WLF 448: Fish & Wildlife Population Ecology

Fall 2006

VIII. POPULATION GROWTH IN UNLIMITED ENVIRONMENTS (Demograf.cpl & Stochgro.cpl)

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A. Introduction

1. Overview (see Peterson 1977 )

2. Examples

B. Exponential Growth

1. Rate of population increase

• finite rate of increase = k (lambda)

2. Populations with discrete breeding periods

• N_t = N₀k^t

3. Continuously breeding populations

• N_t = N₀e^rt

4. Linear relationship between ln N_t and t.

5. Assumptions to use equation 2 or 3 for predictions:

• Unlimited, constant, favorable environment (i.e., population growth rate remains constant).

• Age-specific birth and death rates remain constant (i.e., population has a stable-age distribution).

C. Stochastic Exponential Growth

1. Example: Spring Chinook, Lemhi River

2. Predicting future population size

3. Probability of extinction

4. Time to extinction

D. Life Tables and Population Growth

1. Net reproductive rate (per generation)

• or the mean number of female offspring produced by a female during her lifetime

• R_o = [sum (l_x m_x)] ÷ l₀

2. Mean length of generation

• or the mean reproductive age of females

• G = (x l_xm_x) / R_o

3. Instantaneous rate of change

• r ~ = ln (R_o) / G

• Lotka's or Euler's equation: 1 = sum (e^-rx l_xm_x )

4. Finite rate of population change (in one time step)

• k = e^r

5. Stable age distribution

• The proportion of individuals in each age class remains constant over time.

• In theory, a geometrically growing population will assume and maintain a stable-age distribution defined as:

Cx = k^-x l_x ÷ (sum k^-x l_x )

where Cx = proportion of animals in the age category x to x+1

k = lambda = finite rate of population increase

r = instantaneous rate of increase

l_x = age-specific survivorship

x = subscript indicating the age category

E. Instantaneous and Finite Rates (also see Peterson 1977)

1. Relationship between r and k (lambda)

2. Subdividing instantaneous rates

3. Relationship between finite survival rate (S) and instantaneous mortality rate (z)

• S = e^-z

• z = -ln (S)

F. Student Responsibilities

• If given the proper data, be able to apply the formula N_t = N₀e^rt

• Be able to compute a finite rate of growth (k).

• Be able to compute an instantaneous or intrinsic growth rate (r).

• Under what conditions might you expect to observe exponential growth?

• How are r, R_o, k, and G related?

• Be able to compute an instantaneous mortality rate from an interval survival rate.

• Be able to change from one size of interval survival rate to another size interval survival rate using an instantaneous mortality rate.

• Be able to compute an overall total mortality rate from two interval survival rates within a period.

G. References

Begon, M., and M. Mortimer. 1986. Population ecology: A unified study of animals and plants. Blackwell Scientific Publ., Boston, Mass. 220pp.

Caughley, G., and L. C. Birch. 1971. Rate of increase. J. Wildl. Manage. 35:658-663.

Elseth, G. D., and K. D. Baumgardner. 1981. Population biology. D. Van Nostrand Co., New York. 623pp.

Johnson, D. H. 1994. Population analysis. Pages 419-444 in T. A. Bookhout, ed. Research and management techniques for wildlife and habitats. Fifth ed. The Wildlife Society, Bethesda, Md.

Krebs, C. J. 1989. Ecological methodology. Harper & Row, Publ., New York. 654pp.

Wilson, E. O., and W. H. Bossert. 1971. A primer of population biology. Sinauer Assoc., Inc., Sunderland, Mass. 192pp.

Updated 06 August 1996