458 Age-structured models (continued): Estimating from Leslie matrix models Fish 458, Lecture 4

458 The facts on Finite rate of population increase =e r & r=ln( ), therefore N t =N t A dimensionless number (no units) Associated with a particular time step (Ex: =1.2/yr not the same as = 0.1/mo) >1: pop.; <1 pop

458 Matrix Population Models: Definitions Matrix- any rectangular array of symbols. When used to describe population change, they are called population projection matrices. Scalar- a number; a 1 X 1 matrix State variables- age or stage classes that define a matrix. State vector- non-matrix representation of individuals in age/stage classes. Projection interval- unit of time define by age/stage class width.

458 4x 1 + 3x 2 + 2x 3 = 0 2x 1 - 2x 2 + 5x 3 = 6 x 1 - x 2 - 3x 3 = –2 5 1 –1 3 x1x2x3x1x2x3 = Basic Matrix Multiplication

458 What does this remind you of? n(t + 1) = An(t) Where: A is the transition/projection matrix n(t) is the state vector n(t + 1) is the population at time t + 1 This is the basic equation of a matrix population model.

458 Eigenvectors & Eigenvalues Aw = w v,w = Eigenvector = Eigenvalue When matrix multiplication equals scalar multiplication Note: “Eigen” is German for “self”. vA = v Rate of Population Growth ( ): Dominant Eigenvalue Stable age distribution (w): Right Eigenvector Reproductive values (v): Left Eigenvector

458 Example: Eigenvalue = = No obvious relationship between x and y A x = y Obvious relationship between x and y: x is multiplied by -3 Thus, A acts like a scalar multiplier. How is this similar to ?

458 Characteristic equations From eigenvalues, we understand that Ax = x We want to solve for, so Ax - x = 0 (singularity) or (A- I)x = 0 “I” represents an identity matrix that converts into a matrix on the same order as A. Finding the determinant of (A- I) will allow one to solve for. The equation used to solve for is called the Characteristic Equation

458 Solution of the Projection Equation Solution of the Projection Equation n(t+1) = An(t) 4 - P 1 F P 1 P 2 F 3 - P 1 P 2 P 3 F 4 = 0 or alternatively (divide by 4 ) 1 = P 1 F P 1 P 2 F P 1 P 2 P 3 F 4 -4 This equation is just the matrix form of Euler’s equation: 1 = Σ l x m x e -rx

458 Constructing an age-structured (Leslie) matrix model Build a life table Birth-flow vs. birth pulse Pre-breeding vs. post-breeding census Survivorship Fertility Build a transition matrix

458 Birth-Flow vs. Birth-Pulse Birth-Flow (e.g humans) Pattern of reproduction assuming continuous births. There must be approximations to l(x) and m(x); modeled as continuous, but entries in the projection matrix are discrete coefficients. Birth-Pulse (many mammals, birds, fish) Maternity function and age distribution are discontinuous, matrix projection matrix very appropriate.

458 Pre-breeding vs. Post-breeding Censuses Pre-breeding (P  1) Populations are accounted for just before they breed. Post-breeding (P  0) Populations are accounted for just after they breed

458 Calculating Survivorship and Fertility Rates for Pre- and Post-Breeding Censuses Different approaches, yet both ways produce a of

458 The Transition/Population Projection Matrix 4 age class life cycle graph

458 Example: Example: Shortfin Mako (Isurus oxyrinchus) Software of choice: PopTools

458 Mako Shark Data Mortality: M 1-6 = 0.17 M 7-  = 0.15 Fecundity: 12.5 pups/female Age at female maturity: 7 years Reproductive cycle: every other 2 years Photo: Ron White

458 Essential Characters of Population Models  Asymptotic analysis: A model that describes the long-term behavior of a population.  Ergodicity: A model whose asymptotic analyses are independent of initial conditions.  Transient analysis: The short-term behavior of a population; useful in perturbation analysis.  Perturbation (Sensitivity) analysis: The extent to which the population is sensitive to changes in the model. Caswell 2001, pg. 18

458 Uncertainty and hypothesis testing Characterizing uncertainty Series approximation (“delta method”) Bootstrapping and Jackknifing Monte Carlo methods Hypothesis testing Loglinear analysis of transition matrices Randomization/permutation tests Caswell 2001, Ch. 12

458 References Caswell, H Matrix Population Models: Construction, Analysis, and Interpretation. Sunderland, MA, Sinauer Associates. 722 pp. Ebert, T. A Plant and Animal Populations: Methods in Demography. San Diego, CA, Academic Press. 312 pp. Leslie, P. H On the use of matrices in certain population mathematics. Biometrika 33: Mollet, H. F. and G. M. Cailliet Comparative population demography of elasmobranch using life history tables, Leslie matrixes and stage-based models. Marine and Freshwater Research 53: PopTools: