WLF 448: Fish & Wildlife Population Ecology

Lab 10: LESLIE MATRICES

I. Projecting population growth (density-independent):

A. Observe past trends in population size

• estimate lambda

• project future growth, e.g., N_t = N₀ (lambda)^t

B. Life-table approach (cohort analysis)

• follow cohort through time or approximate with time-specific techniques

• calculate the population's rate of change (~ r, Ro, lambda)

• rate of change x population size = future population size

• N_t = N₀ (lambda)^t

• N_t = r N_t-1 or N_t = N_o e^rt

• assumptions?

C. Leslie matrix approach

• Age-specific vital rates x data on population size and structure

II. Leslie Matrix Approach

based on the multiplication of 2 matrices: Leslie Matrix × Population Vector

A. Leslie Matrix or Population Projection Matrix (denoted A)

• Contains age-specific fecundity and survival rates.

• For example, with 3 age classes (ages 0-2, females only) the Leslie matrix would take the form:

where Fx = age-specific fecundity x survival

      Sx = age-specific survival rate

• The equation for F_x depends on where sampling occurs relative to the breeding season:

1. Pre-birth pulse sampling (young-of-year not present): F_x = S₀ m_x

2. Post-birth pulse sampling: F_x = S_x m_x+1

3. Birth-flow fertility: more complicated situation (we will not discuss)

F_x is used to calculate the number of recruits (young) in the next generation

B. Population Vector (n_t )

• Contains the number of individuals in the population in each age class (or stage) at time t

• For 3 age classes (females only):

C. Population Projections:

• n₁ = A x n_o and N₁ = sum of n_i's in population vector n₁

• n₂ = A x n₁ and N₂ = sum of n_i's in population vector n₂

• n_t+1 = A x n_t and N_t = sum of n_i's in population vector n_t

• theoretically: n_t = A^t x n₀ and lambda = N_t+1 / N_t

D. Post-birth pulse example:

Remember that F_x = S_x m_x+1 in post-birth pulse sampling.

Age-specific vital rates for female component of population:

m_x = average female offspring per female of a given age in the population

Leslie matrix (A):

Population vector (n_t=0 ):

where 20 = young-of-year, 10 = one-yr olds, ...

n₁ = Leslie matrix (A) x population vector (n₀ )

n_t=1 = x

n_t=1 = =

Calculate population size and finite rate of change

N₁ = 74 + 10 + 8 + 20 = 112

Lambda = N_t+1 / N_t = 112 / 100 = 1.12

Projecting population growth of both female and male segments of the population

We can make similar projections using both female and male components of the population if we have age-specific data for both sexes. The Leslie matrix becomes slightly more complicated when you include males (i.e., it incorporates sex-specific survival as well as information on sex ratio at birth).

Alternatively, we can double the population projection for females to estimate total population size (females + males). The later estimate is valid only if we assume a 50:50 sex ratio at birth and equal age-specific survival for females and males.

III. Assumptions of age-based Leslie projections:

1. It is appropriate to classify individuals by age, i.e., properties other than age, such as size or developmental stage, are irrelevant to an individual's demographic fate.

2. The discrete nature of the model discards all information on age dependency of vital rates within age classes.

3. Fertility and survival rates remain constant, i.e., the environment is constant and density effects are unimportant. Note: stochastic Leslie models are possible -- we will use a stochastic-based Leslie model in a few weeks.

4. Most demographic models are based on one sex, usually the female. The implicit assumption is that life cycles of the sexes are identical or the dynamics of the population are determined by one sex independent of the relative abundance of the other (e.g., there will always be enough males to fertilize the females).

IV. Advantages and Disadvantages of Leslie Matrices

A. Advantages of Leslie Matrix Approach

1. Stable-age distribution is not required for valid population projections.

2. Can conduct sensitivity analysis to see how changing certain age-specific vital rates affects population size and age structure.

3. Can incorporate density-dependence, i.e., can dampen values in the matrix to account for density-dependent factors limiting population growth.

4. Can derive useful mathematical properties from the matrix formulas, including stable-age distribution and finite rate of population change (i.e., lambda).

B. Disadvantages

1. Requires a large amount of data (i.e., age-specific data on survival, fecundity, and population structure).

2. In practice, the estimation of F_x is difficult at best (Taylor and Carley 1988).

V. In-Class Exercise

    Program LESLIE - instructions will be given in lab.

VI. Problem Set

VII. Application of Leslie Matrices to Columbia River Spring/Summer Chinook

• Kareiva et al (2000),  used Leslie matrices in an innovative way to evaluate whether eliminating downstream mortality of smolts passing down the Snake and Columbia Rivers as well as eliminating upstream mortality of spawners would lead to recovery of these populations.  You can download a PowerPoint presentation on this paper by clicking on the title above which is a link to the PowerPoint file.

VIII. Selected References

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

• Caswell, H. 1989. Matrix population models. Sinauer Assoc., Inc., Sunderland, Mass. 328pp.

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

• Kareiva, P., M. Marvier and M. McClure. 2000.  Recovery and management options for Spring/SummerChinook Salmon in the Columbia River Basin.  Science 290:977-979.

• Krebs, C. J. 1972. Ecology: The experimental analysis of distribution and abundance. Harper & Row, Publ., New York. 694pp.

• Ricklefs, R. E. 1979. Ecology, second edition. Chiron Press, New York. 966pp.

• Taylor, M. and J. S. Carley. 1988. Life table analysis of age structured populations in seasonal environments. J. Wildl. Manage. 52:366-373.

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

Revised: 19 August 2002