BIOL 4150 Lecture Notes - Lecture 5: Soay Sheep, Leslie Matrix, Diagonal
2 May 2018
School
Department
Course
Professor
10/05/17
= 0.6790, 0.6790, 0.8532
!
p -expected survival rates for each age group
â—‹
= 0.0000, 0.0000, 0.5303
!
m -expected reproductive rate for each age group
â—‹
In order to develop an age-specific model, we need 2 demographic
parameters:
!
Starting value = 0, 3 rows, 3 columns
!
A<-matrix(0,nr=3,ncol=3)
â—‹
X [row, column]
!
Row1: probability they survive and produce an
offspring
!
Row2&3: yearlings and adults that remain in
following years
!
A[2,1] <-p[1]
â—‹
*see R coding on slide
!
Involves multiplying age-specific population densities by a
transition matrix (A)
â—‹
The top row in A reflects the probability of survival from
the previous age class multiplied by fecundity at age 'x'
â—‹
The subdiagonal reflects the age-specific survival
probabilities (0.679,0.679)
â—‹
Leslie Matrix Model:
!
2nd row: 0.679*10 + 0*10 + 0*10 = 6.79
â—‹
3rd row: 0*10 + 0.679*10 + 0.853*10 = 15.32
â—‹
*taken initial population of 30 individuals (equally divided
between age groups) and created new population (with a
smaller number of individuals)
1st row: 0*10 + 0*10 + 0.45209*10 = 4.5209
!
Nt=30
â—‹
New variable: sum of all the different age groups in the population
(N) = 26.4
!
tmax<-30 (population size=30)
â—‹
n<-matrix(0,nr=3,ncol=30)
â—‹
Means… n[1,1]<-30, n[2,1]<-2, n[3,1]<-3
!
n[,1]<-(30,2,3)
â—‹
*means we will do this 29x, n for the next year =
matrix*column of n values in that year
!
% = matrix multiplication
!
for(i in 1:(tmax-1)) n[,i+1] <-A%*%n[,i]
â—‹
N<-numeric(tmax) *too see full population, not just 'n' in
each age group
â—‹
for(i in 1:tmax) N[i] <-sum(n[,i])
â—‹
Over time -> geometric growth model
!
Usually takes 2 generations for population to
set into a rhythm
"
Initial drop -> period of instability where yearlings
are not reproducing
!
plot(t-1,N,type='l',xlab="t",ylab="N")
â—‹
R coding:
!
Ex. Age-structured model of cheetahs
10/17/17
Stability of simple, naturally regulated populations depends on the
magnitude of demographic parameters, whether there is linear or
nonlinear density-dependence, and age structure
!
Fluctuates from 1955 to 1995
â—‹
Threshold effect (steepest part of curve)
!
Offspring/female declines with increasing female
population density
â—‹
Slight drops in food abundance leads to sheep
population crashes (inversely related)
!
Less food availability?
"
Higher energy expenditure (thermoregulation -
metabolism)
"
During a 'crash' year yearling, adults and young lose
weight from August -March
!
Survival decreases as female population density
increases
!
Declining body condition due to food scarcity is main
causal factor
â—‹
Slightly changes shape
!
Alpha = 0.005
"
Bigger = steeper sigmoid effect
(sensitivity to density dependence)
!
*parameters --> minimize square
deviations (most parsimonious)
!
Beta = 15
"
*without pmax, goes from 1 -0
"
Fitting parameters:
!
Gives range from max to zero with population
size
"
= 1/ (1 + (alpha*N)^beta)
"
Logistic regression function:
!
Expected maximum survival rates for each
group (pmax)
"
Give sigmoid shape
!
Age-specific density-dependent survival
parameters (alpha and beta
"
= pmax / (1+ (alphai*N)^betai)
!
Logistic survival function for each age group:
"
All parameters for 3 age groups:
!
Expected maximum reproductive rate for each
age group (mmax)
"
Age specific density-dependent reproductive
parameters (alphaalpha & betabeta)
"
= mmax/ (1+ (alphalpha*N)^betabeta)
!
Logistic reproductive function for each age
group:
"
More parameters:
!
i=0..2
"
j=1..2
"
t=0..50
"
Omega = birth
!
Psi = survival
!
*see equation for matrix on slide
"
Young=f(survival*fecundity) * population size
"
Yearlings = survival rate*young from previous
year
"
From previous years
!
Adults = (survival*yearlings) + (survival*adult
numbers)
"
Initial age distribution: n=(10,4,2)
!
*see slide for equations
â—‹
tmax=50
!
time<-1:tmax
!
Sheep<-numeric(tmax)
!
Pmax<-c(0.88,0.94,0.96)
!
Alpha1<-c(…)
!
Beta1<-(c…)
!
Mmax<-c(…)
!
Alpha2<-c(…)
!
Beta2<-c(…)
!
Psi<-numeric(3)
!
Omega<-numeric(3)
!
n<-matrix(nrow=3,ncol=tmax)
!
n[,1]<-c(40,15,6)
!
Density<-sum(n[,1])
!
Sheep<-density
!
For(t in 2:tmax){
!
For(i in 1:3) psi[i]<-pmax[i]/(1
+(alpha1[i]*density)^beta1[i])
!
For(i in 1:3) omega[i] <-mmax[i]/(1
+(alpha2[i]*density)^beta2[i])
!
n[1,t]<-
n[2,t-1]*psi[2]*omega[2]+n[3,t[1]*psi[3]*omega[3]
!
n[2,t]<-n[1,t-1]*psi[1]
!
n[3,t]<-n[2,t-1]*psi[2]+n[3,t-1]*psi[3]
!
Density<-sum(n[,t])
!
Sheep[t]<-density}
!
Plot(time,sheep,type='l')
!
rm(list=ls())
!
R-coding:
â—‹
Sheep -levels off (oscillates over time) --> density-
dependent
!
Cheetahs -grew exponentially over time (leslie matrix)
â—‹
A good example of a population with extremely non-linear per
capita recruitment is the herd of feral Soay sheep on the Scottish
island of Hirta
!
Such demographic parameters are common, suggesting that
complex dynamics should be the norm (not the exception)
â—‹
Survival rates go down in years with rainier and
colder weather
!
Stochastic weather effects the add to the complex pattern of
population dynamics
â—‹
Short term dynamics depends on age structure, weather and
density
â—‹
Any combination of high instrinsic growth rates, non-linear
density dependence, protracted time lags and delays due to age
structure effects predispose a population to cyclic or even chaotic
fluctuations over time
!
Structure population dynamics of Soay sheep
Age Structured Populations
#$%&'()*+, -./012&, 3+,4567
66849,:;
10/05/17
= 0.6790, 0.6790, 0.8532
!
p -expected survival rates for each age group
â—‹
= 0.0000, 0.0000, 0.5303
!
m -expected reproductive rate for each age group
â—‹
In order to develop an age-specific model, we need 2 demographic
parameters:
!
Starting value = 0, 3 rows, 3 columns
!
A<-matrix(0,nr=3,ncol=3)
â—‹
X [row, column]
!
Row1: probability they survive and produce an
offspring
!
Row2&3: yearlings and adults that remain in
following years
!
A[2,1] <-p[1]
â—‹
*see R coding on slide
!
Involves multiplying age-specific population densities by a
transition matrix (A)
â—‹
The top row in A reflects the probability of survival from
the previous age class multiplied by fecundity at age 'x'
â—‹
The subdiagonal reflects the age-specific survival
probabilities (0.679,0.679)
â—‹
Leslie Matrix Model:
!
2nd row: 0.679*10 + 0*10 + 0*10 = 6.79
â—‹
3rd row: 0*10 + 0.679*10 + 0.853*10 = 15.32
â—‹
*taken initial population of 30 individuals (equally divided
between age groups) and created new population (with a
smaller number of individuals)
1st row: 0*10 + 0*10 + 0.45209*10 = 4.5209
!
Nt=30
â—‹
New variable: sum of all the different age groups in the population
(N) = 26.4
!
tmax<-30 (population size=30)
â—‹
n<-matrix(0,nr=3,ncol=30)
â—‹
Means… n[1,1]<-30, n[2,1]<-2, n[3,1]<-3
!
n[,1]<-(30,2,3)
â—‹
*means we will do this 29x, n for the next year =
matrix*column of n values in that year
!
% = matrix multiplication
!
for(i in 1:(tmax-1)) n[,i+1] <-A%*%n[,i]
â—‹
N<-numeric(tmax) *too see full population, not just 'n' in
each age group
â—‹
for(i in 1:tmax) N[i] <-sum(n[,i])
â—‹
Over time -> geometric growth model
!
Usually takes 2 generations for population to
set into a rhythm
"
Initial drop -> period of instability where yearlings
are not reproducing
!
plot(t-1,N,type='l',xlab="t",ylab="N")
â—‹
R coding:
!
Ex. Age-structured model of cheetahs
10/17/17
Stability of simple, naturally regulated populations depends on the
magnitude of demographic parameters, whether there is linear or
nonlinear density-dependence, and age structure
!
Fluctuates from 1955 to 1995
â—‹
Threshold effect (steepest part of curve)
!
Offspring/female declines with increasing female
population density
â—‹
Slight drops in food abundance leads to sheep
population crashes (inversely related)
!
Less food availability?
"
Higher energy expenditure (thermoregulation -
metabolism)
"
During a 'crash' year yearling, adults and young lose
weight from August -March
!
Survival decreases as female population density
increases
!
Declining body condition due to food scarcity is main
causal factor
â—‹
Slightly changes shape
!
Alpha = 0.005
"
Bigger = steeper sigmoid effect
(sensitivity to density dependence)
!
*parameters --> minimize square
deviations (most parsimonious)
!
Beta = 15
"
*without pmax, goes from 1 -0
"
Fitting parameters:
!
Gives range from max to zero with population
size
"
= 1/ (1 + (alpha*N)^beta)
"
Logistic regression function:
!
Expected maximum survival rates for each
group (pmax)
"
Give sigmoid shape
!
Age-specific density-dependent survival
parameters (alpha and beta
"
= pmax / (1+ (alphai*N)^betai)
!
Logistic survival function for each age group:
"
All parameters for 3 age groups:
!
Expected maximum reproductive rate for each
age group (mmax)
"
Age specific density-dependent reproductive
parameters (alphaalpha & betabeta)
"
= mmax/ (1+ (alphalpha*N)^betabeta)
!
Logistic reproductive function for each age
group:
"
More parameters:
!
i=0..2
"
j=1..2
"
t=0..50
"
Omega = birth
!
Psi = survival
!
*see equation for matrix on slide
"
Young=f(survival*fecundity) * population size
"
Yearlings = survival rate*young from previous
year
"
From previous years
!
Adults = (survival*yearlings) + (survival*adult
numbers)
"
Initial age distribution: n=(10,4,2)
!
*see slide for equations
â—‹
tmax=50
!
time<-1:tmax
!
Sheep<-numeric(tmax)
!
Pmax<-c(0.88,0.94,0.96)
!
Alpha1<-c(…)
!
Beta1<-(c…)
!
Mmax<-c(…)
!
Alpha2<-c(…)
!
Beta2<-c(…)
!
Psi<-numeric(3)
!
Omega<-numeric(3)
!
n<-matrix(nrow=3,ncol=tmax)
!
n[,1]<-c(40,15,6)
!
Density<-sum(n[,1])
!
Sheep<-density
!
For(t in 2:tmax){
!
For(i in 1:3) psi[i]<-pmax[i]/(1
+(alpha1[i]*density)^beta1[i])
!
For(i in 1:3) omega[i] <-mmax[i]/(1
+(alpha2[i]*density)^beta2[i])
!
n[1,t]<-
n[2,t-1]*psi[2]*omega[2]+n[3,t[1]*psi[3]*omega[3]
!
n[2,t]<-n[1,t-1]*psi[1]
!
n[3,t]<-n[2,t-1]*psi[2]+n[3,t-1]*psi[3]
!
Density<-sum(n[,t])
!
Sheep[t]<-density}
!
Plot(time,sheep,type='l')
!
rm(list=ls())
!
R-coding:
â—‹
Sheep -levels off (oscillates over time) --> density-
dependent
!
Cheetahs -grew exponentially over time (leslie matrix)
â—‹
A good example of a population with extremely non-linear per
capita recruitment is the herd of feral Soay sheep on the Scottish
island of Hirta
!
Such demographic parameters are common, suggesting that
complex dynamics should be the norm (not the exception)
â—‹
Survival rates go down in years with rainier and
colder weather
!
Stochastic weather effects the add to the complex pattern of
population dynamics
â—‹
Short term dynamics depends on age structure, weather and
density
â—‹
Any combination of high instrinsic growth rates, non-linear
density dependence, protracted time lags and delays due to age
structure effects predispose a population to cyclic or even chaotic
fluctuations over time
!
Structure population dynamics of Soay sheep
Age Structured Populations
#$%&'()*+, -./012&, 3+,4567 66849,:;
Document Summary
In order to develop an age-specific model, we need 2 demographic parameters: p - expected survival rates for each age group. = 0. 6790, 0. 6790, 0. 8532 m - expected reproductive rate for each age group. Starting value = 0, 3 rows, 3 columns. Row1: probability they survive and produce an offspring. Row2&3: yearlings and adults that remain in following years. Involves multiplying age-specific population densities by a transition matrix (a) The top row in a reflects the probability of survival from the previous age class multiplied by fecundity at age "x" The subdiagonal reflects the age-specific survival probabilities (0. 679,0. 679) 1st row: 0*10 + 0*10 + 0. 45209*10 = 4. 5209. 2nd row: 0. 679*10 + 0*10 + 0*10 = 6. 79. 3rd row: 0*10 + 0. 679*10 + 0. 853*10 = 15. 32. *taken initial population of 30 individuals (equally divided between age groups) and created new population (with a smaller number of individuals)
