user048590/logistic_growth

R scripts for a reproducible analysis of logistic growth

logistic_growth

R scripts for a reproducible analysis of logistic growth

Script to estimate the model parameters using a linear approximation

install.packages("dplyr")
library(dplyr)

The dplyr package enables us to manipulate data via efficient filtering, grouping and arranging. It will therefore be useful in managing this set of data.

growth_data <- read.csv("experiment1.csv")

Firstly, we are reading in our data and calling it growth_data

Case 1. K >> N0, t is small

data_subset1 <- growth_data %>% filter(t<1800) %>% mutate(N_log = log(N))

model1 <- lm(N_log ~ t, data_subset1)

summary(model1)

We first want to determine estimates for the carrying capacity, initial population size and the population growth rate. We define a linear model for when t is small, and where we expect the growth to be exponential, by filtering the data such that it only includes rows where t is less than 1800 (beyond this point the growth is expected to plateau, based on the raw data).

Then, we have mutated the population (N) to log transform it, because this results in 2 distinct growth stages - an initial increasing linear growth stage and a plateau whereby growth rate is constant - in order to create 2 separate linear models. We can consequently use these models to estimate growth rates, carrying capacity and population size as they will provide us with estimates of both the gradient and the y-intercept. For example, when t<1800, the intercept is 992, the gradient is 9.990e-03.

Case 2. N(t) = K

data_subset2 <- growth_data %>% filter(t>1800)

model2 <- lm(N ~ 1, data_subset2)

summary(model2)

Next, we want to create a model for the stage of growth where we expect to observe a plateau, hence have filtered the data to only include values of t greater than 1800.

We require 2 separate linear models because we predict there to be 2 distinct growth stages.

Script to plot the logistic growth data

growth_data <- read.csv("experiment1.csv")

install.packages("ggplot2")
library(ggplot2)

ggplot(aes(t,N), data = growth_data) +
  
  geom_point() +
  
  xlab("t") +
  
  ylab("N") +
  
  theme_bw()

This chunk of code plots a graph of the raw data, with time (t) on the x-axis and population size (N) on the y.

ggplot(aes(t,N), data = growth_data) +
  
  geom_point() +
  
  xlab("t") +
  
  ylab("N ") +
  
  scale_y_continuous(trans='log10')

Here, we have plotted a graph of the raw data, except with a log transformation of the y-axis.

Script to plot data and model

Next, we will plot the model and compare it to the raw data in order to determine how accurately our model can predict variables of interest.

growth_data <- read.csv("experiment1.csv")

logistic_fun <- function(t) {
  
  N <- (N0*K*exp(r*t))/(K-N0+N0*exp(r*t))
  
  return(N)
  
}

N0 <- 992
  
r <- 9.990e-03
  
K <- 5.903e+10

Here we are defining a function for logistic growth (logistic_fun) which we will use to calculate values of N, based on the logistic growth equation.

ggplot(aes(t,N), data = growth_data) +
  
  geom_function(fun=logistic_fun, colour="red") +
  
  geom_point()+
  scale_y_continuous(trans='log10')

We can then plot our logistic model against the raw data to determine how closely our estimates represent the dataset.

Results

Our model provides us with the following values:

N0 = 992

r = 9.990e-03

K = 5.903e+10

where N0 = starting population size, r = growth rate and K = carrying capacity.

Using the values we obtained from the linear models above and applying them to our logistic equation, we can compare our model with the raw data. Given that we are using the data to create our model and form our estimates, it is perhaps unsurprising that the model perfectly aligns with

2. Calculating population size at t=4980

Where population growth rate is exponential, N(t) = N0e^rt

Substituting in the previous values as defined by our model, this gives N(t) = 992e^(9.990e-03 x 4980) = 4.00e+24

The logistic growth model is defined as N(t) = (N0Kexp(rt))/(K-N0+N0exp(r*t))

We know that the value of N will be equivalent to the carrying capacity, as this model assumes population growth is limited by the maximum number of individuals the population can support (k), which, in this case = 5.903e+10

Therefore, the difference between the output generated by these 2 models is large (3.98e+24). This is expected, given that the exponential model assumes that growth is unlimited, therefore this estimation will be far larger than the logistic model (which states that growth will be limited and eventually plateau).

3. Graph comparing the exponential and logistic growth curves

This graph shows the exponential function (blue) versus the logistic function (red).

The code for this can be found in the logistic_vs_exponential.md file