Wildlife Population Analysis Matrix Models for Population Biology
Wildlife Population Analysis Matrix Models for Population Biology Lecture 12
Resources: n n n Caswell, H. 2001. Matrix population models. 2 nd ed. Sinauer Associates, Inc. Sunderland, Mass. Tuljapurkar, S. , and H. Caswell (eds. ) 1997. Structuredpopulation models in marine, terrestrial, and freshwater systems. Chapman & Hall, New York. Manley, B. F. J. 1990. Stage-structured population: sampling, analysis, and simulation. Chapman & Hall, New York.
Matrix addition and subtraction n n Simply addition (subtraction) of the corresponding elements in the matrices. Matrices must be of the same rank (dimension).
Matrix multiplication
Matrix multiplication n Matrices: n n n do not have to be of the same rank; must have the same “inner” dimension; dimensions are specified as rows x columns. Thus a 4 x 3 matrix can be multiplied by a 3 x 1 matrix, But a 1 x 3 matrix cannot be multiplied by a 4 x 3 matrix.
Example:
The identity matrix. n For any square matrix, the identity is a diagonal matrix of equal rank with all of the diagonal elements = 1.
Leslie and Lefkovitch Projection Matrices
History n Application of age-specific survival and fertility rates dates back to the late 19 th century n Use of matrix models was developed independently by Bernardelli (1941), Lewis (1942), and Leslie (1945). n Leslie’s works published in 1945, 1948, 1959, and 1966 were apparently the most influential. n Even so matrix models were not mentioned in many notable ecology texts or in ecological research prior to the 1970 s. n Lefkovitch worked on the dynamics of agricultural pests published a series of influential papers using the matrix models described by Leslie in the early 1960 s. Among these was a 1965 paper that introduced the idea of stagestructured models that classified insects by life-stage rather than age.
Age-structured models n n The goal of population modeling is to develop equations that allow us to understand the processes that govern population dynamics. Consider the equation: where Nt is the population size in year t and Nt+1.
Age-structured models n In the absence of emigration and immigration, the population growth rate, , subsumes the processes of mortality and recruitment. Thus, one could more explicitly write this equation as where F is fertility, the number of offspring recruited per adult and P is the probability of surviving from year t until year t+1.
Age-structured models n Now consider a population of size N with 3 1 -year age classes where ni is the number of individuals in age-class i and age class one is the youngest age class. The dynamics of this population could be expressed as three separate equations: and since individuals in this population do not live beyond age 3, all of the n 3 s die before the next time step (year).
Age-structured models n The age-structured transition matrix model representing this system of equations is a square matrix with one column for each age-class: The population N, composed of individuals of three age classes n 1 -3 is represented by the vector:
Age-structured models n This population is projected through time using matrix multiplication by the equation: (Note that the inner dimensions of the matrices (3 x 3; 3 x 1) agree. )
Life-cycle graph n Life-cycle diagram, where each node represents an age class, the straight lines connecting the nodes represent the survival (transition) probabilities (P) and the curved lines extending back to the first node represent the fertilities (F). F 1 F 3 F 2 1 2 P 1 3 P 2
Example: n Birth and Survival Rates for Female New Zealand Sheep [from G. Caughley, "Parameters for Seasonally Breeding Populations, " Ecology 48(1967)834 -839]
Life-cycle diagram:
The Leslie matrix:
Projection n 10 -years starting with 100 2 -year olds: Note the rapidly (exponential) increasing population and the initial fluctuations in due to starting conditions (age distribution).
Age distributions n 10 -year projection starting with 100 2 -year olds.
Assumptions of age structured models: n Individuals progress through the life-cycle by discrete timesteps (e. g. , years) n Age-specific fertility n Age-specific survival n Single sex models
Stage-based models n n Lefkovitch relaxed the assumptions of the age-structured models described by Leslie Useful for animals that had stage-dependent vital rates. n n Discrete time-steps (e. g. , years) Individuals allowed to remain in life-stages longer than one year n Stage-specific fertility n Stage-specific survival n Single sex models
Stage-based matrix model (3 stages): n n n F i is still the fertility, the number of offspring recruited per adult; Pi is the probability of surviving from year t until year t+1 and remaining in stage i ; and Gi is the probability of growing to stage i during the next time step.
Life cycle graph n Generalized 4 -stage population:
Examples: n n n Arthropods with discrete developmental stages. Plants, crustaceans, and fish with size-dependent ages of maturity. Angiosperms, kelp, molluscs, decapods, insects, isopods, amphibians, and reptiles with reproductive rates that vary with adult body size. n Plants with mortality rates that vary with size. n Plants and animals with size related sex changes.
Another example n n Brault, S. and Caswell, H. 1973. Pod-Specific Demography of Killer Whales (Orcinus orca). Ecology, 74: 1444 -1454. Classified the population into 4 stages of females: yearlings (1 year olds), juveniles (up to 18 yrs), reproductive (up to 45 yrs), and post-reproductive. Thus the life-cycle graph looks like:
Matrix
Projection
Projection stacked
Pre-breeding vs. post-breeding n Pre-breeding n n Post-breeding n n State vector represents population structure immediately prior to breeding season State vector represents population structure after reproduction occurs Transition rates change by adjusting the fertilities and survivals.
Pre-breeding vs. post-breeding n n Generally, speaking Pis in age-based and Gis in stage-based models are annual (single time step) rates and will not vary. However n n Fis in a pre-breeding census model include productivity and survival of offspring until the end of the first time step (e. g. , year). Fis in post-breeding census model are discounted for annual survival of adults Also P 1 s in a prebreeding census reflect survival of individuals between the first and second time step. P 1 s in the postbreeding model are survival from postbreeding until the next postbreeding census.
Example: n Hypothetical bird population n Postbreeding age-structured matrix n Fi = cs*sr*ns*gs*S 1+ = 7*0. 5*0. 35*0. 45*0. 76 = 0. 42
Example: n Hypothetical bird population n Prebreeding age-structured matrix n Fi = cs*ns*gs*S 0 = 7*0. 5*0. 35*0. 45*0. 60 = 0. 33
Four questions from Caswell (2001) n n Asymptotic behavior—Is long-term behavior Ergodicity—Is the behavior of the model dependent upon the initial state vector Transient behavior—What are the short term dynamics of the model? Perturbation analysis—How does the model respond to changes in the vital rates (i. e. , what are the relative sensitivities)?
Four questions from Caswell (2001) n Asymptotic behavior— n What happens if model processes operate for a very long time? n Does it grow or decline? n Does it persist or go extinct? n Does it converge to an equilibrium, oscillate, or behave chaotically? n Most deterministic matrices reach an asymptotic growth rate and stable age distribution.
Population growth rate ( ) n n n Will the population grow or decline? Project population for >20 years, then calculate rate of change (Nt+1/Nt). Project population for >20 years, then calculate average rate of change (Heyde, C. C. , and J. E. Cohen. 1985)
Population growth rate ( ) n Calculate dominant eigenvalue n n No. of eigenvalues = no. of stages Fortunately, most single population matrices have only one real, positive (dominant) eigenvalue.
Stable age (stage) distribution (SAD) n n n What is the predicted structure of the population? Project the population for >20 years, determine the percentage of the population in each age (stage) class. Calculate the right eigenvector of the dominant eigenvalue and normalize. Age/stage structure R eigenvector Final SAD 36. 4% 19. 8% 43. 9%
Ergodicity n n Is the behavior of the model dependent upon the initial state vector? Project model >20 years with different initial age distributions. Does the population reach (or approach) the expected and SAD?
Transient behavior n What are the short term dynamics of the model? n Does it grow or decline? n How rapidly does it converge to an equilibrium? n Does it oscillate, or behave chaotically? n Can be very useful in understanding population responses to perturbations.
Example Spectacled Eider population on the Y-K Delta at Kashunuk River Study site Age 1 Age 2 Age 3 Nest success 0. 47 Clutch size (females hatched) 2. 15 Breeding propensity 0. 56 1 Duckling survival 0. 34 Survival of immature 0. 49 0. 44 0. 82 0 0. 1764 0. 315 0. 70 Survival of adults - exposed to lead - not exposed - lead exposure weighted average
Example A= 0. 00 0. 094 0. 168 0. 82 0 0. 75 0. 70 Eigenvalues 1 2 Eigenvectors (R&L) Real Imaginary Age/stage struct 0. 86 0 15. 3% 0. 95 -0. 08 -0. 23 14. 7% 0. 99 -0. 08 0. 23 70. 0% 1. 012 Reprod val 3+
Example numerical projection with SAD
Example transient after breeding failure
Example numerical projection w/loss of 80% breeders
Example transient w/loss of 80% breeders
Transient dynamics n n The rate of convergence on a stable population growth rate is governed by the relative size of the subdominant eigenvalues. That is, the larger 1 is in relation to i>1 the more rapidly the population will converge on stability. This property often referred to as the damping ratio is defined as:
Example: n Hypothetical population matrix with high Fi and low annual survival, similar to a small mammal or a passerine bird: A= 0 3 4 0. 2 0 0. 4 Eigenvalues Eigenvectors (R&L) Real Imaginary Age/stage struct 1. 046434 0 76. 4% 0. 479315 -0. 15583 0 14. 6% 2. 507855 -0. 49061 0 9. 0% 2. 965901 Reprod val
Example – transient dynamics
Lower F 3 A= 0 3 0. 1 0. 2 0 0. 4 Eigenvalues Real Eigenvectors (R&L) Imaginary Age/stage struct Reprod val 0. 7878 0 66. 0% 0. 733113 0. 382371 0 16. 7% 2. 887734 -0. 770171 0 17. 3% 0. 189044
Lower F 3
Perturbation analysis n How does the model respond to changes in the vital rates? n What are the relative sensitivities? n Estimates of vital rates always are subject to uncertainty. n Conclusions dependent upon exact values are always suspect.
Perturbation analysis - sensitivity of matrix models n Prospective analysis – forward looking. n n What could happen to the population if changes occur in vital rates? Retrospective analyses – examining the past. n How has variation in vital rates affected population growth?
Motivation n n Determining which rate(s) have the greatest affect on population growth Predicting results of future changes in vital rates Quantifying the effects of past changes in vital rates Predicting the actions of natural selection (if changes in phenotypes result in changes to vital rates) Designing sampling schemes. (i. e. choosing which vital rates are the most important to measure accurately)
Prospective Analysis n Two Metrics: n Sensitivities - the effect on population growth rate, 1, of unit changes in the vital rates. n n e. g. , 1 egg increase in mean clutch size Elasticities – the relative effect of vital rates on population growth rate, 1. n e. g. , 1% increase in mean clutch size.
Sensitivity n n Sensitivity refers to the effect on population growth rate, 1, of unit changes in the vital rates. Rate of change in for a unit change in aij while holding all other vital rates constant.
Example – desert tortoise model n n 1 Doak, D. P. Kareiva, and B. Klepetka. 1994. Modeling population viability for the desert tortoise in the western Mojave Desert. Ecological Applications 4: 446 -460. Stage (size) based matrix model 2 3 4 5 6 7 8
Example – desert tortoise model 1 2 3 4 5 6 7 8
Example – eigen analysis
Sensitivity n n Sensitivity refers to the effect on population growth rate, 1, of unit changes in the vital rates. Rate of change in for a unit change in aij while holding all other vital rates constant.
Example a 88 – survival rate of individuals in the largest size class
Sensitivity n Measure of the effect of a change in aij holding all else constant. n The slope of as a function of aij. 0. 989 0. 988 0. 987 Actual 0. 986 Slope Observed values 0. 985 0. 984 0. 983 0. 982 0. 981 0. 98 0. 82 0. 84 0. 86 0. 88 a(8, 8) 0. 90 0. 92 0. 94
Sensitivity matrix
Elasticities n n n Relative effect on population growth rate, 1, of small changes in the vital rates. Sum to 1. Interpreted as the relative contributions of the vital rates to .
Elasticities n n Relative effect on population growth rate, 1, of small changes in the vital rates. Slope of ln( ) as a function of ln(aij)
Elasticities n n Elasticities can be calculated from projections as: where * is the population growth rate after a proportionate change in aij , and p is the change in aij. . Since elasticities are scaled with respect to they sum to 1. 0 and thus are directly comparable. Elasticities also can be summed to determine the relative contributions of more than one vital rate.
Desert Tortoise example: n n P 7 the probability of surviving and remaining in stage 7 has nearly 2. 25 times as much of an effect on as does P 6 Elasticity of transition probabilities (Ps and Gs) = 0. 95, Elasticity of Fs =. 05 Population is 19 times as sensitive to growth and survival as productivity.
Lower-level elasticities n Relative sensitivities of to parameters contributing to the aij. n Example: n Fertility( females recruited per female) n Product of: n clutch size n nest survival n hatchling survival n sex ratio
Lower-level elasticities n Relative sensitivities of parameters (xk) contributing to the aij. or
Lower-level elasticities n n Do not sum to 1, so they can not be interpreted as contributions to . Values are relative n If lle(x 1) = 0. 4 and lle(x 2) = 0. 8, then n lle(x 2) has twice the influence on
Lower-level elasticities – example n Desert tortoise F 8 = 4. 38 (females/female) n Identical because F 8 is the product of parameters Estimate Clutch size Sex ratio Sensitivity Elasticity 32. 00 0. 0002 0. 005 0. 50 0. 0107 0. 005 Nest survival Breeding propensity Offspring survival 0. 61 0. 0087 0. 005 0. 80 0. 0067 0. 005 0. 56 0. 0096 0. 005 Product 4. 38
Retrospective analysis n n n Life Table Response Experiments (LTRE) Set of vital rates (matrix) is the response variable in an experimental design. Treatments affect the various vital rates most frequently used statistic to evaluate the effect of the treatments. Often used to examine the effect of past variation in vital rates on population growth rates.
LTRE designs n Analogous to analysis of variance n one-way, two-way, or factorial n Random effects n Regression analysis.
Example – n One-way fixed design n One treatment (t) n One control (c) n Vital rates are used to populate the matrices:
Calculate the mean matrix
Calculate the sensitivities of Am
Calculate the difference matrix (D) n Difference between At and Ac
Calculate the difference matrix (D) n Multiply by the sensitivities The result is the contributions of the differences in the vital rates to the change in the population growth rate.
Prospective versus Retrospective n Prospective analysis – forward looking. n n n Effect of future management actions Retrospective analyses – examining the past. n n n What could happen to the population if changes occur in vital rates? How has variation in vital rates affected population growth? Effect of environmental variation or past actions Frequently don’t have information for retrospective
Prospective versus Retrospective n n Not a panacea Parameters with greatest elasticities will have the greatest relative impact on n n May not be “manageable” Parameters with greatest contribution to past variation can be indicative of management opportunity n Valuable to look at both when possible n Neither incorporates cost or risk
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