WLF 448: Fish & Wildlife Population Ecology

Lab 8

Population Growth: Simple Models

I. Introduction

A. Characteristics of a good model

1. general
2. realistic
3. precise / accurate
4. simple

B. Deterministic vs. stochastic models (see Braun 2005:155)

C. Population growth: exponential versus logistic growth equations

D. Fundamental population growth parameters (see Life Table notation)

1. Ro = (Sum l_x m_x) / l₀ = net reproductive rate per generation or the mean number of female offspring produced by a female during her lifetime.

2. G (or T) = Sum (x l_x m_x) / Sum (l_x m_x) = mean length of a generation or the mean reproductive age of females.

3. = lambda = Ro^1/G = finite rate of increase of the population in one time step (often 1 yr).

   • Note: in the case of seasonal breeders (i.e., birth-pulse fertility), Ro is the same as lambda expressed in terms of a generation interval. For example, if Ro = 0.700, G = 8.470, and lambda = 0.959, then lambda expressed per generation = lambda^G = 0.959 ^8.470 = 0.701.

   • lambda = e^r , where e = 2.71828 (= natural log or log to the base e).

4. r ~ = ln (Ro) / G = instantaneous rate of change or r ~ = ln (lambda)

5. S_x or p_x = finite survival rate.

6. q_x = finite mortality rate.

II. Exponential or Geometric Population Growth Models

A. Assumptions

1. Population growth in an unlimited, constant, and favorable environment.

2. Growth rates are constant.

3. Populations have a stable-age distribution (i.e., constant birth and death rates in age classes).

B. Reproductive strategies:

1. Semelparity: discrete generations where organisms breed once and die (e.g., insects, annual plants, salmon).

2. Iteroparity: overlapping generations where an organism has more than one reproductive effort during a lifetime, which can be either discrete or cont.

• Discrete reproduction (birth-pulse fertility of Caughley 1977) - e.g., seasonal breeders.

• Continuous reproduction (birth-flow fertility of Caughley 1977).

C. Discrete Generations with a Single Breeding Period.

For a population with discrete generations with a single breeding period, the population size at generation g (denoted N_g ) is calculated as:

N_g = N₀ R_o^g

where N₀ = initial population size, and g = number of generations into the future.

Ro varies with age at reproduction, litter size, and survival, thus affecting N_g.

Discrete Generations Example: Salmon

Age (yrs)	lx	mx	.	.
0	100,000	0	.	.
1	250	0	.	.
2	50	0	.	.
3	10	20,000	.	.

Ro =

G =

N_g=1 =

N₂ =

N₃ =

D. Overlapping Generations w/ Discrete Breeding Periods (Birth Pulse).

For a population with overlapping generations with discrete breeding periods, the population size at time t (denoted N_t ) is calculated as:

N_t = N₀ (lambda)^t

where N0 = initial population size,
       t = time steps into the future (often expressed as years). 

Birth Pulse Example: Wood thrush

Age (yrs)	lx	mx	.	.
0	100	0.0	.	.
1	40	2.0	.	.
2	10	4.0	.	.

Ro =

G =

lambda =

N_{t =1} =

N₂ =

N₃ =

E. Overlapping Generations w/ a Cont. Breeding Period (Birth Flow).

For a population with overlapping generations and a continuous breeding period, the population size at time t (denoted N_t ) is calculated as:

N_t = N₀ e^rt

where N0 = initial population size, 
       t = number of time steps in the future, 
       r = instantaneous rate of increase. 

Note: with continuous breeding the age interval (x) often is set at the midpoint of the age interval. For example, a microtus life table might look like this:

Age (months)	x	lx	mx	.	.
0-3	1.5	100	0.0	.	.
3-6	4.5	60	2.1	.	.
6-9	7.5	30	3.2	.	.
9-12	10.5	10	3.2	.	.

Ro =

G =

r =

N_{t = 6} =

N₁₂ =

N₂₄ =

assume critical habitat was lost or degraded at this point in time... and resulted in r = -0.03.

N₂₅ =

F. Doubling Time

• If you know r for a population, then you can calculate the time it will take for the population to double in size (assuming constant growth rate in an unlimited environment):

t = ln(2) / r

• How long would it take the original microtus population (N₀ = 100) just described to double in size?

G. Common situations in nature where exponential growth may occur over appreciable periods of time:

1. Where a species has naturally colonized or has been introduced by humans into a new and acceptable geographic area.

2. Where a species has been greatly depressed by human activities and such activities cease.

3. Where a species naturally undergoes marked fluctuations and population growth consequently begins from densities that are very low relative to environmental carrying capacity.

III. Instantaneous versus Finite Rates

A. Population Growth Rates

• A logarithmic plot of the discrete overlapping generations population growth equation N_t = N₀ (lambda)^t results in a straight line. We call this graph a semi-log plot because the ordinate (y axis) is logarithmic and the abscissa (x axis) is arithmetic. The slope of the straight line is the instantaneous rate of change (r) when natural logarithms (base e logs) are used.

• Although several of the population parameters we discussed today are expressed as finite rates (e.g., lambda, S_x, q_x), it is often useful to be able to convert to their instantaneous rate equivalents.

• Example: Instantaneous Growth Rate

If a population doubles each year, then:

lambda = 2 (finite rate of growth yearly)

lambda = e^r = 2

r = ln (lambda) = 0.693 (instantaneous rate of growth yearly).

To calculate daily growth (change):

1. Use the basic formula N_t = N₀ e^rt
2. Divide the instantaneous rate of growth yearly by 365.

For example, a population of 1000 individuals having an instantaneous growth rate of 0.693 yearly would increase daily as follows:

N_1/365 = 1000 (e^{0.693 x 1/365}) = 1001.9

B. Survival and Mortality Rates

Recall that:

• Finite survival rate (S) = N_t / N_t-1, where S can range from 0 to 1.

• Finite mortality rate = 1.0 - S

Conversion Formulas (between finite and instantaneous rates)

• Instantaneous mortality rate (z) = - ln (S)

• Finite survival rate (S) = e^-z , where z = instantaneous mortality rate

Note: although instantaneous rates are useful mathematically, it is clearly easiest to think about finite survival rates or finite mortality rates, and survival data should be reported in this form (Krebs 1989:412).

Example:

One common problem in calculating survival rates is to convert observed rates to a standardized time base. For example, you might watch a mallard brood for 38-days and get a finite survival rate of 0.60; however, you will probably want to report your results according to a standard observation period (e.g., 30 days). In other words, you want to convert S₃₈ to S₃₀.

1. Convert the observed survival rate to an instantaneous mortality rate (see conversion equations described above).

   • z₃₈ = - ln (0.60) = 0.51

2. Adjust the instantaneous mortality rate to the standard time period:

   • z₃₀ = z₃₈ (t_s / t₀),

     where ts = standardized time interval  

           t0 = observed time interval  

   • z₃₀ = 0.51 (30 / 38) = 0.403

3. Convert the adjusted instantaneous mortality rate back to the finite rate.

   • S₃₀ = e^-0.403 = 0.668

Remember that instantaneous mortality rates are additive, whereas finite survival rates are multiplicative. However, you cannot subdivide finite survival rates like you can instantaneous mortality rates. In other words, if you want a survival estimate in some other unit of time than currently expressed, then you must convert to an instantaneous rate before subdividing.

IV. Problem Set

V. Selected References

Begon, M., J. L. Harper and C. R. Townsend. 1996. Ecology: Individuals, populations and communities. 3rd ed. Blackwell Scientific Ltd., Cambridge, Mass. 1068pp

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

Revised: 25 August 2011