lect_6_exponential growth

```Overview of
population growth:
New Concepts:
discrete
continuous
density
independent
Geometric
Exponential
density
dependent
Discrete
Logistic
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
= finite rate
of increase
N
4
3
1
2
time
Continuous (exponential) growth
5
Nt = N0ert
N
4
3
1
2
time
r = intrinsic
growth rate
Continuous (exponential) growth
5
population
growth rate
N
4
3
1
per capita
growth rate
dN = rN;
dt
1 dN = r
N dt
2
time
dN
dt
Read as change in N (density)
over change in time.
1 dN = r
N dt
1 dN
N dt
Y
N
= b + mX
Per capita growth is constant and
independent of N
Comparison
Discrete
Continuous
Nt = N0t
Nt = N0ert
Where:  = er
Increasing:  &gt; 1
Decreasing:  &lt; 1
r = ln 
r&gt;0
r&lt;0
Every time-step
(e.g., generation)
None
Compounded instantaneously
Applications:
Populations w/
discrete breeding season
No breeding season - at any time
there are individuals in all stages
of reproduction
Examples:
Most temperate
vertebrates and plants
Humans, bacteria, protozoa
Often intractable;
simulations
Mathematically convenient
Time lag:
Mathematics:
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
population growth:
New Concepts:
discrete
continuous
density
independent
Geometric
X
Exponential
X
density
dependent
Discrete
Logistic
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

Source-sink structure
In time
Variability in space

Source-sink structure
 (arithmetic)
Source-sink structure
with the rescue effect
In time
Variability in space

Source-sink structure
 (arithmetic)
Source-sink structure
with the rescue effect
In time
 (geometric)
G &lt; A
G declines with
increasing variance
Variability in space

Source-sink structure
 (arithmetic)
Source-sink structure
with the rescue effect
In time
 (geometric)
G &lt; A
G declines with
increasing variance
 (arith &amp; geom)
Increase the number of
subpopulations increases
the growth rate (to a point),
and slows the time to
extinction
Temporal variability reduces population growth rates
Cure – populations decoupled with respect to variability,
but coupled with respect to sharing individuals
```