# Exponential Growth

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```					   Overview of                              discrete        continuous
population growth:           density     Geometric        Exponential
independent

density      Discrete           Logistic
New Concepts:              dependent     Logistic

- Stability
- DI (non-regulating)
vs.
DD (regulating) growth

- equilibrium
Variability in growth

(1) Individual variation in births and deaths
(2) Environmental (extrinsic variability)
(3) Intrinsic variability
How do populations grow – a derivation of geometric growth

Growth rate (r) = birth rate – death rate
(express as per individual)

N1 = N0 + rN0

N0 = initial population density (time = 0)
N1 = population density 1 year later (time =1)
How do populations grow?

Growth rate (r) = birth rate – death rate

N1 = N0 + rN0 = N0 (1 + r)
How do populations grow?

Growth rate (r) = birth rate – death rate

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)
How do populations grow?

Growth rate (r) = birth rate – death rate

N1 = N0 + rN0 = N0 (1 + r)

N2 = N1 + rN1 = N1 (1 + r)

Can we rewrite N2 in terms of N0 ???
How do populations grow?

Growth rate (r) = birth rate – death rate

N1 = N0 + rN0 = N0 (1 + r)
substitute
N2 = N1 + rN1 = N1 (1 + r)
How do populations grow?

Growth rate (r) = birth rate – death rate

N1 = N0 + rN0 = N0 (1 + r)
substitute
N2 = N1 + rN1 = N1 (1 + r)

rewrite: N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2
How do populations grow?

Growth rate (r) = birth rate – death rate

N1 = N0 + rN0 = N0 (1 + r)
substitute
N2 = N1 + rN1 = N1 (1 + r)

N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2

or

Nt = N0 (1 + r)t
}

= , finite rate of increase
Discrete (geometric) growth

5

Nt = N0t
N
= finite rate                    4

of increase                3

2
1

time
Continuous (exponential) growth

5

Nt = N0ert
N                4
r = intrinsic
3            growth rate
2
1

time
Continuous (exponential) growth
5

population             per capita
growth rate           growth rate

N                  4            dN = rN;               1 dN = r
3
dt                     N dt
2
1
dN    Read as change in N (density)
time          dt        over change in time.

1 dN = r
1 dN                                      N dt
N dt
Y   = b + mX

N             Per capita growth is constant and
independent of N
Comparison              Discrete                       Continuous
Nt = N0t                      Nt = N0ert
Where:  = er                              r = ln 
Increasing:  > 1                               r>0
Decreasing:  < 1                               r<0

Every time-step                   None
Time lag:     (e.g., generation)       Compounded instantaneously

Populations w/        No breeding season - at any time
Applications:   discrete breeding season   there are individuals in all stages
of reproduction
Most temperate
Examples:       vertebrates and plants
Humans, bacteria, protozoa

Often intractable;
Mathematics:          simulations
Mathematically convenient
Geometric (or close to it)
growth in wildebeest population
of the Serengeti following
Rinderpest inoculation
Exponential growth in the total human population
The Take Home Message:
Simplest expression of population growth:
1 parameter, e.g., r = intrinsic growth rate

Population grows geometrically/exponentially,
but the Per capita growth rate is constant

First Law of Ecology: All populations possess
the capacity to grow exponentially

Exponential/geometric growth is a model
to which we build on
Overview of                              discrete        continuous
population growth:           density        X
Geometric            X
Exponential
independent

density      Discrete           Logistic
New Concepts:              dependent     Logistic

- Stability
- DI (non-regulating)
vs.
DD (regulating) growth

- equilibrium
Variability in growth

(1) Individual variation in births and deaths
(2) Environmental (extrinsic variability)
(3) Intrinsic variability
Variability in space   In time
Variability in space    In time



Source-sink structure
Variability in space     In time



Source-sink structure

 (arithmetic)

Source-sink structure
with the rescue effect
Variability in space             In time

 (geometric)
                        G < A
G declines with
Source-sink structure       increasing variance

 (arithmetic)

Source-sink structure
with the rescue effect
Variability in space                     In time

 (geometric)
                               G < A
G declines with
Source-sink structure              increasing variance

 (arith & geom)
 (arithmetic)                  Increase the number of
subpopulations increases
Source-sink structure           the growth rate (to a point),
and slows the time to
with the rescue effect          extinction

Temporal variability reduces population growth rates

Cure – populations decoupled with respect to variability,
but coupled with respect to sharing individuals

```
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 views: 4 posted: 6/30/2012 language: pages: 25