BIOL 4150 Lecture Notes - Lecture 3: Rinderpest
2 May 2018
School
Department
Course
Professor
Chapters 6 & 8
09/21/17
Geometric growth
!
Modeling geometric growth in R
!
Estimating growth rates from census data
!
Outline:
Nt+τ= Nt*λτ
○
Nt= number or density of individuals in year t (state variable)
○
Nt+τ = number or density of individuals τyears later
○
λ = annual rate of growth (parameter; constant)
○
τ = number of years between observations
○
When time is considered in discrete units, growth can be modeled
geometrically
!
First, we declare a variable (N) that is going to a vector with
10 element. Same with year. Then, we set the initial density
and the annual rate of growth
○
N <-numeric (10)
!
Year <-numeric (10) *or Year <-
c(1,2,3,4,5,6,7,8,9,10)
!
N[1] <-7.89mo
!
lambda <-1.39
!
Year <-1:10
"
for(t in 2:10){N[t] <-N[t-1]*lamba}
"
*to multiply lambda by population size 9 times
(updating population density each time):
!
plot(Year,N,'b')
"
plot(Year,N,'p')
"
plot(Year,N,'l')
"
*graphing results, plot both point and a line between
each point ('b') or plot point ('p') or lines without points
('l')
!
rm(list=ls())
"
To clear all variables:
!
Computing:
○
Modeling geometric growth in R:
!
Geometric Growth
r = ln(λ)
!
λ = er
!
*it is often convenient to convert their annual growth rates to their
exponential equivalent
Nt+τ= Nt*er*τ
○
Nt+τ/ Nt= er*τ
○
rt= ln(Nt+τ/Nt) / τ
○
A series of sequential population censuses can be used to estimate
the average exponential growth rate (r)
!
The population grew considerably during 1954-1984
○
rt= ln(15000/4700) / 4 = 0.29
!
The caribou increased from 4700 to 15000 over first 4 years
of study
○
Mean r = 0.147
!
Note: although the average growth rate over time was
remarkably consistent, it varied from year to year due to
environmental stochasity
○
George Rive Caribou -heart is thought to have collapsed in early
1900s, perhaps due to overhunting or vegetation disappearance
!
Estimating Exponential Growth Rates
09/26/17
All populations experience a mix of density-dependent and density-
independent processes
!
While all processes that influence rates of birth, death, immigration
or emigration limit population size, only density-dependent
processes regulate population size
!
Nt+1= Nt*exp(g(Nt))
○
Exponential growth rate
!
g(Nt) = rmax*(1-(Nt/K)) *as population gets larger Nt/K gets
closer to zero --> no growth
○
Logistic Growth:
!
Natural regulation occurs when density dependent changes in the
exponential rate of population change lead to negative feedback
(negative growth rate)
!
Nt+1= Nt*exp[rmax*(1-(Nt/K))]
○
*most generic model to use
○
Ricker Logistic Growth Model:
!
f(N) = N*exp[rmax*(1-(Nt/K))] - N
○
*initially increases and then decreases into negative net
recruitment values
○
Aka what population size allows the addition of the
most number of individuals (think of the carrying
capacity) -where the curve is the steepest
!
Hump shape -interest rates
!
The net number of individuals added(/subtracted) to the
population size over time
○
~ (growth rate)*(population size)
○
Net Recruitment Function:
!
*E-F has largest hump --> most amount of individuals added
to the population
○
Hump-shaped growth curve produces sigmoid density curve over
time
!
Density-Dependent Population Growth
*see Figure on slide
!
*carrying capacity when r=0 (equilibrium of population)
○
Regression line = best fit line measured by deviations around
it
○
Rate of population growth decreases (~0.3 to -0.3) as the density of
elk increased
!
There could be another argument for the decline in 'r' as N
increases
○
'r' could vary by chance (or by environmental factors)
○
Need to test relationship between N and r using a t-test
!
N<-c(……)
○
r<-c(…..)
○
model<-lm(r~N)
○
summary(model)
○
plot(N,r)
○
abline(model)
○
rm(list=ls()) *to clear data
○
lm(formula = r~N)
!
Call:
○
Min, 1Q, Median, 3Q, Max
!
Residuals:
○
Estimate, std. error, t value, Pr(>ltl)…etc
!
Coefficients:
○
Regression test in R:
!
*see slide for equation
○
b= -3.917*10^-5
○
Slope:
!
*see slide for equation
○
a= -0.494
○
Intercept:
!
*see slides for equations
○
The total sum of squared refers to deviations between each
observation and the mean value
○
The residual sum of squared refers to deviations between
each observation and the regression line
○
It is a measure of unexplained variability in y, once we
have removed the linear effect of x
!
The regression sum of squared deviations refers to deviations
between the mean y and the regression line, measured at each
observed y
○
Total & residual sum of squared deviations:
!
Regression DF = 1
○
Residual DF = n-2
○
Degrees of Freedom:
!
F = (SSregression/DFregression) / (Ssresidual/DFresidual)
○
F=9.64
○
Tabular F for 1 df in numerator, and 12 df in denominator =
6.55
○
Because 9>6.55, there is little chance null hypothesis can be
accepted
○
The chance that the data shows no relationship (with
this variation) is 1/100
!
Determines that there is sufficient evidence to accept
the hypothesis
!
This implies that the probability of seeing a slope of this
magnitude given the variation in the data is <0.05
○
F-test:
!
SSregression/Sstotal = 0.445
!
Calculated as the ratio of the regression sums of squares
divided by total sums of squares
○
This implies that 45% of the observed variation in population
abundance is explained by the model
○
Higher R^2 --> more precision
○
Coefficient of Determination (R2) -how much of the variation of
one variable is explained by the other
!
Ex. Population of Elk in Yellowstone
N: 12000 (1958), 19750 (1961), 28300 (1967)
!
N[1] <-12000
!
N[2] <-19750
!
N[3] <-28300
!
Or r[1] <-(log(N[2]/N[1]))/3
○
r[1] <-(log(19750/12000))/3
!
*type in r[1] and press 'enter' to yield answer
!
Ex. In R-Software
09/28/17
λ = (Nt+1 / Nt ) = (B -D + Nt ) / Nt= er= eb-d
○
r=b-d
○
Nt+1 = B -D + Nt
!
B = total births
○
D = total deaths
○
b = instantaneous birth rate
○
d = instantaneous death rate
○
Where…
!
Death rate increases with population size
○
As population size increases, r decreases (to a negative value)
○
Option 1: death rates are density-dependent
!
Birth rate decreases with population size
○
As population size increases, r decreases (to a negative value)
○
Option 2: birth rates are density-dependent
!
Death rate increases with population size while birth rate
decreases
○
As population size increases, r decreases (to a negative value)
○
Option 3: both birth and death rates are density-dependent
!
An increase in death rate leads to a new, lower population
equilibrium (decreased K)
○
Changes in either density-dependent or density-independent factors
influence expected abundance
!
This is because r becomes smaller as N decreases
○
*If N is high it will decrease (in a curved fashion) over time
!
Density-Dependent Population Growth (continued)
Rainfall magnitude is highly unpredictable
○
The Serengeti is an ecosystem of open plains and savannah
woodlands that is highly seasonal with wet periods from
November-May
!
Rinderpest introduced from Russia in 1880
○
Killed most cattle and many native wildlife
○
Population decreased ~1980 due to drought (with
variation)
"
Resulted in large increase in wildebeest population
!
Inoculation program during 1960s removed rinderpest
○
Wildebeest ecology:
!
Used population size from 2000 to calculate 'r' in 1999
○
*see slide with data from 1958-1999
!
'r' decreases linearly (from ~0.2 to -0.05) as the number of
wildebeest increased
○
Displays density dependence
○
Model 1: Ricker Logistic Model
!
73% non-predation
○
8% lion
○
8% hyena
○
1% cheetah
○
8% un-id predator
○
2% miscellaneous
○
Causes of wildebeest mortality:
!
Per capita availability of dry season food regulated the
Serengeti wildebeest population
!
Per capita availability of dry season food declined as the
Serengeti wildebeest population increased over time
○
Fertility loss
!
Neonatal mortality
!
Late calf mortality
!
Yearling mortality
!
Adult mortality
!
*see slides
○
Both fertility and adult mortality were density-dependent
!
One can complete a regression test to determine if a factor is
density-dependent
!
Populations have the ability to self-regulate
○
Natural regulation allows population to 'bounce back' to equilibrium
!
Ex. Serengeti Wildebeest
*for = a function
(…) = an argument
<- = assignment operator
*note: c=concatenation
*discrete vs continuous
growth
Birth rate
!
Death rate
!
Immigration
!
Emigration
!
Vital Rates:
*Carrying Capacity (K)
when b-d = r= 0
Population Growth & Regulation
#$%&'()*+, -./0.12.&, 34+,3546
44738,9:
Chapters 6 & 8
09/21/17
Geometric growth
!
Modeling geometric growth in R
!
Estimating growth rates from census data
!
Outline:
Nt+τ= Nt*λτ
○
Nt= number or density of individuals in year t (state variable)
○
Nt+τ = number or density of individuals τyears later
○
λ = annual rate of growth (parameter; constant)
○
τ = number of years between observations
○
When time is considered in discrete units, growth can be modeled
geometrically
!
First, we declare a variable (N) that is going to a vector with
10 element. Same with year. Then, we set the initial density
and the annual rate of growth
○
N <-numeric (10)
!
Year <-numeric (10) *or Year <-
c(1,2,3,4,5,6,7,8,9,10)
!
N[1] <-7.89mo
!
lambda <-1.39
!
Year <-1:10
"
for(t in 2:10){N[t] <-N[t-1]*lamba}
"
*to multiply lambda by population size 9 times
(updating population density each time):
!
plot(Year,N,'b')
"
plot(Year,N,'p')
"
plot(Year,N,'l')
"
*graphing results, plot both point and a line between
each point ('b') or plot point ('p') or lines without points
('l')
!
rm(list=ls())
"
To clear all variables:
!
Computing:
○
Modeling geometric growth in R:
!
Geometric Growth
r = ln(λ)
!
λ = er
!
*it is often convenient to convert their annual growth rates to their
exponential equivalent
Nt+τ= Nt*er*τ
○
Nt+τ/ Nt= er*τ
○
rt= ln(Nt+τ/Nt) / τ
○
A series of sequential population censuses can be used to estimate
the average exponential growth rate (r)
!
The population grew considerably during 1954-1984
○
rt= ln(15000/4700) / 4 = 0.29
!
The caribou increased from 4700 to 15000 over first 4 years
of study
○
Mean r = 0.147
!
Note: although the average growth rate over time was
remarkably consistent, it varied from year to year due to
environmental stochasity
○
George Rive Caribou -heart is thought to have collapsed in early
1900s, perhaps due to overhunting or vegetation disappearance
!
Estimating Exponential Growth Rates
09/26/17
All populations experience a mix of density-dependent and density-
independent processes
!
While all processes that influence rates of birth, death, immigration
or emigration limit population size, only density-dependent
processes regulate population size
!
Nt+1= Nt*exp(g(Nt))
○
Exponential growth rate
!
g(Nt) = rmax*(1-(Nt/K)) *as population gets larger Nt/K gets
closer to zero --> no growth
○
Logistic Growth:
!
Natural regulation occurs when density dependent changes in the
exponential rate of population change lead to negative feedback
(negative growth rate)
!
Nt+1= Nt*exp[rmax*(1-(Nt/K))]
○
*most generic model to use
○
Ricker Logistic Growth Model:
!
f(N) = N*exp[rmax*(1-(Nt/K))] - N
○
*initially increases and then decreases into negative net
recruitment values
○
Aka what population size allows the addition of the
most number of individuals (think of the carrying
capacity) -where the curve is the steepest
!
Hump shape -interest rates
!
The net number of individuals added(/subtracted) to the
population size over time
○
~ (growth rate)*(population size)
○
Net Recruitment Function:
!
*E-F has largest hump --> most amount of individuals added
to the population
○
Hump-shaped growth curve produces sigmoid density curve over
time
!
Density-Dependent Population Growth
*see Figure on slide
!
*carrying capacity when r=0 (equilibrium of population)
○
Regression line = best fit line measured by deviations around
it
○
Rate of population growth decreases (~0.3 to -0.3) as the density of
elk increased
!
There could be another argument for the decline in 'r' as N
increases
○
'r' could vary by chance (or by environmental factors)
○
Need to test relationship between N and r using a t-test
!
N<-c(……)
○
r<-c(…..)
○
model<-lm(r~N)
○
summary(model)
○
plot(N,r)
○
abline(model)
○
rm(list=ls()) *to clear data
○
lm(formula = r~N)
!
Call:
○
Min, 1Q, Median, 3Q, Max
!
Residuals:
○
Estimate, std. error, t value, Pr(>ltl)…etc
!
Coefficients:
○
Regression test in R:
!
*see slide for equation
○
b= -3.917*10^-5
○
Slope:
!
*see slide for equation
○
a= -0.494
○
Intercept:
!
*see slides for equations
○
The total sum of squared refers to deviations between each
observation and the mean value
○
The residual sum of squared refers to deviations between
each observation and the regression line
○
It is a measure of unexplained variability in y, once we
have removed the linear effect of x
!
The regression sum of squared deviations refers to deviations
between the mean y and the regression line, measured at each
observed y
○
Total & residual sum of squared deviations:
!
Regression DF = 1
○
Residual DF = n-2
○
Degrees of Freedom:
!
F = (SSregression/DFregression) / (Ssresidual/DFresidual)
○
F=9.64
○
Tabular F for 1 df in numerator, and 12 df in denominator =
6.55
○
Because 9>6.55, there is little chance null hypothesis can be
accepted
○
The chance that the data shows no relationship (with
this variation) is 1/100
!
Determines that there is sufficient evidence to accept
the hypothesis
!
This implies that the probability of seeing a slope of this
magnitude given the variation in the data is <0.05
○
F-test:
!
SSregression/Sstotal = 0.445
!
Calculated as the ratio of the regression sums of squares
divided by total sums of squares
○
This implies that 45% of the observed variation in population
abundance is explained by the model
○
Higher R^2 --> more precision
○
Coefficient of Determination (R2) -how much of the variation of
one variable is explained by the other
!
Ex. Population of Elk in Yellowstone
N: 12000 (1958), 19750 (1961), 28300 (1967)
!
N[1] <-12000
!
N[2] <-19750
!
N[3] <-28300
!
Or r[1] <-(log(N[2]/N[1]))/3
○
r[1] <-(log(19750/12000))/3
!
*type in r[1] and press 'enter' to yield answer
!
Ex. In R-Software
09/28/17
λ = (Nt+1 / Nt ) = (B -D + Nt ) / Nt= er= eb-d
○
r=b-d
○
Nt+1 = B -D + Nt
!
B = total births
○
D = total deaths
○
b = instantaneous birth rate
○
d = instantaneous death rate
○
Where…
!
Death rate increases with population size
○
As population size increases, r decreases (to a negative value)
○
Option 1: death rates are density-dependent
!
Birth rate decreases with population size
○
As population size increases, r decreases (to a negative value)
○
Option 2: birth rates are density-dependent
!
Death rate increases with population size while birth rate
decreases
○
As population size increases, r decreases (to a negative value)
○
Option 3: both birth and death rates are density-dependent
!
An increase in death rate leads to a new, lower population
equilibrium (decreased K)
○
Changes in either density-dependent or density-independent factors
influence expected abundance
!
This is because r becomes smaller as N decreases
○
*If N is high it will decrease (in a curved fashion) over time
!
Density-Dependent Population Growth (continued)
Rainfall magnitude is highly unpredictable
○
The Serengeti is an ecosystem of open plains and savannah
woodlands that is highly seasonal with wet periods from
November-May
!
Rinderpest introduced from Russia in 1880
○
Killed most cattle and many native wildlife
○
Population decreased ~1980 due to drought (with
variation)
"
Resulted in large increase in wildebeest population
!
Inoculation program during 1960s removed rinderpest
○
Wildebeest ecology:
!
Used population size from 2000 to calculate 'r' in 1999
○
*see slide with data from 1958-1999
!
'r' decreases linearly (from ~0.2 to -0.05) as the number of
wildebeest increased
○
Displays density dependence
○
Model 1: Ricker Logistic Model
!
73% non-predation
○
8% lion
○
8% hyena
○
1% cheetah
○
8% un-id predator
○
2% miscellaneous
○
Causes of wildebeest mortality:
!
Per capita availability of dry season food regulated the
Serengeti wildebeest population
!
Per capita availability of dry season food declined as the
Serengeti wildebeest population increased over time
○
Fertility loss
!
Neonatal mortality
!
Late calf mortality
!
Yearling mortality
!
Adult mortality
!
*see slides
○
Both fertility and adult mortality were density-dependent
!
One can complete a regression test to determine if a factor is
density-dependent
!
Populations have the ability to self-regulate
○
Natural regulation allows population to 'bounce back' to equilibrium
!
Ex. Serengeti Wildebeest
*for = a function
(…) = an argument
<- = assignment operator
*note: c=concatenation
*discrete vs continuous
growth
Birth rate
!
Death rate
!
Immigration
!
Emigration
!
Vital Rates:
*Carrying Capacity (K)
when b-d = r= 0
Population Growth & Regulation
#$%&'()*+, -./0.12.&, 34+,3546 44738,9:
Chapters 6 & 8
09/21/17
Geometric growth
!
Modeling geometric growth in R
!
Estimating growth rates from census data
!
Outline:
Nt+τ= Nt*λτ
○
Nt= number or density of individuals in year t (state variable)
○
Nt+τ = number or density of individuals τyears later
○
λ = annual rate of growth (parameter; constant)
○
τ = number of years between observations
○
When time is considered in discrete units, growth can be modeled
geometrically
!
First, we declare a variable (N) that is going to a vector with
10 element. Same with year. Then, we set the initial density
and the annual rate of growth
○
N <-numeric (10)
!
Year <-numeric (10) *or Year <-
c(1,2,3,4,5,6,7,8,9,10)
!
N[1] <-7.89mo
!
lambda <-1.39
!
Year <-1:10
"
for(t in 2:10){N[t] <-N[t-1]*lamba}
"
*to multiply lambda by population size 9 times
(updating population density each time):
!
plot(Year,N,'b')
"
plot(Year,N,'p')
"
plot(Year,N,'l')
"
*graphing results, plot both point and a line between
each point ('b') or plot point ('p') or lines without points
('l')
!
rm(list=ls())
"
To clear all variables:
!
Computing:
○
Modeling geometric growth in R:
!
Geometric Growth
r = ln(λ)
!
λ = er
!
*it is often convenient to convert their annual growth rates to their
exponential equivalent
Nt+τ= Nt*er*τ
○
Nt+τ/ Nt= er*τ
○
rt= ln(Nt+τ/Nt) / τ
○
A series of sequential population censuses can be used to estimate
the average exponential growth rate (r)
!
The population grew considerably during 1954-1984
○
rt= ln(15000/4700) / 4 = 0.29
!
The caribou increased from 4700 to 15000 over first 4 years
of study
○
Mean r = 0.147
!
Note: although the average growth rate over time was
remarkably consistent, it varied from year to year due to
environmental stochasity
○
George Rive Caribou -heart is thought to have collapsed in early
1900s, perhaps due to overhunting or vegetation disappearance
!
Estimating Exponential Growth Rates
09/26/17
All populations experience a mix of density-dependent and density-
independent processes
!
While all processes that influence rates of birth, death, immigration
or emigration limit population size, only density-dependent
processes regulate population size
!
Nt+1= Nt*exp(g(Nt))
○
Exponential growth rate
!
g(Nt) = rmax*(1-(Nt/K)) *as population gets larger Nt/K gets
closer to zero --> no growth
○
Logistic Growth:
!
Natural regulation occurs when density dependent changes in the
exponential rate of population change lead to negative feedback
(negative growth rate)
!
Nt+1= Nt*exp[rmax*(1-(Nt/K))]
○
*most generic model to use
○
Ricker Logistic Growth Model:
!
f(N) = N*exp[rmax*(1-(Nt/K))] - N
○
*initially increases and then decreases into negative net
recruitment values
○
Aka what population size allows the addition of the
most number of individuals (think of the carrying
capacity) -where the curve is the steepest
!
Hump shape -interest rates
!
The net number of individuals added(/subtracted) to the
population size over time
○
~ (growth rate)*(population size)
○
Net Recruitment Function:
!
*E-F has largest hump --> most amount of individuals added
to the population
○
Hump-shaped growth curve produces sigmoid density curve over
time
!
Density-Dependent Population Growth
*see Figure on slide
!
*carrying capacity when r=0 (equilibrium of population)
○
Regression line = best fit line measured by deviations around
it
○
Rate of population growth decreases (~0.3 to -0.3) as the density of
elk increased
!
There could be another argument for the decline in 'r' as N
increases
○
'r' could vary by chance (or by environmental factors)
○
Need to test relationship between N and r using a t-test
!
N<-c(……)
○
r<-c(…..)
○
model<-lm(r~N)
○
summary(model)
○
plot(N,r)
○
abline(model)
○
rm(list=ls()) *to clear data
○
lm(formula = r~N)
!
Call:
○
Min, 1Q, Median, 3Q, Max
!
Residuals:
○
Estimate, std. error, t value, Pr(>ltl)…etc
!
Coefficients:
○
Regression test in R:
!
*see slide for equation
○
b= -3.917*10^-5
○
Slope:
!
*see slide for equation
○
a= -0.494
○
Intercept:
!
*see slides for equations
○
The total sum of squared refers to deviations between each
observation and the mean value
○
The residual sum of squared refers to deviations between
each observation and the regression line
○
It is a measure of unexplained variability in y, once we
have removed the linear effect of x
!
The regression sum of squared deviations refers to deviations
between the mean y and the regression line, measured at each
observed y
○
Total & residual sum of squared deviations:
!
Regression DF = 1
○
Residual DF = n-2
○
Degrees of Freedom:
!
F = (SSregression/DFregression) / (Ssresidual/DFresidual)
○
F=9.64
○
Tabular F for 1 df in numerator, and 12 df in denominator =
6.55
○
Because 9>6.55, there is little chance null hypothesis can be
accepted
○
The chance that the data shows no relationship (with
this variation) is 1/100
!
Determines that there is sufficient evidence to accept
the hypothesis
!
This implies that the probability of seeing a slope of this
magnitude given the variation in the data is <0.05
○
F-test:
!
SSregression/Sstotal = 0.445
!
Calculated as the ratio of the regression sums of squares
divided by total sums of squares
○
This implies that 45% of the observed variation in population
abundance is explained by the model
○
Higher R^2 --> more precision
○
Coefficient of Determination (R2) -how much of the variation of
one variable is explained by the other
!
Ex. Population of Elk in Yellowstone
N: 12000 (1958), 19750 (1961), 28300 (1967)
!
N[1] <-12000
!
N[2] <-19750
!
N[3] <-28300
!
Or r[1] <-(log(N[2]/N[1]))/3
○
r[1] <-(log(19750/12000))/3
!
*type in r[1] and press 'enter' to yield answer
!
Ex. In R-Software
09/28/17
λ = (Nt+1 / Nt ) = (B -D + Nt ) / Nt= er= eb-d
○
r=b-d
○
Nt+1 = B -D + Nt
!
B = total births
○
D = total deaths
○
b = instantaneous birth rate
○
d = instantaneous death rate
○
Where…
!
Death rate increases with population size
○
As population size increases, r decreases (to a negative value)
○
Option 1: death rates are density-dependent
!
Birth rate decreases with population size
○
As population size increases, r decreases (to a negative value)
○
Option 2: birth rates are density-dependent
!
Death rate increases with population size while birth rate
decreases
○
As population size increases, r decreases (to a negative value)
○
Option 3: both birth and death rates are density-dependent
!
An increase in death rate leads to a new, lower population
equilibrium (decreased K)
○
Changes in either density-dependent or density-independent factors
influence expected abundance
!
This is because r becomes smaller as N decreases
○
*If N is high it will decrease (in a curved fashion) over time
!
Density-Dependent Population Growth (continued)
Rainfall magnitude is highly unpredictable
○
The Serengeti is an ecosystem of open plains and savannah
woodlands that is highly seasonal with wet periods from
November-May
!
Rinderpest introduced from Russia in 1880
○
Killed most cattle and many native wildlife
○
Population decreased ~1980 due to drought (with
variation)
"
Resulted in large increase in wildebeest population
!
Inoculation program during 1960s removed rinderpest
○
Wildebeest ecology:
!
Used population size from 2000 to calculate 'r' in 1999
○
*see slide with data from 1958-1999
!
'r' decreases linearly (from ~0.2 to -0.05) as the number of
wildebeest increased
○
Displays density dependence
○
Model 1: Ricker Logistic Model
!
73% non-predation
○
8% lion
○
8% hyena
○
1% cheetah
○
8% un-id predator
○
2% miscellaneous
○
Causes of wildebeest mortality:
!
Per capita availability of dry season food regulated the
Serengeti wildebeest population
!
Per capita availability of dry season food declined as the
Serengeti wildebeest population increased over time
○
Fertility loss
!
Neonatal mortality
!
Late calf mortality
!
Yearling mortality
!
Adult mortality
!
*see slides
○
Both fertility and adult mortality were density-dependent
!
One can complete a regression test to determine if a factor is
density-dependent
!
Populations have the ability to self-regulate
○
Natural regulation allows population to 'bounce back' to equilibrium
!
Ex. Serengeti Wildebeest
*for = a function
(…) = an argument
<- = assignment operator
*note: c=concatenation
*discrete vs continuous
growth
Birth rate
!
Death rate
!
Immigration
!
Emigration
!
Vital Rates:
*Carrying Capacity (K)
when b-d = r= 0
Population Growth & Regulation
#$%&'()*+, -./0.12.&, 34+,3546 44738,9:
Document Summary
When time is considered in discrete units, growth can be modeled geometrically. Nt = number or density of individuals in year t (state variable) Nt+ = number or density of individuals years later. = annual rate of growth (parameter; constant) First, we declare a variable (n) that is going to a vector with. Then, we set the initial density and the annual rate of growth. Year
