SLIDE 1
The muskox
Original distribution: North America, Greenland Depleted by hunting from 1700 to 1850 Last individuals in Alaska: 1850-60
Population growth in ideal habitats How does a population grow when - - PowerPoint PPT Presentation
Population growth in ideal habitats How does a population grow when - - PowerPoint PPT Presentation
Population growth in ideal habitats How does a population grow when colonizing an habitat under ideal physical and biological conditions ? The muskox Original distribution: North America, Greenland Depleted by hunting from 1700 to 1850
SLIDE 2
SLIDE 3
Nunivak Island
Nunivak Island
31 animals, 1936
SLIDE 4
Geometric growth
Boi almiscarado na Ilha de Nunivak (Alaska)
200 400 600 800 1930 1940 1950 1960 1970 1980
Anos
Núm indivíduos
Muskox (Ovibos moschatus)
Initial population at the Nunivak reserve: 31 individuals
SLIDE 5
Measures of variation in N
t t+ 1
∆N = Nt+ 1-Nt
Absolute variation Nt Nt+ 1
∆t ∆N > 0 growth ∆N = 0 no change ∆N < 0 decline
t N t N N
t t
∆ ∆ = ∆ −
+1
Mean variation over ∆t variation time-1
≡
t N Ni ∆ ∆ 1
Mean relative variation % variation
≡
SLIDE 6
Finite rate of increase
λ =
+ t t
N N
1
= finite rate of increase
λ
What happens if λ remains constant ?
t n n t t t t t t t t t
N N N N N N N N N N λ λ λ λ λ λ λ = = = = = =
+ + + + + +
...
3 2 3 2 2 1 2
Nt+ n = Nt
n
λ
SLIDE 7
Geometric growth
50 100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
n
N t+n
Do you recognize the muskox here ? = 1 > 1 < 1
t n n t
N N λ =
+
λ λ λ
SLIDE 8
t n n t
N N λ =
+
50 100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
n
N t+n
May λ remain constant ?
t t
N N
1 +
= λ
Contribution of each individual in t, for the population in t+ 1
SLIDE 9
Biological meaning of λ ?
What is the biological meaning of λ ? Are newborns envolved ? deaths ? both ?
SLIDE 10
Why does Nt change ?
Nt
Natural mortality
Mortality by human activities I mmigrants Emigrants
Births Closed population
Nt+ 1= Nt-Dt+ Bt
Open population
SLIDE 11
Reproduction timing
11000000 1
Tempo
Núm nascimentos
- 20
Tempo
Seasonal breeding Continuos breeding
SLIDE 12
Census and reproduction in seasonal breeders
Bt= newborns
Nt Nt+1
∆t
Dt= deaths
Pre-breeding census
SLIDE 13
Survival rate
Nt Nt+1
Dt Bt
t t t t t t t t t
B N N B N D B N S + = + − + =
+1
SLIDE 14
Birth rate
bt= Number of newborns Number of parents
Nt Nt+1
Dt
Bt
t t t
N B b =
SLIDE 15
Biological meaning of λ
t t t
N N
1 +
= λ
remember Substituting Nt+ 1 Using:
( )
t t t t t t t t
B N S N B N N S + = ∴ + =
+ + 1 1
We get:
( )
t t
b S + = 1 λ
Survival rate Birth rate
SLIDE 16
Newborns in Portugal, 1994, INE 1995
2000 4000 6000 8000 10000
Jan Fev Mar Abr Mai Jun Jul Ago Set Out Nov Dez
meses
Núm de nados-vivos
Femininos Masculinos
SLIDE 17
Continuous reproduction
N changes continuously ! Any time interval ∆ t= [t, t+ 1] will be arbitrary
t N t N N
t t
∆ ∆ = ∆ −
+1
Remember mean variation :
Nt Nt+ 1 t t+ 1
dt dN t N
Lim
t
= ∆ ∆
→ ∆
= Instantaneous variation at t
SLIDE 18
Instantaneous variation
Instantaneous variation at time t:
t t
D B dt dN − =
Instantaneous rates,
t t t
b N B parents newborns = =
Birth rate = Mortality rate =
t t t
d N D population deaths = =
SLIDE 19
Instantaneous rate of growth
( )
t t t t t t t
d b N d N b N dt dN − = − =
r
Instantaneous rate of growth
(Malthusian parameter)
rN dt dN =
r units: Individuals per individual per unit time
Given an initial Nt what is Nt+ ∆t ?
SLIDE 20
Solution
rN dt dN =
Ordinary differential equation of 1st degree Assuming r is constant,
Solution, by separable variables:
t r t t t
e N N
∆ ∆ +
=
For any ∆ t Dependent variable Parameter Independent variable
SLIDE 21
Exponential growth
50 100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
tempo
N
r > 0 r < 0 r = 0
t r t
e N N =
SLIDE 22
Unregulated growth
Discrete time:
λ
t t
N N =
+1 Continuous: t r t t t
e N N
∆ ∆ +
=
If λ applies to the time interval ∆ t= 1,
Relationship between instantaneous rate of growth and finite rate of increase
λ =
r
e
SLIDE 23
Unregulated growth cannot last long
600 1200 1800 2400 3000
1 3 5 7 9 11 13 15 17 19 21 Tempo Número indivíduos (Nt) r =1 r =0.5 r =0.25
r= 1 year -1 Nt = 10 initial individuals = 10 years
Nt+ 10 = 10 e 1 x 10 = 220 265 individuals
t r t t t
e N N
∆ ∆ +
=
t ∆
SLIDE 24
Survival and reproduction depend upon Nt
t d b t t r t t t
e N e N N
∆ − ∆ ∆ +
= =
) (
Survival, birth rate = f (Nt)
SLIDE 25
What good is the unregulated growth model if it does not apply to most populations ?
1. Illustrates the consequence of assuming constant survival and birth rates 2. The model describes the initial stages of population growth, showing the enormous potential of populations to grow 3. It is a good starting point for the introduction of other components that confer greater realism to population growth
SLIDE 26
Human population 1
SLIDE 27
Human population 2
Source: Demographic yearbook. Annuaire démographique. New York Dept. of Economic and Social Affairs, Statistical Office, United Nations
SLIDE 28