Last updated: 2021-08-03

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Rmd b5ce8a9 mabarbour 2021-08-03 Publish code for reproducing common theory figures.

The code below reproduces Fig. 2B–G in the manuscript “An empricist’s guide to ecological theory”.

Setup

# load required libraries 
library(deSolve)
library(seqinr)
library(tidyverse)
library(cowplot)
library(patchwork)

# set plot theme
theme_set(theme_cowplot())

# colors for plotting
color_palette <- c("#ffe900","#658c61") 
color_states <- c("#436340","#658c61","#86c97f")

# source in Rosenzweig-MacArthur model
source('code/Models.R') # source in R-M C-R model

Model parameters

# state variable values (initial values at beginning of "experiments")
R <- 0.6
C <- 0.1
i.state <- c(R=0.6,C=0.1) 

# parameter values
r <- 1.0 # per capita rate of increase in resource
K <- 3.5 # prey carrying capacity
# e.exp1 <- 0.45
e.exp2 <- 0.65
e.exp3 <- 0.75
h <- 0.8 # handling time
m <- 0.5 # mortality rate of predator
a <- 1.3 # attack rate

# set simulation duration
Time <- 150 

## create resource isocline, which is always the same
Rx <- seq(0.1,K,0.1) # manipulating different Resource densities to solve R isocline.
Riso <- expression(r/a * (1 + a*h*Rx) * (1 - Rx/K)) # set R = 0 and solved algebraically
RisoStable <- eval(Riso)

Time series plot and phase plane diagram when e = 0.65 (Fig. 2B,D)

##  adjust parameters
p.rm2 <- c(r = r, e = e.exp2, a = a, K = K, h = h)
rm2 <- ode(i.state,1:Time, rmcr_v2, p.rm2)

# adjust resource and consumer isoclines
Ciso2 <- expression(m / (a * (e.exp2 - m*h))) #expression(m * Ro / (e.exp2 * a - m)) # adjusted attack rate
CisoStable2 <- eval(Ciso2)

## plot dynamics around consumer and resource isoclines.

# make arrow data
plot_arrows <- as.data.frame(cbind(rm2[-Time, ], rm2[-1, ]))
colnames(plot_arrows)[4:6] <- c("time.n1","R.n1","C.n1")

## time series
plot_exp2_temporal_dynamics <- ggplot(as.data.frame(rm2), aes(x = time)) +
  geom_line(aes(y = R, color = "Resource")) + 
  geom_line(aes(y = C, color = "Consumer")) +
  ylab("Abundance") +
  xlab("Time") +
  scale_color_manual(values = color_palette, name = "")
  #scale_color_viridis_d(name = "")
plot_exp2_temporal_dynamics

# phase plane
plot_exp2_isocline_dynamics <- data.frame(Rx = Rx, RisoStable = RisoStable) %>%
  ggplot(., aes(x = Rx, y = RisoStable)) +
  geom_line(aes(color = "R isocline")) + 
  geom_vline(aes(xintercept = CisoStable2, color = "C isocline")) +
  scale_color_manual(values = color_palette, name = "") +
  #scale_color_viridis_d(name = "") +
  geom_segment(data = plot_arrows, aes(x = R, xend = R.n1, y = C, yend = C.n1), arrow = arrow(length = unit(0.1,"cm"))) +
  geom_point(aes(x = i.state[1], y = i.state[2]), shape = 21, fill = "white") +
  xlab("Resource") +
  ylab("Consumer")
plot_exp2_isocline_dynamics

Time series plot and phase plane diagram when e = 0.75 (Fig. 2C,E)

## adjust parameter values
p.rm3 <- c(r = r, e = e.exp3, a = a, K = K, h = h)
rm3 <- ode(i.state,1:Time, rmcr_v2, p.rm3)

## adjust consumer isocline
Ciso3 <- expression(m / (a * (e.exp3 - m*h))) #expression(m * Ro / (e.exp3 * a - m)) 
CisoStable3 <- eval(Ciso3)

## plot dynamics around consumer and resource isoclines.

# make arrow data
plot_arrows <- as.data.frame(cbind(rm3[-Time, ], rm3[-1, ]))
colnames(plot_arrows)[4:6] <- c("time.n1","R.n1","C.n1")

## time series
plot_exp3_temporal_dynamics <- ggplot(as.data.frame(rm3), aes(x = time)) +
  geom_line(aes(y = R, color = "Resource")) + 
  geom_line(aes(y = C, color = "Consumer")) +
  ylab("Abundance") +
  xlab("Time") +
  scale_color_manual(values = color_palette, name = "")
  #scale_color_viridis_d(name = "")
plot_exp3_temporal_dynamics

# phase plane
plot_exp3_iscoline_dynamics <- data.frame(Rx = Rx, RisoStable = RisoStable) %>%
  ggplot(., aes(x = Rx, y = RisoStable)) +
  geom_line(aes(color = "R isocline")) + 
  geom_vline(aes(xintercept = CisoStable3, color = "C isocline")) +
  scale_color_manual(values = color_palette, name = "") +
  #scale_color_viridis_d(name = "") +
  geom_segment(data = plot_arrows, aes(x = R, xend = R.n1, y = C, yend = C.n1), arrow = arrow(length = unit(0.1,"cm"))) +
  geom_point(aes(x = i.state[1], y = i.state[2]), shape = 21, fill = "white") +
  xlab("Resource") +
  ylab("Consumer")
plot_exp3_iscoline_dynamics

Bifurcation plot (Fig. 2F)

Shows how consumer-resource dynamics change along a continuous gradient of conversion efficiency.

# much of this code was derived from: http://www.r-bloggers.com/r-tools-for-dynamical-systems-bifurcation-plot-in-r%C2%A0for%C2%A0system%C2%A0of%C2%A0odes/)

Time <- 300 # need bigger time range to make sure I get the dynamics

param.name <- "e" # choose parameter to perturb
e.init <- 0.5
param.seq <- seq(e.init, 1,length=50) # choose range of parameters
p.rm4 <- c(r = r, e = e.init, a = a, K = K, h = h, m=m) # set starting parameters.
param.index <- which(param.name == names(p.rm4)) # tells the loop which parameter in "p.rm4" to grab for manipulation.

out <- list()
for (i in 1:length(param.seq))
  out[[i]] <- matrix(0, Time, length(i.state))

for (i in 1:length(param.seq)) {
  # set params
  p.rm4.loop <- p.rm4
  p.rm4.loop[param.index] <- param.seq[i] # changes the parameter value for manipulation in the "init" function below.
  # converge
  init <- ode(i.state, 1:Time, rmcr_v2, p.rm4.loop)
  # get converged points
  out[[i]] <- ode(init[Time,-1], 1:Time, rmcr_v2, p.rm4.loop)[,-1]
}

range.lim <- lapply(out, function(x) apply(x, 2, range)) # get abundance ranges for each set of parameters
range.lim <- apply(do.call("rbind", range.lim), 2, range) # get abundance ranges across parameters 

##  consumer bifurcation plot
plot.variable <- "C" # choose which variable to show

# get max and min for consumer and resource abundances once transient dynamics have settled down
sim.list <- list()
for (i in 1:length(param.seq)) {
  sim.list[[i]] <- data.frame(rep = i,
                              R_max = max(out[[i]][200:300,"R"]),
                              R_min = min(out[[i]][200:300,"R"]),
                              C_max = max(out[[i]][200:300,"C"]),
                              C_min = min(out[[i]][200:300,"C"]))
}

# gather into a data frame
sim.df <- plyr::ldply(sim.list) %>%
  mutate(e = param.seq) %>% 
  gather(state, value, -rep, -e) %>%
  separate(state, into = c("species","max_min"))

# Consumer bifurcation plot
bifur_C_plot <- sim.df %>%
  filter(species == "C") %>%
  ggplot(., aes(x = e, y = value, group=max_min)) +
  geom_line(color = color_palette[1]) +
  ylab("Consumer (max and min)") +
  xlab("Conversion efficiency (e)") +
  # plot points directly from sim
  geom_point(aes(x=e.exp2, y=max(rm2[100:150,"C"])), size = 2) + 
  geom_point(aes(x=e.exp3, y=max(rm3[100:150,"C"])), size = 2) + 
  geom_point(aes(x=e.exp3, y=min(rm3[100:150,"C"])), size = 2) + 
  theme_cowplot(font_size = 12)
bifur_C_plot

Summary plot (Fig. 2G)

Shows how changing consumer conversion efficiency (e) and the resource’s carrying capacity (K) qualitatively alters the dynamics of the system.

param.name.1 <- "e" # choose parameter to perturb
param.name.2 <- "K"
grain <- 300
e.init <- 0.45
K.init <- 0.1
param.seq.1 <- seq(e.init, 1, length = grain) # choose range of parameters
param.seq.2 <- seq(K.init, 4, length = grain) # choose second range of parameters
param.seq <- expand.grid(e = param.seq.1, K = param.seq.2) %>%
  mutate(r = r, a = a, h = h) %>%
  mutate(R.eq = m/(a*(e-m*h))) %>%
  mutate(C.eq = (r/a)*(1-R.eq/K)*(1+a*h*R.eq),
         Stability = K - (e + h*m)/(a*h*(e-h*m))) %>%
  mutate(State = ifelse(C.eq < 0, "R only",
                        ifelse(Stability < 0, "C-R Stable Coexistence", "C-R Limit Cycle")))

df_geom_text <- param.seq %>%
  group_by(State) %>%
  summarise_at(vars(e, K), list(mean))

plot_e_r_manip <- param.seq %>%
  ggplot(., aes(x = e, y = K)) +
  geom_tile(aes(fill = State)) +
  scale_fill_manual(values = color_states, name = "") +
  geom_point(aes(x = e.exp2, y = 3.5)) + # new K
  geom_point(aes(x = e.exp3, y = 3.5)) + # new K
  xlab("Conversion efficiency (e)") +
  ylab("Carrying capacity (K)") +
  geom_text(data = df_geom_text, aes(x = e, y = K, label = State), size = 3) +
  theme_cowplot(font_size = 12) +
  theme(legend.position = "none")
plot_e_r_manip

Patch plots together to reproduce Fig. 2 in Box 3

# use patchwork for multi-panel plot
plot_temporal <- plot_exp2_temporal_dynamics + ggtitle("e = 0.65") + theme_cowplot(font_size = 12) + theme(legend.position = "none", plot.title = element_text(hjust = 0.5)) +
  plot_exp3_temporal_dynamics + ylab("") + ggtitle("e = 0.75") + theme_cowplot(font_size = 12) + 
  theme(plot.title = element_text(hjust = 0.5))

plot_isocline <- plot_exp2_isocline_dynamics + theme_cowplot(font_size = 12) + theme(legend.position = "none") +
  plot_exp3_iscoline_dynamics + ylab("") + theme_cowplot(font_size = 12)

# final plot
plot_temporal / plot_isocline / bifur_C_plot / plot_e_r_manip

ggsave(filename = "figures/rosenzweig-macarthur.png", width = 6, height = 8)

sessionInfo()
R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 16.04.7 LTS

Matrix products: default
BLAS:   /usr/lib/libblas/libblas.so.3.6.0
LAPACK: /usr/lib/lapack/liblapack.so.3.6.0

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] patchwork_1.1.1 cowplot_1.1.1   forcats_0.5.1   stringr_1.4.0  
 [5] dplyr_1.0.7     purrr_0.3.4     readr_1.4.0     tidyr_1.1.3    
 [9] tibble_3.1.3    ggplot2_3.3.5   tidyverse_1.3.1 seqinr_4.2-8   
[13] deSolve_1.28    workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.6        lubridate_1.7.10  assertthat_0.2.1  rprojroot_2.0.2  
 [5] digest_0.6.27     utf8_1.2.2        plyr_1.8.6        R6_2.5.0         
 [9] cellranger_1.1.0  backports_1.2.1   reprex_2.0.0      evaluate_0.14    
[13] highr_0.9         httr_1.4.2        pillar_1.6.1      rlang_0.4.11     
[17] readxl_1.3.1      rstudioapi_0.13   whisker_0.4       jquerylib_0.1.4  
[21] rmarkdown_2.9     labeling_0.4.2    munsell_0.5.0     broom_0.7.8      
[25] compiler_4.1.0    httpuv_1.6.1      modelr_0.1.8      xfun_0.24        
[29] pkgconfig_2.0.3   htmltools_0.5.1.1 tidyselect_1.1.1  fansi_0.5.0      
[33] crayon_1.4.1      dbplyr_2.1.1      withr_2.4.2       later_1.2.0      
[37] MASS_7.3-54       grid_4.1.0        jsonlite_1.7.2    gtable_0.3.0     
[41] lifecycle_1.0.0   DBI_1.1.1         git2r_0.28.0      magrittr_2.0.1   
[45] scales_1.1.1      cli_3.0.1         stringi_1.7.3     farver_2.1.0     
[49] fs_1.5.0          promises_1.2.0.1  xml2_1.3.2        bslib_0.2.5.1    
[53] ellipsis_0.3.2    generics_0.1.0    vctrs_0.3.8       tools_4.1.0      
[57] ade4_1.7-17       glue_1.4.2        hms_1.1.0         yaml_2.2.1       
[61] colorspace_2.0-2  rvest_1.0.0       knitr_1.33        haven_2.4.1      
[65] sass_0.4.0