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
○
= 0.0000, 0.0000, 0.5303
!
m -expected reproductive rate for each age group
○
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)
○
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]
○
*see R coding on slide
!
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)
○
Leslie Matrix Model:
!
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)
1st row: 0*10 + 0*10 + 0.45209*10 = 4.5209
!
Nt=30
○
New variable: sum of all the different age groups in the population
(N) = 26.4
!
tmax<-30 (population size=30)
○
n<-matrix(0,nr=3,ncol=30)
○
Means… n[1,1]<-30, n[2,1]<-2, n[3,1]<-3
!
n[,1]<-(30,2,3)
○
*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]
○
N<-numeric(tmax) *too see full population, not just 'n' in
each age group
○
for(i in 1:tmax) N[i] <-sum(n[,i])
○
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")
○
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
○
Threshold effect (steepest part of curve)
!
Offspring/female declines with increasing female
population density
○
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
○
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
○
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:
○
Sheep -levels off (oscillates over time) --> density-
dependent
!
Cheetahs -grew exponentially over time (leslie matrix)
○
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)
○
Survival rates go down in years with rainier and
colder weather
!
Stochastic weather effects the add to the complex pattern of
population dynamics
○
Short term dynamics depends on age structure, weather and
density
○
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
○
= 0.0000, 0.0000, 0.5303
!
m -expected reproductive rate for each age group
○
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)
○
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]
○
*see R coding on slide
!
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)
○
Leslie Matrix Model:
!
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)
1st row: 0*10 + 0*10 + 0.45209*10 = 4.5209
!
Nt=30
○
New variable: sum of all the different age groups in the population
(N) = 26.4
!
tmax<-30 (population size=30)
○
n<-matrix(0,nr=3,ncol=30)
○
Means… n[1,1]<-30, n[2,1]<-2, n[3,1]<-3
!
n[,1]<-(30,2,3)
○
*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]
○
N<-numeric(tmax) *too see full population, not just 'n' in
each age group
○
for(i in 1:tmax) N[i] <-sum(n[,i])
○
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")
○
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
○
Threshold effect (steepest part of curve)
!
Offspring/female declines with increasing female
population density
○
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
○
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
○
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:
○
Sheep -levels off (oscillates over time) --> density-
dependent
!
Cheetahs -grew exponentially over time (leslie matrix)
○
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)
○
Survival rates go down in years with rainier and
colder weather
!
Stochastic weather effects the add to the complex pattern of
population dynamics
○
Short term dynamics depends on age structure, weather and
density
○
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)
