter(!Study == "This study") %>% 
  mutate(Length = list(seq(1, 400))) %>% 
  unnest() %>% 
  mutate(Weight = a*(Length^b),
         This = 10^(coefficients(model)[1] + coefficients(model)[2]*log10(Length))) %>%
  group_by(Length) %>% 
  mutate(min = min(Weight),
         max = max(Weight)) %>% 
  ungroup() %>% 
  ggplot(aes(x = Length)) +
  geom_line(aes(y = Weight, color = Study, linetype = Gender, group = Study_id), size = 1) +
  geom_line(aes(y = This), size = 1.2, color = "black", linetype = "dashed") +
  labs(x = "Total length (mm)", y = "Total Weight (gr)") +
  theme_cowplot() +
  theme(text = element_text(family = "Times"),
        legend.justification=c(0,1),
        legend.position=c(0.01, 1.05),
        legend.background = element_rect(fill = "transparent"),
        legend.box = "horizontal",
        legend.text = element_text(size = 9),
        legend.title = element_text(size = 10)) +
  guides(color = guide_legend(order = 1),
         linetype = guide_legend(order = 0, ncol = 2),
         size = "none") +
  scale_color_brewer(palette = "Paired", direction = -1) +
  scale_linetype_manual(values=c("solid", "dashed", "dotted"), name = "Sex") +
  scale_size_manual(values = c(1, 1.5))
```

```{r fit2, fig.height = 3.8, fig.pos = "H", fig.cap = "\\label{fig:all_allo}Length-weight relationships (n = 18) for 12 studies and this study. Colors indicate studies from which the parameters were extracted. Dotted, dashed and solid lines show models for males, females, and combined sexes, respectively. The dashed black line represents the relationship estimated in this study."}
fig_all_allo
```

```{r}
groups$groups%<>%
  mutate(Study_id = rownames(.)) %>% 
  select(-Biomass_ratio)

Predicted_biomass %<>% 
  left_join(groups$groups, by = "Study_id") %>% 
  mutate(groups = as.character(groups),
         groups = ifelse(Length == 34, groups, ""))

fig_bio_ratio <- Predicted_biomass %>% 
  mutate(Study = paste0(Study, " (", Gender_site, ")")) %>%
  transform(Study = reorder(Study, -Biomass_ratio),
            Study_id = reorder(Study_id, -Biomass_ratio)) %>%
  rename(Sex = Gender) %>% 
  ggplot(aes(x = Study, y = Biomass_ratio, group = Study_id)) +
  geom_violin(aes(fill = Sex), alpha = 0.8) +
  stat_summary(geom = "point", fun.y = "median", size = 2, color = "black", fill = "steelblue", shape = 21) +
  geom_hline(yintercept = 1, linetype = "dashed") +
  labs(y = "Weight ratio (Predicted/Observed)", x = "") +
  coord_flip() +
  theme_cowplot() +
  theme(text = element_text(family = "Times")) +
  scale_fill_brewer(palette = "Greys") +
  theme(text = element_text(family = "Times"),
        legend.position = "top",
        legend.text = element_text(size = 10),
        legend.background = element_rect(fill = "transparent"),
        axis.text = element_text(size = 10),
        axis.title = element_text(size = 12)) +
  geom_text(aes(y = 3.7, label = groups), family = "Times", fontface = "plain") +
  scale_y_continuous(limits = c(NA, 3.7)) +
  guides(fill = guide_legend(title.position = "top"))
```

```{r pred_obs, fig.height = 5.2, fig.width = 5.5, fig.cap = "\\label{fig:bio_ratio}Violin plot of predicted-to-observed weight ratios for 18 pairs of allometric parameters. Blue circles indicate median values and Like letters indicate values that do not differ significantly."}
fig_bio_ratio
```

\clearpage

\section*{Discussion}

Our results suggest that lionfish exhibit highly variable allometric relationships across the invaded range, and that this variation is related to space. Moreover, we shot that the use of *ex situ* parameters may lead to highly biased weight estimates. Our comparison of observed weights to those predicted with locally-informed parameters and *ex situ* parameters showed that weight can be overestimated by more than a three-fold, and highlights the need to use local information. Here we discuss the implications of our findings and highlight potential future research directions.

We detected substantial differences in weight-at-length between organisms from the Caribbean, Gulf of Mexico, and North-Western Atlantic. Groupings of predicted-to-observed weight ratios aligned with the spatial distribution of the examined studies, suggesting that these differences are mediated by space. These regional allometric differences mirror similar patterns in age-at-length of lionfish across both their invaded and native regions \citep{pusack_2016}. Variation may be driven by genetics or by organisms' exposure to distinct environmental conditions. For example, \citet{betancurr_2011} used mitochondrial DNA to demonstrate the existence of two distinct population groups, identified as the “Caribbean group” and “Northern Group”, and \citet{fogg_2015} alternatively suggested that age-at-length differences may be climate-driven. Differences in weight-at-length could also reflect differential energy input or usage, or a combination of both. Future research is needed to determine which processes are at work here.

Differences in length-weight relationships have traditionally been highlighted as potential pitfalls to fishery management. For example, \citet{wilson_2012} show that small-scale variations in length-at-age and fishing mortality in other Scorpaeniformes translate to differential landings, effort, and catch per unit effort in the live fish fishery of California, and that these differences must be taken into account in management plans. The lionfish case poses the opposite scenario, where the manager desires to eradicate the species. To accurately gauge both the effectiveness of lionfish removal efforts and the resources needed to successfully manage an invasion, we must acknowledge and understand regional biological differences in important variables such as allometric growth parameters.

The results presented here have major implications for management. For example, \citet{edwards_2014} simulated a lionfish culling program under two scenarios, one using length-at-age and length-to-weight parameters from North Carolina and one using parameters from Little Cayman. Their results show that using different parameters caused up to a four-year difference in the time required for the simulated lionfish population to recover to 90% of its initial biomass after removals ceased. Here, we show that using one set of length-weight parameters versus another for a given length can result in more than a threefold under- or overestimation of total weight. These spatially-driven differences become especially important when allocating resources for lionfish removal programs, incentivizing lionfish fisheries as a source of alternative livelihoods, or estimating ecosystem impacts. Research efforts focused on invasive lionfish populations need to use parameters calculated for their region to the extent possible, or at least use reasonable sets of different parameters that provide upper and lower bounds in their results.

\section*{Acknowledgements}

The authors would like to thank thank Nils Van Der Haar and Michael Doodey from Dive Aventuras as well as Guillermo Lotz-Cador who provided help to collect samples.

Conflict of Interest: The authors declare that they have no conflict of interest.

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