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Population dynamics

• 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).

• nx = number alive at age x

• lx = proportion of individuals alive at age x

Age (x) nx lx

0 200 1.00

1 180 0.90

2 175 0.88

3 120 0.60

4 50 0.25

5 3 0.02

6 0 0.00

Life History Tables

• dx = proportion of original population dying during the age interval x to x+1

• qx = proportion of existing population dying during age interval x to x+1; qx = dx/lx

Age (x) nx lx dx qx

0 200 1.000 0.100 0.100

1 180 0.900 0.025 0.028

2 175 0.875 0.275 0.314

3 120 0.600 0.350 0.583

4 50 0.250 0.235 0.940

5 3 0.015

6 0 0.000

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Life expectancy

• 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) nx lx dx qx Lx Tx ex

0 200 1.000 0.100 0.100 0.950 3.130 3.130

1 180 0.900 0.025 0.028 0.888 2.180 2.422

2 175 0.875 0.275 0.314 0.738 1.293 1.477

3 120 0.600 0.350 0.583 0.425 0.555 0.925

4 50 0.250 0.240 0.960 0.130 0.130 0.520

5 2 0.010

6 0 0.000

Birth Rates and population growth

• 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)

Reproductive Rate

• 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) nx lx dx qx Lx Tx ex mx fx lxmx

0 200 1.000 0.100 0.100 0.950 3.130 3.130

1 180 0.900 0.025 0.028 0.888 2.180 2.422 2 360.00 1.80

2 175 0.875 0.275 0.314 0.738 1.293 1.477 3 525.00 2.63

3 120 0.600 0.350 0.583 0.425 0.555 0.925 4 480.00 2.40

4 50 0.250 0.240 0.960 0.130 0.130 0.520 5 250.00 1.25

5 2 0.010

6 0 0.000

R0 = 8.05

Future population size

• 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

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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 (Lotka-Volterra model):

N N et

rt=

0

dN

dtrN= or

Sample calculations

• Nt = (No * Ro)

– N1 = ?

– N0 = 100

– R = 0.75

– N1 = 75

• Assume T=1, then r = ln 0.75 / 1.0

– r = -0.288

– N4 = 100 e (-0.288*4)

– N4 = 31.6

– N16 = 100 e (-0.288*16)

– N16 = 0.99

N N et

rt=

0

0

2000

4000

6000

8000

10000

nu

mb

er

of in

div

idu

als

0 2 4 6 8 10 12 14 16 time

r = .1 r = .2 r = .3

Exponential Growth Human Population Growth

Given current growth rates, what will the world population be in 30 years??

Nt=N0ert

Nt=6,426,101,450 e0.0125(30)

9,349,922,439

year r doubling time

1970 0.02 35

1991 0.018 39

2000 0.0125 55

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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 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 dependent effects

Two natural populations showing exponential growth until K is approached.

Density independent effects

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Density dependent and independent factors

• A natural population showing density dependent effects.

Incorporating Density dependent factors – Lotka-Volterra Model

• As you approach K, resources more limited, birth rates decrease, death rates increase.

dN

dtrN=

dN

dtrN

K

N= −

1

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

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• 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

Lotka-Volterra Models of Interspecific Competition

• Models change in population size of species 1, accounting for impact of species 2.

• Similarly, affect of species 1 on species 2:

dN

dtr N

K N a N

K

1

1 1

1 1 12 2

1

=− −

dN

dtr N

K N a N

K

2

2 2

2 2 21 1

2

=− −

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

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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

Core area near optima

Ecological Gradient

Area with tolerable conditions

Core habitat near optima

Second gradient

Likely species distribution

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Niche – combination of multiple optima along many gradients

• The role an organism plays in the environment

– All resources, interactions with biotic/abioticcomponents of the environment

– N-dimensional hypervolume• 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

Popu

latio

n S

ize

Niche width and overlap along an ecological gradient

• 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)

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

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Evolutionary trade offs – specialist vs. generalist

• Specialist (+/-)

• Generalist (+/-)

Competition in the intertidal zone

What are some of the relevant ecological gradients in intertidal zones?

What resources might be limiting?

Niche Shift through Character Displacement

• Character displacement –selection for morphological change to relieve competitive pressure.

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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.

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

• Fundamentally, just another form of competition

• Involves energy transfer through consumption

– Carnivory

– Herbivory

– Parasitism

Primary Production

Primary Consumer

Secondary Consumer

Tertiary Consumer

Predation and Natural Selection

• 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.

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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