1. Overview of population growth:
discrete continuous
density
independent
Geometric Exponential
Discrete
Logistic
density
dependent
Logistic
New Concepts:
Stability
DI (non-regulating)
vs.
DD (regulating) growth
equilibrium
Variability in growth
Individual variation in births and deaths
Environmental (extrinsic variability)
Intrinsic variability

2. 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)

3. How do populations grow?
Growth rate (r) = birth rate – death rate
N1 = N0 + rN0 = N0 (1 + r)

4. How do populations grow?
Growth rate (r) = birth rate – death rate
N1 = N0 + rN0 = N0 (1 + r)
N2 = N1 + rN1 = N1 (1 + r)

5. 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 ???

6. How do populations grow?
Growth rate (r) = birth rate – death rate
N1 = N0 + rN0 = N0 (1 + r)
substitute
N2 = N1 + rN1 = N1 (1 + r)

7. 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

8. } 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

9. Discrete (geometric) growth5
Nt = N0t N
= finite rate
of increase 4 3 2 1
time

10. Continuous (exponential) growth5
Nt = N0ert N
r = intrinsic
growth rate 4 3 2 1
time

11. Continuous (exponential) growth5
population
growth rate
per capita
growth rate dN dt
1 dN
N dt N
= r
= rN; 4 3 2 1 dN dt
Read as change in N (density)
over change in time.
time
1 dN
N dt
= r
1 dN
N dt
Y = b + mX
Per capita growth is constant and
independent of N N

12. Comparison Discrete Continuous Nt = N0t Nt = N0ert
Where:  = er r = ln 
Increasing:
Decreasing:
> r > 0
 < r < 0
None
Compounded instantaneously
Every time-step
(e.g., generation)
Time lag:
No breeding season - at any time
there are individuals in all stages
of reproduction
Populations w/
discrete breeding season
Applications:
Most temperate
vertebrates and plants
Examples:
Humans, bacteria, protozoa
Often intractable;
simulations
Mathematics:
Mathematically convenient

13. Geometric (or close to it)
growth in wildebeest population
of the Serengeti following
Rinderpest inoculation

14. Exponential growth in the total human population

15. Exponential/geometric growth is a model
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

16. Overview of population growth:
discrete continuous
density
independent
Geometric Exponential
Discrete
Logistic X X
density
dependent
Logistic
New Concepts:
Stability
DI (non-regulating)
vs.
DD (regulating) growth
equilibrium
Variability in growth
Individual variation in births and deaths
Environmental (extrinsic variability)
Intrinsic variability

17. Variability in space
In time
No migration
migration

19. Variability in space
In time
Source-sink structure
No migration
migration

21. Variability in space
In time
Source-sink structure
No migration
(arithmetic)
Source-sink structure
with the rescue effect
migration

23. Variability in space
In time
(geometric)
G < A
G declines with
increasing variance
Source-sink structure
No migration
(arithmetic)
Source-sink structure
with the rescue effect
migration

25. Source-sink structure No migration
Variability in space
In time
(geometric)
G < A
G declines with
increasing variance
Source-sink structure
No migration
(arith & geom)
Increase the number of
subpopulations increases
the growth rate (to a point),
and slows the time to
extinction
(arithmetic)
Source-sink structure
with the rescue effect
migration
Temporal variability reduces population growth rates
Cure – populations decoupled with respect to variability,
but coupled with respect to sharing individuals