Exponentialpopulationgrowth：人口增长指数

ΔN = dN/dt
• Both ΔN and dN/dt tell us how the population is growing:
dN/dt > 1
population increasing
dN/dt < 1 dN/dt = 1
population decreasing population stable
– Demographic – Environmental
• Population:
– A group of individuals of the same species in the same location that could potentially interact
• Individuals:
• If start with 2 pennies, double every day
– after 1 month 230 = $5.3 million dollars!
Not like population growth!
More like population growth!
•linear growth
– Continuous growth – B and d (therefore r) are constant – No age structure, all females same – Closed population
• Since e is the base of natural logarithm: ln(ex) = x • Take the natural log of both sides of the eqn:
Nt = N0ert ln(Nt) = ln(N0ert)
= ln(N) + ln(ert)
= ln(N) + rt
convenience
– Allows for differential equations (calculus) – Can calculate instantaneous rate of change
• Uses:
– Can describe population growth and predict it – Used in a variety of models (pop growth, species
ln(N)
r = slope
y = b + mx
time
Note: semi-log plot because x-axis is not logged
Geometric pop growth: Continuous model
• What if we also want to be able to predict a future population size based on this growth rate?
Why is population growth geometric?
• Example with cell division:
– Division rate of cells: 2 4 16 etc = 2n – Don’t add fixed amount each time step – Instead, population size is doubled at each time step
– Clearly this is not what see in nature – population growth is not usually unchecked
– Darwin reasoned that whatever factors limit populations also drive natural selection
• r describes how the population is growing:
r>0 Pop ↑
r<0 Pop ↓
r=0 Pop stable
ln(N)
dN/dt > 1 N
time
dN/dt < 1 time
dN/dt = 1 time
Geometric pop growth: Continuous model
Exponential population growth
• Malthus 1798: essay on human population growth
– Food is necessary – Passion between sexes necessary and will remain
unchecked – Population growth is geometric when unchecked
• population growth independent of population size:
dN/dt = B – D
= b(N) – d(N)
= (b - d)N (Substitute r for b - d)
dN/dt = r(N)
(r = per capita growth rate)
Lecture 6: Population growth models
• Background - population models • What is a population? • Exponential growth
– Continuous model – Discrete model
• Stochastic (chance) effects
- Genet – whole genetic organism Example: aspen groves
- Ramet – individual units/organisms Example: individual aspen tree
Why use population models?
• To address these types of questions:
• Tool Kit:
N = population size t = time Nt = population size at time t N0 = population size at start (t=0)
• Example with spiders:
N0 = 500, N1 = 800, N2 = 700
1. Unitary – form is determinant; no budding or cloning; free-living individuals (discrete organisms); each has unique genotype
2. Modular – branched growth of repeating units; can break up into separate clonal units; all units have same genotype
• Continuous means a smooth change in numbers • Births and deaths occurring constantly (e.g. no
seasons) • Not very realistic, but useful for mathematical
NOT compounded instantaneously = 100 ( 2) = $200
compounded instantaneously = 100 ( ert )= 100 (e1(1) ) = $272
Geometric pop growth: Continuous model
• Review - Assumptions:
Geometric pop growth: Continuous model
• If we assume a closed population (no I or E):
ΔN = B – D
• For each time step the change in population size (ΔN) is the same as the population growth rate (i.e. the slope):
- births (B) & deaths (D) - immigration (I) & emigration (E)
• How much does the population size change from one time step to the next?
2
Geometric pop growth: Continuous model
1. How do populations grow? 2. What limits population growth? 3. What factors affect population dynamics? 4. What are the population consequences of
species interactions? 5. How do we manage ecological resources?
•Exponential/
geometric growth
N
•slope constant N
•slope increasing
•fixed # added
each time
•increasing #
added/time
t
t
Modeling geometric population growth
• What factors determine what the population size will be from one time step to the next? - Initial population size (Nt)
dN/dt = rN
• Why use e?
Nt = N0ert
– e is the base of natural logarithm: ln(ex) = x
– e is used to calculate compound interest
– Example: $100 in bank for 1 year with 100% interest
Change in population size for any given time step: slope = dN/dt Population growth rate per individual: r = (dN/dt)/N
• Examples of per capita growth rates (r) in nature:
r
Doubling Time
Virus
110,000 3.3 minutes
Bacteria
21,000 17 minu
Cow
0.365
1.9 years
Humans
0.013
50 years
3
Geometric pop growth: Continuous model
interactions)
Geometric pop growth: Continuous model
• B and D are dependent upon population size: B = bN b is the per female (per capita) birth rate D = dN d is the per capita death rate**
Modeling geometric population growth
• We are interested in the relationship between population size and time – how does the size of the population vary with time?
N t
N = Number of individuals in population
t = time step
1
Exponential population growth
• Darwin’s example with elephants:
– Start with 1 pair of elephants – Elephants breed between 30-90 years of age – Typically have 6 offspring – After 750 years – 19 million elephants!