Fall 2009 IB Workshop Seriessponsored by IB academic advisors

IB Opportunities in C-U

Tuesday, Sept. 154:00-5:00pm 135 Burrill

There are many local opportunities for volunteering/internships! Representatives in fields from medicine to ecology will speak

with students!

Ch 11: Population Growth + Regulation dN/dt = rN dN/dt = rN(K-N)/K

For Thursday lecture,

Bring lecture outline;

Complete Problem Set

2-1 and 2-2 (see pg. 75).

Bring pg. 73 too!

Objectives

• Population Dynamics• Growth in unlimited environment• Geometric growth Nt+1 = Nt

• Exponential growth Nt+1 = Ntert

• dN/dt = rN • Model assumptions• Growth in limiting environment• Logistic growth dN/dt = rN (K - N)/ K• D-D birth and death rates• Model assumptions

1.***Draw a graph of population growth constrasting:

1) Growth with unlimited resources

2) Growth with limited resources

Label axes.

Indicate carrying capacity (K).

3) Add equations representing both types of growth:

A) exponential

B) logistic

Population growth predicted by the exponential (J) vs. logistic (S) model.

Population growth can be mimicked by simple mathematical models of demography.

• Population growth (# ind/unit time) =

recruitment - losses

• Recruitment = Births + Immigrants

• Losses = Deaths + Emigrants

• Growth (g) = (B + I) - (D + E)

• Growth (g) = (B - D) (in practice)

• Geometric growth:• Individuals added at one time of year (seasonal reproduction) • Uses difference equations

• Exponential growth: • individuals added to population continuously (overlapping generations)• Uses differential equations

• Both assume no age-specific birth/death rates

Two models of population growth with unlimited resources :

Difference model for geometric growth with finite amount of time

• ∆N/ ∆t = rate of ∆ = (bN - dN) = gN,• where b = finite rate of birth or per capita birth rate/unit of time• g = b-d, gN = finite rate of growth

Projection model of geometric growth (to predict future (or past) population size)• Nt+1 = Nt + gNt

• =(1 + g)Nt Let (lambda) =(1 + g), then

• Nt+1 = Nt

• = Nt+1/Nt = finite rate of increase, /unit time

Geometric growth over many time intervals:

• N1 = N0

• N2 = N1 = · · N0

• N3 = N2 = · · · N0

• Nt = t N0

• Populations grow by multiplication rather than addition (like compounding interest)

• So if know and N0, can find Nt.

Example of geometric growth (Nt = t N0)

• Let =1.12 (12% per unit time) N0 = 100

• N1 = (1.12) 100 112

• N2 = (1.12 x 1.12) 100 125

• N3 = (1.12 x 1.12 x 1.12) 100 140

• N4 = (1.12 x 1.12 x 1.12 x 1.12) 100 157

2. Problem 2.1 on Pg 75 - due Thursday!

A moth species breeds in late summer and leaves only eggs to survive the winter. The adult dies after laying eggs. One local population of the moth increased from 5000 to 6000 in one year.

1. Does this species have overlapping generations? Explain.

2. What is for this population? Show formula with numbers; don’t solve.

3. Predict the population size after 3 yrs. Show formula with numbers; don’t solve.

4. What is one assumption you make in predicting the future population size?

If know Nt N0 and ,can calculate t (in future and

past)

• Nt = t N0

t = Nt/ N0

= (Nt/ N0)1/t

• t ln = ln (Nt/ N0)

• t = [ln (Nt/ N0)]/ ln

3. Invasive insect:Geometric population growth:Calculate:A. lambdaB. t when n = 1

Geometric growth (dots) andexponential growth (lines) on regular and log scales (when >1 and r >0).

Values of , r, and Ro indicate whether population is: ***

Ro < 1 Ro >1Ro =1

Differential equation model of exponential growth:

dN / dt = r N

rate of contribution numberchange of each of in = individual X

individualspopulation to population in thesize growth

population

4. Calculate daily increase in population.

dN / dt = r N

• r = difference between birth (b) and death (d)• Instantaneous rate of birth and death• r = (b - d) so r is analogous to g, but

instantaneous rates• rates averaged over individuals (i.e. per

capita rates)• r =intrinsic rate of increase

5. Calculate r.

E.g.: exponential population growth

r = .247

Exponential growth: Nt = N0ert

• Continuously accelerating curve of increase

• Slope varies directly with population size.

• (N) (gets steeper as size increases).

r > 0

r < 0

r = 0

6. Calculate A. rB. population size in 6 mo.

Environmental conditions and species influence r, the intrinsic rate of increase.

Population growth rate depends on the value of r; r is environmental- and species-specific.

Value of r is unique to each set of environmental conditions that influenced birth and death rates…

•…but have some general expectations of pattern:

• High rmax for organisms in disturbed habitats

• Low rmax for organisms in more stable habitats

•

Rates of population growth are directly related to body size.

• Population growth:

• increases inversely with mean generation time.

• Mean generation time:

• Increases directly with body size.

Assumptions of the model

• 1. Population changes as proportion of current population size (∆ per capita)

• ∆ x # individuals -->∆ in population;• 2. Constant rate of ∆; constant birth and death rates• 3. No resource limits• 4. All individuals are the same (no age or size structure)

Objectives

• Growth in unlimited environment• Geometric growth Nt+1 = Nt

• Exponential growth Nt+1 = Ntert

• dN/dt = rN • Model assumptions• Growth in limiting environment• Logistic growth dN/dt = rN (K - N)/ K• D-D birth and death rates• Model assumptions

Populations have the potential to increase rapidly…until balanced by extrinsic factors.

Population growth rate =

Intrinsic Population Reduction in

growth X size X growth rate

rate at due to crowding

N close

to 0

Population growth predicted by the *** model.

K = ***

On which island(s) is there evidence oflimited resources affecting population growth ?

Assumptions of the exponential model

• 1. No resource limits• 2. Population changes as proportion of current

population size (∆ per capita)• ∆ x # individuals -->∆ in population;• 3. Constant rate of ∆; constant birth and death

rates• 4. All individuals are the same (no age or size structure)

1,2,3 are violated ***

Population growth rates become lower as population size increases.

• Assumption of constant birth and death rates is violated.• Birth and/or death rates must change as pop. size changes.

Population equilibrium is reached when b = dThose rates can change with density.

Density-dependent factors: + or -? .

Habitat quality affects reproductive variables affected (How is K affected?)

Population size is regulated by density-dependent factors affecting birth and/or death rates.

1) Density-dependence in plants first decreases growth. 2. Size hierarchy develops.

skewed

3) Density-dependence thirdly decreases survival. Intraspecific competition causes “self-thinning”.

r (intrinsic rate of increase) decreases as a function of N .• Population growth is negatively density-dependent.

.

rm

r

r0

N K

slope = rm/K

Positive density-dependenceAllee effect: greater pop size increaseschance of recruitment.

2) (+) density-dependence secondly increases some components of reproduction; decreases others…

• Describes a population that experiences

negative density-dependence.• Population size stabilizes at K = carrying capacity

• dN/dt = rmN(K-N)/K

• where rm = maximum rate of increase w/o

resource limitation

= ‘intrinsic rate of increase’

K = carrying capacity • (K-N)K = environmental break (resistance)

= proportion of unused resources

Logistic equation

Logistic (sigmoid) growth occurs when the population reaches a resource limit.• Inflection point at K/2 separates

accelerating and decelerating phases of population growth; point of most rapid

growth

Logistic curve incorporates influences of decreasing per capita growth rate andincreasing population size.

Specific

Assumptions of logistic model:

• Population growth is proportional to the remaining resources (linear response)

• All individuals can be represented by an average (no change in age structure)

• Continuous resource renewal (constant E)• Instantaneous responses to crowding . ***• K and r are specific to particular organisms

in a particular environment.

7. Complete 2-2 (see pg. 75) due Thursday.