N.t = 30
R = 1.16
N.t1 <- N.t * R
N.t1[1] 34.8
We can use this equation for multiple years and plot the result:
N <- c()
for (i in 1:10) {
N[i] <- N.t * R^i
}
plot(N)
1.1 Blue Whale Recovery
N.1963 <- 10000
N.target <- 50000
R <- 1.1
t <- log(50000/10000)/log(R)
t[1] 16.89
R <- 1.02
t <- log(50000/10000)/log(R)
t[1] 81.27
1.2 Human Population (1800 - 1995)
humans <- data.frame(Year = c(1800, 1850, 1870, 1890, 1910, 1930, 1950, 1970,
1975, 1980, 1985, 1990, 1995))
humans <- cbind(humans, Population = c(0.91, 1.13, 1.3, 1.49, 1.7, 2.02, 2.51,
3.62, 3.97, 4.41, 4.84, 5.29, 5.75))
Time.Interval <- c()
Time.Interval[1] <- NA
for (i in 2:length(humans$Year)) {
Time.Interval[i] <- humans$Year[i] - humans$Year[i - 1]
}
Prev.Population <- humans$Population[1:(length(humans$Population) - 1)]
Growth.Rate <- humans$Population[2:length(humans$Population)]/Prev.Population
Annual.Growth.Rate <- Growth.Rate^(1/Time.Interval[2:length(Time.Interval)])
Table_1.3 <- cbind(humans, Time.Interval, Prev.Population = c(NA, Prev.Population),
Growth.Rate = c(NA, Growth.Rate), Annual.Growth.Rate = c(NA, Annual.Growth.Rate))
Table_1.3 Year Population Time.Interval Prev.Population Growth.Rate
1 1800 0.91 NA NA NA
2 1850 1.13 50 0.91 1.242
3 1870 1.30 20 1.13 1.150
4 1890 1.49 20 1.30 1.146
5 1910 1.70 20 1.49 1.141
6 1930 2.02 20 1.70 1.188
7 1950 2.51 20 2.02 1.243
8 1970 3.62 20 2.51 1.442
9 1975 3.97 5 3.62 1.097
10 1980 4.41 5 3.97 1.111
11 1985 4.84 5 4.41 1.098
12 1990 5.29 5 4.84 1.093
13 1995 5.75 5 5.29 1.087
Annual.Growth.Rate
1 NA
2 1.004
3 1.007
4 1.007
5 1.007
6 1.009
7 1.011
8 1.018
9 1.019
10 1.021
11 1.019
12 1.018
13 1.017
Plot using Growth Rate
qplot(x = Table_1.3$Year[2:13], y = Table_1.3$Growth.Rate[2:13], xlab = "Year",
ylab = "Growth Rate")
Plot using the Annual Growth Rate
qplot(x = Table_1.3$Year[2:13], y = Table_1.3$Annual.Growth.Rate[2:13], xlab = "Year",
ylab = "Annual Growth Rate")
Compare the change in annual growth rate with the absolute increase in number of people
Table_1.4 <- data.frame(Year = c(1975, 1985, 1995), Population.Size = c(3.97,
4.84, 5.75), Annual.Growth.Rate = c(1.01863, 1.018782, 1.016816))
Number.Added <- Table_1.4$Population.Size * (Table_1.4$Annual.Growth.Rate -
1)
Table_1.4 <- cbind(Table_1.4, Number.Added)
Table_1.4 Year Population.Size Annual.Growth.Rate Number.Added
1 1975 3.97 1.019 0.07396
2 1985 4.84 1.019 0.09090
3 1995 5.75 1.017 0.09669
Calculate (based on the number added in 1995) the number added:
- per day:
per.day <- Table_1.4$Number.Added[3]/365
per.day[1] 0.0002649
- per hour:
per.hour <- per.day/24
per.hour[1] 1.104e-05
- per minute:
per.minute = per.hour/60
per.minute[1] 1.84e-07
1.3 Human Population (1995 - 2035)
Table_1.5 <- data.frame(Year = c(1995, 2005, 2015, 2025, 2035))
fec <- 0.0273
R <- 1.016816
s <- R - fec
change10yr <- (R - 1)/4
R05 <- R - change10yr
R15 <- R05 - change10yr
R25 <- R15 - change10yr
Rs <- c(R, R05, R15, R25, 1)
fecs <- Rs - s
Rs10 <- Rs^10
Population <- 5.75
for (i in 2:5) {
Population[i] <- Population[i - 1] * Rs10[i - 1]
}
Table_1.5 <- cbind(Table_1.5, Fecundity = fecs, R = Rs, R_10_yr = Rs10, Population = Population)
Table_1.5 Year Fecundity R R_10_yr Population
1 1995 0.02730 1.017 1.181 5.750
2 2005 0.02310 1.013 1.134 6.793
3 2015 0.01889 1.008 1.087 7.701
4 2025 0.01469 1.004 1.043 8.373
5 2035 0.01048 1.000 1.000 8.732