Lab 2_Exponential growth model

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# Lab 2_Exponential growth model
### Vratika Chaudhary
### Population Ecology.
### 2020/01/17

---

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.pull-left[
Exponential growth model
![](http://bio1510.biology.gatech.edu/wp-content/uploads/2014/08/exponential-OSB-Figure_45_03_01.jpg)
]
.pull-right[
Burmese pythons in Everglades
![](https://miro.medium.com/max/2400/1*-eXWpIVw5c1cxYwfm_e9Ig.jpeg)
]

---

# Expontnial growth models can be expressed as:
Differential equation format
`$$\frac{dN}{dT}=rN$$`
Intergral format
`$$N_t=N_oe^{rt}$$`

where

- r = per capital growth rate

- No = Initial population size

- Nt = Projected population size

- t = Time of projection

- e = natural log base (2.713..)

---

# Realation between r and lambda

r = per capita population growth rate
lambda = finite or discrete time growth rate

`$$\lambda=e^r$$`
`$$r=ln(\lambda)=ln[{\frac{N_t+1}{N_t}}]$$`

*In R, log means natural log
---

# Estimating R and lambda
Using time series data or population sizes

# Method 1- Regression method
`$$y=mx+b$$`
y= output variable (population size)
m= slope of the regression line
x= predictor variable (year in our case)
b=  intercept of the regression line

```r
#regression
population =c(400,475,540,580,598,500,600,610,645,670)#example data
year=1:10 #example data
x= year
y = log(population)
model= lm(y~x)
summary(model)
```

```
# 
# Call:
# lm(formula = y ~ x)
# 
# Residuals:
#       Min        1Q    Median        3Q       Max 
# -0.129456 -0.016554 -0.004167  0.063969  0.109088 
# 
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept) 6.076581   0.059882 101.475 9.93e-14 ***
# x           0.044340   0.009651   4.594  0.00177 ** 
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# 
# Residual standard error: 0.08766 on 8 degrees of freedom
# Multiple R-squared:  0.7252,	Adjusted R-squared:  0.6908 
# F-statistic: 21.11 on 1 and 8 DF,  p-value: 0.001769
```

```r
r= model$coefficients[[2]]
r
```

```
# [1] 0.04433963
```
---

```r
#regression
summary(model)
```

```r
r= model$coefficients[[2]]
r
```

```
# [1] 0.04433963
```
---
# Regression method plot

```r
plot(x,y)
abline(model)
```

![](Lab2_presentation_files/figure-html/cars-1.svg)
---
#Method 2- Ratio method

`$$\lambda= \frac{N_{t+1}}{N_t}$$`
lambda value for each time step 
and 
`$$r=log(mean(\lambda))$$`

```r
#Ratio method
pop<- c(500,524,560,579,589,600,650)#population vecto
pop #population vector
```

```
# [1] 500 524 560 579 589 600 650
```

```r
len<- length(pop)
len #length of population vector
```

```
# [1] 7
```

```r
lambda<- pop[-1]/pop[-len]
```

---

```r
pop<- c(500,524,560,579,589,600,650)#population vecto
pop #population vector
```

```
# [1] 500 524 560 579 589 600 650
```

```r
len<- length(pop)
#Ratio method
lambda<- pop[-1]/pop[-len]
lambda.mean = mean(lambda)
lambda.mean
```

```
# [1] 1.044985
```

```r
r = log(lambda.mean) #per capita growth rate
r
```

```
# [1] 0.0440027
```

---
#Stochastic differential equation method (Dennis et al. 1991)
- Often sampling does not take place every year, sometime there is no money for consecutive sampling

- This method uses a new variable 'tau' which is the time difference between two consecutive sampling occassions.

- Fits linear regression without intercept and uses process mean(slope of regression line) and process variance.

---
`$$y_i= \frac{log(\frac{N_{t+1}}{N_t})}{\tau}$$`

- lambda =  (Nt+1/Nt)

- x =  sqrt(tau)   #square root of tau

- y = log(lambda)/x

- mod.fit= lm(y~ 0+x)

- summary(mod.fit)

- mu = summary(au.fit)$coefficients[1]

- r = mu + var(mod.fit$residuals)/2
---
#Doubling and tripling time

- Doubling time = log(2)/r

- Tripling time = log(3)/r

- Halving time = log(0.5)/r