• However, birth rates, mortality rates, immigration and emmigration

Population dynamics
Life History Tables
• Population size through time should be predictable
• Nt+1 = Nt + B + I - D - E
• Time 1
– N = 100
– 20 births
– 25 deaths
– 10 immigrants
– 15 emmigrants
• Time 2 – 100 + 20 +10 – 25 – 15 = 90
Life History Tables
• However, birth rates, mortality rates, immigration
and emmigration are variable by life stages
• Need to incorporate changing values to account
for and predict age structure
For simplicity, assume I=E
Life History Tables
• Time (x) = time interval used for separating age
categories. For simplicity assume t=1 (discrete
generations).
• dx = proportion of original population dying during
the age interval x to x+1
• nx = number alive at age x
• qx = proportion of existing population dying during
age interval x to x+1; qx = dx/lx
• lx = proportion of individuals alive at age x
Age (x)
0
1
2
3
4
5
6
nx
lx
200
1.00
180
0.90
175
0.88
120
0.60
50
0.25
3
0.02
0
0.00
Age (x)
0
1
2
3
4
5
6
nx
lx
dx
qx
200
1.000
0.100
0.100
180
0.900
0.025
0.028
175
0.875
0.275
0.314
120
0.600
0.350
0.583
50
0.250
0.235
0.940
3
0.015
0
0.000
1
Life expectancy
Birth Rates and population growth
• ex = Tx / lx
• Tx = average life expectancy from current time:
e.g. how much living will be done by cohort from
beginning of period x:
• Tx=Σ(Lx); summed from x to last x
• Lx=(lx+lx+1)/2
Age (x)
0
1
2
3
4
5
6
Lx
Tx
• fx = total natality; number of fertilized eggs
produced in a given year by all individuals of age
x
• mx = average natality of individuals of age x
(fx/nx)
nx
lx
dx
qx
200
1.000
0.100
0.100
0.950
3.130
ex
3.130
180
0.900
0.025
0.028
0.888
2.180
2.422
175
0.875
0.275
0.314
0.738
1.293
1.477
120
0.600
0.350
0.583
0.425
0.555
0.925
50
0.250
0.240
0.960
0.130
0.130
0.520
2
0.010
0
0.000
Reproductive Rate
Future population size
• R0 = rate of change in the population. If below
1.0, population is shrinking
• R0=∑
∑(lxmx)
• Sum of the number of fertilized eggs produced
per original individual during each age
Age (x)
0
1
2
3
4
5
6
Lx
Tx
ex
•
•
•
•
nx
lx
dx
qx
mx
fx
lxmx
200
1.000
0.100
0.100
0.950
3.130
3.130
180
0.900
0.025
0.028
0.888
2.180
2.422
2
360.00
1.80
175
0.875
0.275
0.314
0.738
1.293
1.477
3
525.00
2.63
120
0.600
0.350
0.583
0.425
0.555
0.925
4
480.00
2.40
50
0.250
0.240
0.960
0.130
0.130
0.520
5
250.00
1.25
2
0.010
0
0.000
Nt = (No * Ro) + I - E
Ro incorporates age-specific births and deaths
Usually assume I = E for simplicity
Nt = (No * Ro)
– Nt = 100
– R = 0.75
– N1 = 75
R0 = 8.05
2
Sample calculations
r and Ro
Ro = net reproductive rate; for discrete generations (x=1) a
multiplier allowing us to determine population size at future
generation
r (Malthusian Parameter) = intrinsic rate of increase; also
per capita rate of increase. When r is >0.0 population will
increase, when it is <0.0 population will decrease.
• r= ln Ro/T
– Where T = generation time, time units between
generations. For simplicity we assume this is 1.0
• Intrinsic rate of population growth is defined as (LotkaVolterra model):
dN
= rN
dt
N t = N 0 e rt
or
number of individuals
Exponential Growth
• Nt = (No * Ro)
– N1 = ?
– N0 = 100
– R = 0.75
– N1 = 75
• Assume T=1, then r = ln 0.75 / 1.0
– r = -0.288
rt
t
0
– N4 = 100 e (-0.288*4)
– N4 = 31.6
N = N e
– N16 = 100 e (-0.288*16)
– N16 = 0.99
Human Population Growth
10000
8000
6000
4000
2000
year
1970
1991
2000
0
0
2
4
6
8 10 12 14 16
time
r
0.02
0.018
0.0125
doubling time
35
39
55
Given current growth rates, what will the world population be in 30 years??
Nt=N0ert
r = .1
r = .2
r = .3
Nt=6,426,101,450 e0.0125(30)
9,349,922,439
3
Why don’t we observe continuous exponential growth?
• Competition for limited resources
• Carrying capacity – the number of individuals
of a species that can be supported by available
resources in a habitat
Density dependent effects
Density dependent vs. density independent
• Both negatively impact populations growth/size
• If the impact worsens with greater density it’s
density dependent
– Disease
– Competition
– Famine
• If the impact does not vary with density it’s
density independent
– Disturbance – fire, flood, etc.
Density independent effects
Two natural populations showing exponential growth until K is approached.
4
Density dependent and independent factors
• A natural population showing density dependent
effects.
Incorporating Density dependent factors – Lotka-Volterra Model
dN
= rN
dt
• As you approach K, resources more limited, birth
rates decrease, death rates increase.
dN
K

= rN  1 − 

dt
N
Intra vs. interspecific competition
• As N approaches K resources are more limiting, this is
intraspecific competition
• Interspecific competition = competition among two
species using the same resources
• Ecological equivalents:
– α12 - Number of individuals of species 2 that are
equivalent to one individual of species 1.
– α21 - Number of individuals of species 1 that are
equivalent to one individual of species 2.
Types of Competition
• Types of resources –
• Exploitative – Use a resource more efficiently
before a competitor has a chance
• Interference – physically prevent a competitor
from having access to a resource
• Asymmetric – effect of species 1 on species 2
not the same as species 2 on species 1
• Symmetric – effects of species similar
5
Lotka-Volterra Models of Interspecific Competition
• Asymmetric competition - α12 not equal to α21
• symmetric competition - α12 roughly equal to α21
• Use α12 to calculate affect of one species on
another.
– K1 =1000
– N1 = 600
– N2 = 300
– α12 = 0.8; 0.8 * 300 = 240
– N1 = 600 + 240 equivalent competitors = 840
 K − N 1 − a12 N 2 
dN 1

= r1 N 1  1
dt
K1


• Models change in population size of species 1,
accounting for impact of species 2.
• Similarly, affect of species 1 on species 2:
 K 2 − N 2 − a 21 N 1 
dN 2

= r2 N 2 
dt
K2


Species abundance isoclines
N1/K1=1 – stable, all resources used by species 1
K1/α12 =1 - stable, all resources used by species 2 (equivalent population)
Combine the
isoclines for both
species to produce
a graphical model
of competitive
interactions.
Possible outcomes:
-Stable coexistence
-Dominance by one
species
6
Competition and ecological gradients
• Models are oversimplifications, assume
resources stable and consistent throughout
• Species are distributed across multiple
gradients, should be most competitive (K
maximized) near optima.
Area with tolerable conditions
Ecological Gradient
Core area near optima
Area with tolerable conditions
Core habitat near optima
Likely species distribution
Second gradient
7
Niche – combination of multiple optima along many gradients
• The role an organism plays in the environment
– All resources, interactions with biotic/abiotic
components of the environment
– N-dimensional hypervolume
Population Size
• Each dimension is a biotic or abiotic resource
Niche Width
• Niche Width – range of gradient(s) over which species
occurs and is abundant.
• Generalist – jack of all trades, wider range of optima,
wider niche
• Specialist – narrower range of optima, expect narrow
niche
Gradient
Niche width and overlap along an ecological gradient
Niche space and competition
• Selection favors individuals who get the most
resources
• Individuals that avoid competition will get more
resources
• Competitive pressure leads to
– Niche shift
– Specialization
• Parameters d and w describe niche width and the
amount of overlap among species.
• Non-competing specialists – small w and large d (little or
no overlap)
• Competing generalists – large w and small d (large
overlap)
8
Evolutionary trade offs – specialist vs. generalist
• Specialist (+/-)
Competition in the intertidal zone
What are some of the relevant ecological gradients in intertidal zones?
What resources might be limiting?
• Generalist (+/-)
Niche Shift through Character Displacement
• Character
displacement –
selection for
morphological change
to relieve competitive
pressure.
9
Fundamental vs. Realized Niche
• Fundamental niche – total potential niche
space for a species
• Realized niche – actual niche space used, a
subset of the fundamental niche.
Predation
Convergent Evolution
• Similar niche properties
exert similar selective
pressure, resulting in
similar species.
• Species no the “same”
due to historical factors,
continental isolation in
this case.
Predation and Natural Selection
• Fundamentally, just another form of competition
• Involves energy transfer through consumption
– Carnivory
– Herbivory
– Parasitism
Tertiary Consumer
• Predator – selection for ability to obtain the
most energetically beneficial food at the least
expense.
– Select the most abundant, easiest to catch
(old, young, sick, weak)
• Prey – selection to avoid being eaten, or to
become a less desirable meal.
Secondary Consumer
Primary Consumer
Primary Production
10
Optimal Foraging Theory
• Predators should optimize energetic gains by balancing
the costs/benefits of capturing prey.
• Costs
– Search time
– Handling time
– Digestion
• Benefits
– Calories assimilated
11