STRUCTURED POPULATION MODELS Can subdivide populations by: where individuals occur in space (e.g. metapopulation models) the age of individuals (age-structured models) the size of individuals (size-structured models) the stage of individuals (stage-structured individuals)

Life cycle of a fir

pupae Sexual adult egg larvae, 3rd instar larvae, 2nd instar larvae, diapause, pupation larvae, 3rd instar emergence Sexual adult 2nd molt oviposition larvae, 2nd instar hatching egg 1st molt larvae, 1st instar

Fecundity schedules: average fertility of females a function of their ages. Survivorship schedules: average mortality rates as a function of age.

Fecundity schedules: Semelparous (in animals) or monocarpic (in plants) reproduction: when organisms reproduce only once in their lifetime. Century plant Marine salmon Iteroparous (in animals) or polycarpic (in plants) reproduction: when organisms reproduce more than once in their lifetime. Elephant Cherry tree

Survivorship schedules: In plants we distinguish: annuals, biennials, perennials. A perennial: sequoia An annual: sunflower A biennial: spinach

Three types of survivorship curves: Type I: mammals with much parental care in a low risk environment. Age ln Survivorship Type II: (rare) Individuals of all ages have the same probability of dying. This is equivalent to exponential decay: constant mortality risk throughout a lifetime Type III: Species with many, small and vulnerable young.

To develop an age-structured model: Partition the population into age classes (N1(t), N2(t),….) Formulate rules of transition from one age class into the next. Use probabilities of survival from one age class to the next Pi Use age specific fecundities Fi

(assuming birth-pulse model & post-breeding census) Book keeping: (assuming birth-pulse model & post-breeding census) S(x): the number of survivors at the beginning of age x b(x): the per-capita birth rate for members of the age class x S(0) S(1) S(2) S(3) S(4) S(5) = 0 b(1) b(2) Life history interval Age (x) x = 1 x = 4 x = 3 x = 2 x = 5 x=0 BIRTH DEATH OF OLDEST INDIVIDUAL

Fecundity and mortality schedules are summarized in life history tables: Age x S(x) b(x) 500 1 400 2 200 3 50 4

Age x S(x) b(x) l(x) Pi Fi 500 1 400 2 0.8 1.60 200 3 0.4 0.5 1.50 50 Estimate model parameters from table: Age x S(x) b(x) l(x) Pi Fi 500 1 400 2 0.8 1.60 200 3 0.4 0.5 1.50 50 0.1 0.25 4 0.0 0.00

Generalized:

In Matrix form: = Leslie Matrix n(t+1) = A n(t)

n(t+1) = A n(t) n(t+1) = l n(t) Asymptotic behavior: Any study that does not address asymptotic analysis, transient analysis, ergodicity (dependence of asymptotic behavior on initial conditions) and the results of perturbations has not completely explored its model. -Hal Caswell in Matrix Population Models (2001) Asymptotic behavior: Linear model: no equilibrium points except zero. Exponential decline or increase. One form of asymptotic behavior are stable age distributions: n(t+1) = l n(t) n(t+1) = A n(t) Where n(t) and n(t+1) are in stable age distribution (where the proportion of individuals in the age classes does not change) and l is the asymptotic exponential growth rate.

A n(t) = l n(t) (A -l I)n(t) = 0 In Matrix algebra, in the formula (l not zero): A n(t) = l n(t) l is called the eigenvalue of a matrix n(t) is called the eigenvector of a matrix Solving for the eigenvalue and eigenvector involves solving: (A -l I)n(t) = 0 Where I is the Identity Matrix, a square matrix with 1’s in the diagonal and 0’s everywhere else.

N1(t+1) = 3.0 N2(t) +N3(t) N2(t+1) = 0.3 N1(t) N3(t+1) = 0.5N2(t) Simple example: N1(t+1) = 3.0 N2(t) +N3(t) N2(t+1) = 0.3 N1(t) N3(t+1) = 0.5N2(t) Generalizations: There are as many eigenvalues as there are numbers of equations (unless the matrix is reducible). The asymptotic behavior of the model is determines by the dominant eigenvalue l1, which is the eigenvalue with the largest absolute number. The population grows if l1>1, shrinks if l1<1.

Some more generalizations: Projection matrices can be visualized in “digraphs”, where every age/stage is represented by a symbol and every transition by an arrow. For example: 1 2 3 4 p1 p3 p2 f2 f4 f3

Reducibility: If there is a path from every stage to every other stage, the matrix is irreducible. – The model cannot be reduced to a simpler model. 1 2 3 4 p1 p3 p2 f2 f4 f3 Irreducible: 1 2 3 4 p1 p3 p2 f2 f3 Reducible: Every irreducible, nonnegative matrix has a real, positive dominant eigenvalue and a real positive eigenvector. (Perron-Frobenius theorem)

Primitivity: A nonnegative irreducible matrix is primitive if and only if the greatest common divisor of all circuit lengths is 1. 1 2 3 p1 p2 f2 f1 f3 This matrix has 3 circuits of lengths 1, 2, and 3. The greatest common divisor is 1. The matrix is primitive. f4 1 2 3 p1 p2 f3 4 p3 This matrix has 2 circuits of lengths 2 and 4. The greatest common divisor is 2. The matrix is imprimitive. Imprimitive matrices cycle. They have complex eigenvalues and do not converge on a stable age distribution. They have real, positive eigenvectors.