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Age Structured Leslie Matrix
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# Objectives:

1. To understand Age-structured population and Leslie matrix.
2. To understand the application of Leslie matrix and survivorship curves in population ecology.

### Age Structured  Leslie Matrix:

Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology. It was invented by and named after P. H. Leslie. The Leslie Matrix (also called the Leslie Model) is one of the best known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration and where only one sex, usually the female, is considered. This is also used to model the changes in a population of organisms over a period of time. Leslie matrix is generally applied to populations with annual breeding cycle. In a Leslie Model, the population is divided into groups based on age classes (see Fig. 1) . A similar model which replaces age classes with life stage is called a Lefkovitch matrix, whereby individuals can both remain in the same stage class or move on to the next one. At each time step the population is represented by a vector with an element for each age classes where each element indicates the number of individuals currently in that class. The Leslie Matrix is a square matrix with the same number of rows and columns and the population vector as elements. The (i,j)th cell in the matrix indicates how many individuals will be in the age class, i at the next time step for each individual in stage j. At each time step, the population vector is multiplied by the Leslie Matrix to generate the population vector for the following time step. To build a Leslie matrix, some information must be known from the population: Fig. 1 . Age and Age-class

• nx, the number of individual (n) of each age class x.
• sx, the fraction of individuals that survives from age class x to age class x+1.
• fx, fecundity, the per capita average number of female offspring reaching n0, born from mother of the age class x. More precisely it can be viewed as the number of offspring produced at the next age class mx+ 1 weighted by the probability of reaching the next age class. Therefore, fx = sxmx + 1.

The observations that n0 at time t+1 is simply the sum of all offspring born from the previous time step and that the organisms surviving to time t+1 are the organisms at time t surviving at probability sx , we get nx + 1 = sxnx. This then motivates the following matrix representation:

Where ω is the maximum age attainable in our population.

This can be written as; or; Where the population vector at time t and L is the Leslie matrix.

The characteristic polynomial of the matrix is given by the Euler-Lotka equation.

The Leslie model is very similar to a discrete-time Markov chain. The main difference is that in a Markov model, one would have
fx + sx = 1 for each x, while the Leslie model may have these sums greater or less than 1.

### How to create a Leslie Matrix:

Population vector s+1 rows by 1 column , (s+1) *1 . Here s is the maximum age.

### Birth :

Newborns = (Number of age 1 females) times (Fecundity of age 1 females) + (Number of age 2 females) times (Fecundity of age 2 females) + .... Note: fecundity here is defined as number of female offspring. Also, the term "newborns" may be flexibly defined (e.g., as eggs, newly hatched fry, fry that survive past yolk sac stage, etc. ### Mortality:

Number at age in next year = (Number at previous age in prior year) times (Survival from previous age to current age)

Na,t = Na-1,t-1Sa

Leslie Matrix: i.e. ( s+1) * 1 = (s+1) (s+1) * (s+1) * 1

### Age-specific vital rates for female component of population:

 X nx Sx mx Fx 0 20 0.5 0 0.5 x 1 = 0.5 1 10 0.8 1 . 2 40 0.5 3 . 3 30 0.0 2 0.0 N0 = 100 --- --- ----

mx = Average female offspring per female of a given age in the population.

### Leslie matrix (A): Population vector (nt=0 ): where 20 = young-of-year, 10 = one-yr olds, ...

n1 = Leslie matrix (A) x population vector (n0 )  Calculate population size and finite rate of change:

N1 = 74 + 10 + 8 + 20 = 112

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

### Advantages and Disadvantages of Leslie Matrices:

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).

1. Requires a large amount of data (i.e., age-specific data on survival, fecundity, and population structure).
2. In practice, the estimation of Fx is difficult at best.

### Life Tables and Survivorship Curves:

Life tables describe how mortality varies with age over a time period corresponding to maximum life span. A Life table is constructed by following the fate of a group or cohort of new organisms until all are dead or by using the age at death of a sample of individuals.

Life table construction:

ax = Number of individuals alive at a time x.
lx = The proportion of original cohort surviving to time x.
mx = Number of offsprings produced by the surviving individuals of age x.

R0, the basic reproductive rate describes the overall outcome of the patterns of survivorship and fecundity. Intrinsic rate of natural increase, where T is the generation time.

Cohort generation time .

Barnacle life table:

 X ax lx mx lxmx xlxmx 0 1*10^6 1 0 0 1 62 .000062 4600 .285 .285 2 34 .000034 8700 .296 .592 3 20 .000020 11600 .232 .696 4 15.5 .0000155 12700 .197 .788 5 11 .0000110 12700 .082 .700 6 6.5 .0000065 12700 .025 .492 7 2 .0000020 12700 .025 .175 8 2 .0000020 12700 .025 .200 1.282 3.928

From the given table we can calculate R0, r and Tc; ### Survivorship curves:

Survivorship curves plot the numbers in a cohort still alive at each age. Species having widely different life span have been compared with the help of survivorship curves. To obtain these curves the number of survivors in a population is plotted against age which is represented as percentage of lifespan. There are 4 basic types of survivourship curves as follows: convex curve, concave curve , diagonal curve , stair step type of curve (see Fig. 2). Fig. 2 .Survivorship curves of different species  (Adapted from Gregory M. Erickson)

When the mortality rate is roughly constant, the survivorship curve is more or less straight; when the rate increases, the curve is convex; and when the rate decreases, the curve is concave.

### Behaviours of the curve:

1. Convex curve is the characteristic of species in which most individuals die in old age as in man and many species of large animals in which mortality rate is low until near the end of the lifespan.
2. Concave curve is the characteristic of animals in which mortality is high during young stages as in oysters, most shell fishes and oak trees. In these mortality is high during free swimming larval stages and acorn seeling stage of oak.

When mortality rate is constant at all ages, the curve is a straight diagonal line. This curve is obtained in hydra, mice and many adult birds such as sparrow.

Stair step type of curve is found in some holometabolous insects (insects with complete metamorphosis)  in which survival differs greatly in successive life history stages. Fig. 3 . Survivorship curves for red deer hinds on the island of Rhum(Adapted from 'Ecology From Individuals to Ecosystems (Fourth Edition)'-Michael Begon, Colin R. Townsend, John L. Harper, 200).

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