Chapter 11 Population 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 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