• Exponential Growth
• Birth and Death Model
• Visualizing it All

• Living things are born and die all the time.
• Populations increases and decreases based on the amount of births compared to the amount of deaths.
• We can use models to help us visualize population changes.

• This is the exponential growth formula.
• The larger a population is, the faster it will grow.
• \(N_0\): The Population we start at
• \(r\): Growth Rate
• \(t\): Time

\[ N(t) = N_0 e^{rt} \]

• We are going to use a model that shows births and deaths per individual.
• This will tell us the average contribution each population member makes.
• This is useful when you want to compare different populations and how fast they are growing and changing.
• \(r = b - d\): net growth rate
• \({dN}/{dt}\): rate of change in a population over time

\[ r = b - d \quad \quad \frac{dN}{dt} = rN \]

• Do you notice how it grows faster over time?

library (ggplot2)

time <- 2:30
N0 <- 50
r <- 0.3
N <- N0 * exp(r * time)
df <- data.frame(time, N)

ggplot(df, aes(time, N)) +
  geom_line(color="black") +
  labs(title = "Exponential Growth Graph",
       x="Time",
       y="Population")

-This graph shows us the green populations growing, the blue populations staying stable, and the red populations decreasing. Obviously, it isn’t this rigid in real life.

time <- 2:30
N0 <- 50

df <- data.frame(
  time = time,
  growth = N0 * exp(0.2 * time),
  decline = N0 * exp(-0.2 * time),
  stable = N0 * exp(0 * time))

ggplot() +
  geom_line(data=df, aes(time, growth), color="green") +
  geom_line(data=df, aes(time, decline), color="red") +
  geom_line(data=df, aes(time, stable), color="blue") +
  scale_y_log10() +
  labs(title="Birth and Death Model",
       x="Time",
       y = "Population")

library(plotly)

time <- 2:30
N0 <- 20
r <- 0.3
population <- N0 * exp(r * time)

plot_ly(x = ~time, y = ~population, type = 'scatter', mode = 'lines+markers') %>%
  layout(title = "Population Growth Plotly",
         xaxis = list(title = "Time"),
         yaxis = list(title = "Population"))