Ch 14: Population Growth + Regulation dN/dt = rN dN/dt = rN(K-N)/K

NEXT WEEK:Computer sessions all on MONDAY:R AM 7-9 R PM 4-6 F AM 7-9 Lab: last 1/2 of manuscript due Lab VII Life Table for Human Pop Bring calculator! Will complete Homework 8 in lab

Ch 14: Population Growth + Regulation dN/dt = rN dN/dt = rN(K-N)/K

Sample Exam ?

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

1. Does this species have overlapping generations? Explain.

2. What is for this population? Show calculations.

3. Predict the population size after 3 yrs. Show calculations.

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

Objectives

• Age structure• Life table + Population growth • Growth in unlimited environments• Geometric growth Nt+1 = Nt

• Exponential growth Nt+1 = Ntert

• Model assumptions

Exponential growth of the human population

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

• Population growth (# ind/unit time) =

recruitment - losses• Recruitment = births and immigration• Losses = death and emigration• Growth (g) = (B + I) - (D + E)• Growth (g) = (B - D) (in practice)

How fast a population grows depends on its age structure.

• When birth and death rates vary by age, must know age structure

• = proportion of individuals in each age class

Age structure varies greatly among populations with large implications for population growth.

Population Growth: (age structure known) • How fast is a population growing?

• per generation = Ro

• instantaneous rate = r

• per unit time = • What is doubling time?

•

Life Table: A Demographic Summary Summary of vital statistics (births + deaths) by age class; Used to determine population growth

See printout for Life Table for example…

Values of , r, and Ro indicate whether population is decreasing, stable, or increasing

Ro < 1 Ro >1Ro =1

Life Expectancy: How many more years can an individual of a given age expect to live?

How does death rate change through time?

Both are also derived from life table…

Use Printout for Life Table for example…

Survivorship curves: note x scale…

+plants

death rate constant

Sample Exam ?

In the population of mice we studied, 50% of each age class of females survive to the following breeding season, at which time they give birth to an average of three female offspring. This pattern continues to the end of their third breeding season, when the survivors all die of old age.

1. Fill in this cohort life table.

2. Is the population increasing or decreasing?

Show formula used.

3. How many female offspring does a female mouse have in her lifetime?

4. At what precise age does a mouse have her first child? Show formula used.

5. Draw a graph showing the surivorship curve for this mouse population. Label axes carefully.

Explain how you reached your answer.

x nx lx mx

lxmx xlxmx

0-1

Etc…

1000 1.0 0

0

Cohort life table: follows fate of individuals born at same time and followed throughout their lives.

mx

Survival data for a cohort (all born at same time) depends strongly on environment + population density.

What are advantages and disadvantages of a cohort life table?

Advantages: • Describes dynamics of an identified cohort• An accurate representation of that cohort’

behaviorDisadvantages:• Every individual in cohort must be identified and

followed through entire life span - can only do for sessile species with short life spans

• Information from a given cohort can’t be extrapolated to the population as a whole or to other cohorts living at different times or under different conditions

Static life table: based on individuals of known age censused at a single time.

Static life table: avoids problem of variation in environment; can be constructed in one day (or season)

n = 608

E.g.: exponential population growth

= 1.04

• 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 bN = 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 population size)• Nt+1 = Nt + gNt

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

• Nt+1 = Nt

• = Nt+1 /Nt

• Proportional ∆, as opposed to finite ∆, as above• Proportional rate of ∆ / time• = finite rate of increase, proportional/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 x 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

Geometric growth:

N

N0

> 1 and g > 0

= 1 and g = 0 < 1 and g < 0

time

Differential equation model of exponential growth:dN/dt = rN

rate of contribution numberchange of each of in = individual X

individualspopulation to population in thesize growth

population

dN / dt = r N

• Instantaneous rate of birth and death• r = difference between birth (b) and

death (d)• 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

Exponential growth: Nt = N0 ert

• Continuously accelerating curve of increase

• Slope varies directly with population size (N)

r > 0

r < 0

r = 0

Exponential and geometric growth are related:

• Nt = N0 ert

• Nt / N0 = ert

• If t = 1, then ert = • N1 / N0 = = er

ln = r

The two models describe the same data equally well.

TIME

Exponential

Environmental conditions 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 habitatsLow rmax for organisms in more stable habitats

Rates of population growth are directly related to body size.

• Population growth:

• increases directly with the natural log of net reproductive rate (lnRo)

• increases inversely with mean generation time

• Mean generation time:

• Increases directly with body size

Rates of population growth and rmax are directly related to body size.Body Size Ro T r

small 2 0.1 6.93

medium 2 1.0 0.69

large 2 10 0.0693

0.1 1 10

if Ro=2 6.9

.69

.069

r

T

Generation time decreasesw/ increase in r;T increases w/ decrease in r

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)

Sample Exam ?

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

1. Does this species have overlapping generations? Explain.

2. What is for this population? Show calculations.

3. Predict the population size after 3 yrs. Show calculations.

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

Sample Exam ?

In the population of mice we studied, 50% of each age class of females survive to the following breeding season, at which time they give birth to an average of three female offspring. This pattern continues to the end of their third breeding season, when the survivors all die of old age.

1. Fill in this cohort life table.

2. Is the population increasing or decreasing?

Show formula used.

3. How many female offspring does a female mouse have in her lifetime?

4. At what precise age does a mouse have her first child? Show formula used.

5. Draw a graph showing the surivorship curve for this mouse population. Label axes carefully.

Explain how you reached your answer.

x nx lx mx

lxmx xlxmx

0-1

Etc…

1000 1.0 0

0

Objectives

• Age structure• Life table + Population growth • Growth in unlimited environments• Geometric growth Nt+1 = Nt

• Exponential growth Nt+1 = Ntert

• Model assumptions

Vocabulary

Chapter 14 Population Growth and Regulation demography exponential growth* geometric growth per capita age structures* stable age distribution life tables fecundity survival survivorship cohort life table static life table* intrinsic rate of increase* net reproductive rate generation time doubling time carrying capacity (K) logistic equation* inflection point density-dependent factors density-independent factors self-thinning curve -3/2 power law r max* arithmetic* geometric* survivorship curves* doubling time model assumptions time lag size hierarchy Leslie matrix projection matrix transition probabilities life cycle figure life expectancy little r lambda (