Abstract

When we think about south of the equator countries, often vast amount of wilderness and beautiful sceneries come to mind. But recently most of the news about wilderness and beautiful scenery is followed by a cry for help, since recent human actions have made countless animal and plant species extinct. One of the most majestic animals an African forest Elephant is no exception to this phenomenon. According to this article the population of this elephant has been drastically decreasing. In the 1800 the population was about 26 million, in 1989 the population is around the 150 thousands, now days the species is considered a critically endangered species, due to its low population of less than 50 thousand.

The extinction of this elephant is due mostly to an increase in poaching activity (The illegal trafficking and killing of wildfire) by humans. Another reason is the loss of suitable habitat due to human expansion. Because the human population has not found a suitable predator to control our population, it keeps growing faster than the mortally rate. This means more habitats and food supply that we have to take from nature (Also killing elephants for their ivory tusk to make ivory decor is definitely not helping).

Data

This Data is exponential decay because the population of African forest elephant is decreasing exponentially. Since my data covers 3 decades my initial year is 1989 and ends 2021 with a total of 32 years.

the population for 1989 is 172000 the expected half life is 16 years (2005), with an estimated population of 86000

The formula to calculate the rate of decay is:

\(N = (Ni) e^(kt)\)

N = the population at a specific time

Ni = the initial population at the 1 time. (1.72* 10^5)

e = 2.71828

k ~ how quickly the decay happens (-0.0217)

t = time (32 years)

the formula to find k is:

\(k= {ln(50000/172000)}/{t}\)

\(k= {ln(0.290698)}/(32)\) = -0.03861

Using this k this is the expected population for each year for 32 years.

Year /T /N

year Time N Data
1989 0 172000 172000
1990 1 165000
1991 2 159000
1992 3 153000
1993 4 147000
1994 5 142000
1995 6 136000
1996 7 131000
1997 8 126000
1998 9 122000
1999 10 117000
2000 11 112000
2001 12 108000
2002 13 104000
2003 14 100000 120000
2004 15 96400
2005 16 (Expected HL) 92700
2006 17 89200
2007 18 (Actual HL) 85800
2008 19 82600
2009 20 79500
2010 21 76500
2011 22 73600
2012 23 70800
2013 24 68100 70000
2014 25 65500
2015 26 63000
2016 27 60600
2017 28 58300
2018 29 56100
2019 30 54000
2020 31 52000
2021 32 50000 50000 
    Estimate Std. Error t value Pr(>|t|)    
k -3.864e-02  2.704e-05   -1429   <2e-16 
x <- c(0:32)

    y <- c(1.72E+05, 1.65E+05, 1.59E+05, 1.53E+05, 1.47E+05, 1.42E+05, 1.36E+05, 1.31E+05, 1.26E+05, 1.22E+05, 1.17E+05, 1.12E+05, 1.08E+05, 1.04E+05, 1.00E+05, 9.64E+04, 9.27E+04, 8.92E+04, 8.58E+04, 8.26E+04, 7.95E+04, 7.65E+04, 7.36E+04, 7.08E+04, 6.81E+04, 6.55E+04, 6.30E+04, 6.06E+04, 5.83E+04, 5.61E+04, 5.40E+04, 5.20E+04, 5.00E+04 )

    plot (x,y, xlab = "Years", ylab = "Population of elephants", main = "Exponential decay")
    N <- 172000*exp(0.03861*x)
    
tryfit <- nls(y ~ 172000*exp(k*x), start =c(k=0))
summary(tryfit)
## 
## Formula: y ~ 172000 * exp(k * x)
## 
## Parameters:
##     Estimate Std. Error t value Pr(>|t|)    
## k -3.864e-02  2.704e-05   -1429   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 207.5 on 32 degrees of freedom
## 
## Number of iterations to convergence: 5 
## Achieved convergence tolerance: 5.906e-08
lines(x,predict(tryfit), col='red')

A <- c(0:3)

B <- c(172000, 120000, 70000, 50000)



plot (A,B, xlab = "Years", ylab = "Population of elephants", main = "Exponential decay")
    N <- 172000*exp(0.03861*x)
    
tryfit <- nls(B ~ 172000*exp(k*A), start =c(k=0))
summary(tryfit)
## 
## Formula: B ~ 172000 * exp(k * A)
## 
## Parameters:
##   Estimate Std. Error t value Pr(>|t|)    
## k -0.41415    0.01959  -21.14 0.000232 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4705 on 3 degrees of freedom
## 
## Number of iterations to convergence: 5 
## Achieved convergence tolerance: 1.369e-06
lines(A,predict(tryfit), col='red')