Geometric and Exponential Growth


Applied Population Dynamics
WILD 5700/7700

Today’s learning objectives

The equations for geometric and exponential growth.


The relationship between geometric growth and the BIDE model.


The difference between continuous and discrete time models of population growth.


The definition of density independent population growth.

What is Population Dynamics?

The study of spatial and temporal variation in population size and structure.

Fundamental Question

How does abundance go from \(N_t\) to \(N_{t+1}\)?


Answer: The BIDE Model

\[N_{t+1} = N_t + B_t + I_t - D_t - E_t\]

B=Births, I=Immigrations, D=Deaths, E=Emigrations


Geometric growth is a simplification of BIDE.


Exponential growth is a continuous time version of geometric growth.

Geometric Growth

Reverend Thomas Malthus, 1766–1834

Relevance To Wildlife Biology and Management


Charles Darwin Origin of Species

“There is no exception to the rule that every organic being increases at so high a rate, that if not destroyed, the earth would soon be covered by the progeny of a single pair.”

“Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence…”

Aldo Leopold, Game Management 1946

“Every wild species has certain fixed habits which govern the reproductive process, and determine its maximum rate. Thus one pair of quail, if entirely unmolested in an ‘ideal’ environment, would increase at this rate:”

At End of Young Adults Total
1st year 14 2 16
       
       
       
At End of Young Adults Total
1st year 14 2 16
2nd year (16/2)14=112 16 128
       
       
At End of Young Adults Total
1st year 14 2 16
2nd year (16/2)14=112 16 128
3rd year (128/2)14=896 128 1024
       

“The maximum rate of increase is of course never attained in nature. Part of it never takes place, part of it is absorbed by natural enemies, and part of it absorbed by hunters.”

So What Is Geometric Growth?


For an arbitrary time step (\(t\)):

\[N_t = N_0(1+r)^t\]

Or, for one time step:

\[N_{t+1} = N_t + N_tr\]

\(r\) = discrete-time version of intrinsic rate of increase

Example

\[N_{t+1} = N_t + N_t r\]

Time Population size (\(N\))
0 10
1
2
3
4
5
6
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2
3
4
5
6
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3
4
5
6
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4
5
6
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5
6
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5 320
6
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5 320
6 640
7
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5 320
6 640
7 1280
8
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5 320
6 640
7 1280
8 2560
9
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5 320
6 640
7 1280
8 2560
9 5120
10
Time Population size (\(N\))
0 10
1 20
2 40
3 80
4 160
5 320
6 640
7 1280
8 2560
9 5120
10 10240

Three Possible Outcomes

\[N_{t+1} = N_t + N_t r\]

\(-1 \leq r < 0\)
population goes extinct

\(r = 0\)
population is at equilibrium

\(r > 0\)
population grows to \(\infty\)

Connection to BIDE

From BIDE To Geometric Growth

\[N_{t+1} = N_t + B_t + I_t - D_t - E_t\]

\(N_t\) = Abundance at year \(t\)
\(B\) = Births
\(I\) = Immigrations
\(D\) = Deaths
\(E\) = Emigrations

From BIDE To Geometric Growth

Step 1: Ignore immigration and emigration

\[ N_{t+1} = N_t + B_t - D_t \]

Step 2: Divide both sides by \(N_{t}\).

\[ \frac{N_{t+1}}{N_t} = \frac{N_t + B_t - D_t}{N_t}\]

Step 3: Write in terms of birth and death rates, with \(r=b-d\).

\[ \begin{aligned} \frac{N_{t+1}}{N_{t}} \quad &= \quad 1 + b - d \\ \quad &= \quad 1 + r %\\ % \quad &= \quad \lambda \end{aligned} \]

Step 4: Multiply through by \(N_t\) to get the solution:

\[N_{t+1} = N_t + N_tr\]

Note

The finite rate of increase is defined as: \(\lambda = N_{t+1}/N_t\), which for geometric growth is equivalent to \(\lambda = 1+r\).

Exponential Growth

Density Independent Growth


Geometric and exponential growth are examples of density independent growth.


Definition: Population growth rate (\(r\) or \(\lambda\)) is not affected by population size (\(N\)).


Implications: Resources are unlimited and there is no carrying capacity!

Model Assumptions

  1. Population is geographically closed.
    1. No immigration
    2. No emigration
  1. Reproduction occurs seasonally (for geometric growth).
  1. Constant birth rate (\(b\)) and death rate (\(d\)).
    1. No genetic variation among individuals
    2. No age- or stage-structure
    3. No time lags
  1. No stochasticity.
    1. No random variation in birth or death
    2. No random variation in environmental conditions

Can We Apply The Model To Real Data?

All models are wrong, but some are useful. (George Box)

Is exponential growth a useful model?


Possibly for describing some populations during short time periods.

And as foundation for more realistic models.

Looking Ahead

Is the human population exhibiting exponential growth?

Our world in data

Assignment


Read pages 15–19 in Conroy and Carroll.


Be prepared for a quiz.