Chapter 11 Population Growth. Modeling Geometric Growth Geometric growth represents maximal population growth among populations with non-overlapping generations.

Chapter 11Population Growth

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:

= 2408 = 2.4177

996

• To determine the growth of a non-overlapping population: by multiply by the size of the population at each beginning generation

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 2.4177 = 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

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

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

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

Examples of Sigmoidal Growth

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.

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:

• 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

• 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

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

Same proportion of individuals are affected at any density

• Density independent factors