## Population Models

Population models are designed for projecting population size, composition, and distribution over time through births, deaths, and migration.

### Age-Classified Population

Since birth and death processes are age-dependent, the individuals of a population are categorized into age classes. The state of population is described by the age-distribution vector (i.e., n x 1 matrix)

where x(t) denotes the size of ith age class (i.e., the number of individuals in ith class) at time t. The age distribution vector at time t + 1 is projected by using projection matrix x(t + 1) = Px(t)

The projection matrix is the sum of survival matrix and fertility matrix

The element s; of the survival matrix defines the portion of individuals in class i who will survive and move to the class ;+ 1:

and the element f of the fertility matrix defines the number of newborns per i-class individual so that the total of newborns is

If there are j juvenile classes and ; post-reproductive classes, then f = f = ... = j = o and fn _ i+1 =

Inherent growth rate of population

The total size of population

xk may either increase or decrease with time depending on the balance between births and deaths. This balance is characterized by the positive-dominant eigenvalue p of the projection matrix, which is said to be inherent growth rate. A population goes extinct if p <1 and grows exponentially if p > 1.

Some projection matrices do not have a positive, strictly dominant eigenvalue, and so the concept ofinher-ent growth rate is generally applicable to populations where the number of adult classes is more than one, and there is no post-reproductive adult classes (i.e., fertilities of adult classes are positive).

Stable age distribution

If the initial age distribution vector x(0) is equal to the eigenvector q associated with positive strictly dominant eigenvalue p, then x(') = P' q = P' -1 (Pq) — P'-1 (pq) — p' q and the normalized age distribution

When x(0) is not equal to q, the normalized age distribution converges to

which is said to be stable age distribution.

After the stable age distribution has been reached, the number of individuals increases p times over the projection period. To keep the constant size of population one must remove 100(p — 1)/p percentage of the population.

Equilibrium age distribution

When the growth rate of the population is equal to 1 (p — 1), there exists equilibrium age distribution x — Px

The equilibrium age distribution is more than often the target of population control or exploitation. The effect of harvesting is modeled by multiplying the age distribution vector by harvest matrix

The harvested portion of ith class is equal to (1 — hi). Removing 100(p — 1)/p percentage of each class, one will keep both the constant size and the constant age distribution of a population.

Consider, for example, the projection matrix

This matrix has a positive, strictly dominant eigenvalue, p — 1.01815, and thus applying this matrix to an initial age distribution vector x(0) =

one may reveal that relative proportions of the age classes eventually converge to constant values

Removing 1.78% of each class / 0.9822 0 H — 0 0.9822 0

keeps the constant size and age distribution of the population x(' + 1) « 0.3883 x 1.01815'+1

0.9822 0

0.95

0.28

0 0.9822 0 \ 0 0 0.9822 / \ 0.1375 j — 0.3883 x 1.01815t x 1.01815

0 0.9822 0

0.28

 /1 0 0\ 0.95 — 0.3883 x 1.01815' x 0 1 0 0.28 \0 0 1 / 0.1375 0.95 — 0.3883 x 1.01815' x 0.28 « x(') 0.1375

Stage-Classified Population

Variability in developmental rates makes age to be a poor predictor of the birth and death processes. For example, reproductive maturity often depends on the interaction of several factors, not only of age. Therefore, size or developmental stage is sometimes better predictor than age.

The projection matrix of stage-classified population is the sum of transition and fertility matrix:

The diagonal element i of the transition matrix is the fraction of individuals who survive and remain in stage-class i, and tj + 1,i is the fraction of individuals who survive and move to next developmental stage.

Like in case of age-classified population, the concepts of inherent growth rate and stable stage-distribution are applicable if the number of adult classes is more than one, and there are no post-reproductive adult classes (i.e., fertilities of adult classes are positive). 