1. Chapter 11 Population Growth

2. Modeling Geometric Growth
Geometric growth represents maximal population growth among populations with non-overlapping generations
Successive generations differ in size by a constant ratio
To construct a model for geometric population growth, recall the formula for geometric rate of increase:
 = Nt+1 Nt
e.g., geometric rate of increase for Phlox:
 = =
996
To determine the growth of a non-overlapping population: by multiply  by the size of the population at each beginning generation

3. Geometric growth for a hypothetical population of Phlox:
Initial population size = 996
Number of offspring produced by this population during the year:
N1 = N0 x  or
996 x = 2408
Calculating geometric growth from generation to generation:
Nt = N0 x t
Where, Nt is the number of individuals at time t; N0 is the initial population size;  is the geometric rate of increase; t is the number of generations

4. Modeling Exponential Growth
Overlapping populations growing at their maximal rate can be modeled as exponential growth:
dN/dt = rN
The change in the number of individuals over time is a function of r, the per capita rate of increase (a constant), times the population size (N) which is variable
Recall that we can interpret r as b – d; also, we can calculate r using the following formula: lnR0/T
To determine the size of the exponentially growing population and any specified time (t):
Nt = N0 ert
Where, e is a constant; the base of the natural logarithms, r is the per capita rate of increase, t is the number of time intervals

6. Exponential Growth in Nature
Example: Scots pine
Bennett (1983) estimated population sizes and growth of postglacial tree populations by counting pollen grains from sediments (e.g., pollen grains/meter2/year)
Assumption of this method?
Results: populations of the species grew at exponential rates for about 500 years

7. Logistic Population Growth
Environmental limitation is incorporated into another model of population growth called logistic population growth; characterized by a sigmoidal growth curve
The population size at which growth has stopped is called carrying capacity (K), which is the number of individuals of a particular population that the environment can support

8. Examples of Sigmoidal Growth

9. What causes populations to slow their rates of growth and eventually stop growing at carrying capacity?
A given environment can only support a certain number of individuals of a species population
The population will grow until it reaches some kind of environmental limit imposed by shortage of food, space, accumulation of waste, etc.

10. Logistic Growth Equation
Logistic growth equation was proposed to account for the patterns of growth shown by populations as the begin to use up resources:
dN/dt = rmN (K-N/K)
where, rm is the maximum per capita rate of increase (intrinsic rate of increase) achieved by a population under ideal conditions
Rearranged equation:

11. Thus, as population size increases, the logistic growth rate becomes a smaller and smaller fraction of the exponential growth rate and when N=K, population growth stops
The N/K ratio is sometimes called the “environmental resistance” to population growth
As the size of a population (N) gets closer and closer to K, environmental factors increasingly affect further population growth

12. The realized per capita rate of increase [r = rm N(1-N/K)] depends on population size
When N is small, r approximates rm
As N increases, realized r decreases until N = K; at that point realized r is zero

14. Limits to Population Growth: Density Dependent vs
Limits to Population Growth: Density Dependent vs. Density Independent Factors
Density dependent factors
Birth rate or death rate changes as a function of population density
Density independent factors
Same proportion of individuals are affected at any density