Exponential 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..)

Relation 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

- Using time series data or population sizes

Estimating R and lambda

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

#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)

Method 1- Regression method

#regression
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

Method 1- Regression method

r= model$coefficients[[2]]
r #per capita growth rate

[1] 0.04433963

Regression method plot

plot(x,y)
abline(model)

Method 2- Ratio method

\[ \lambda= \frac{N_{t+1}}{N_t} \]

• lambda value for each time step and \[ r=log(mean(\lambda)) \]

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

# [1] 500 524 560 579 589 600 650

Method 2- Ratio method

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

# [1] 1.044985

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

# [1] 0.0440027

Method 3- 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.

\[ y_i= \frac{log(\frac{N_{t+1}}{N_t})}{\tau} \]

Method 3- Stochastic differential equation method (Dennis et al. 1991)

\[ y_i= \frac{log(\frac{N_{t+1}}{N_t})}{\tau} \]

• lambda = (Nt+1/Nt)

• x = sqrt(tau) #square root of tau

• y = log(lambda)/x

• mod.fit= lm(y~ 0+x)

Method 3- Stochastic differential equation method (Dennis et al. 1991)

• 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

Quiz et al.

• Lab quiz 1 will be assigned at 10:30am, due next Thurday
• Based on in class assignment and Krebs 2015
• Next week Stochastic population growth