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# Objectives:

- To understand and determine stable stage distribution and Lefkovitch matrix.
- To estimate the finite rate of increase from Lefkovitch matrix calculations.
- To analyze and interpret stage distribution plots.

A population is growing at a constant rate reaches a stable age distribution in which the proportion of individuals in each age class remain the same from year one to the next. The Leslie matrix is used to predict this stable age distribution and to calculate the population growth rate after this distribution is established, this technique was developed by Patrick Leslie, a British mathematician, around 1940. Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology. 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. Leslie matrix is generally applied to populations with annual breeding cycle. It also used in population ecology to model the changes in a population of organisms over a period of time. In Leslie Matrix Model, the population is divided into groups based on age classes (ground squirrels are very good example).

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. Stage structured Leslie matrix are used when individuals are censused (rather than marked from after the birth) and the age of individual cannot be known for some reason. In such population individuals are described by size classes (e.g., fishes, deer, frogs etc).

Age is not an accurate predictor of birth or death rates in many organisms. For e.g., a small sugar maple in a northeastern forest can be 50 years old and yet have low levels of reproduction. In this species, the size is a better predictor of birth rate than age. In other species, birth and death rates may be function of the stage in the life cycle of an organism. Such organisms are best modeled with stage-structured matrix models (for example moths, elk).

The Lefkovitch model has been used as the solution to a variety of problems in modeling population dynamics. A stage structured matrix is preferred when

- Demography depends on physiological stages and development into these stages is not consistent among individuals.
- Some individuals exhibit retarded or accelerated development or regression.
- There are sub classes of the population that have different demographic characteristics.
- The ages of individuals cannot be determined and equal width classes lead to computational inefficiency.

The principal drawback of stage structured modeling is the potentially large number of parameters that must be computed. Lefkovitch matrix population projection technique uses the same matrix equation of Leslie matrix model; however the elements in Lefkovitch’s matrix are divided into stage classes. With an assumption that all individuals in each stage class are subject to the same mortality, fecundity, and growth rates.

### Construction of The Stage-Structured Matrix

In our study we are taking Tawny Elk as an example, suppose we can only distinguish newborns,1-,2-,3-year-olds, but from adult(age through 13-17) we can only grouped as middle-aged and elderly. To construct a stage-structured Leslie matrix, it is useful to construct a flow diagram of the movement of individuals through a life table. In Fig 1; each circle in the diagram represents either an age category or a stage category.

Figure 1.Flow diagram for organizing a stage-structured Leslie matrix

G_{m} in the diagram represent the probability that a middle aged adult lives to next year, multiplied by proportion of middle-aged adults that remain in that stage. The concept may be more clear if we think like this, before the birthdays of elk; stage m consists of individuals from age 4 – 12; after their birthdays, stage m consists of the same individuals except the 12-years-old. The proportion of individuals remain in the group is the number of elk of age 4-11 divided by the individuals of age 4-12.

The proportion remains in stage m is given by:

P_{m} is the probability that a middle-aged adult will survive to next year, enter to next stage class. It is equal to the probability of surviving from one year to the next times the proportion that leave the middle-age stage, which is equal to 1- (the proportion that stay in the middle-aged stage).

G_{e} is the probability of an elderly elk living from one year to the next (P_{e}) multiplied by the proportion remain in stage e.

The number of newborns is the sum of fecundity for each stage classes,

Table 1. The generic version of our stage-structured Leslie matrix.

For calculating number of individuals in each stage category in next year is achieved by multiplying the created matrix (Table .1) to number of individuals in each stage category of this year.

An excellent example for stage structured matrix is logger head turtle. Loggerhead turtles are distributed across the world can be found in Pacific, Atlantic, and Indian oceans. An adult loggerhead's major part of the life is spent in saltwater, except for the females, as they have to come ashore for a short period to lay eggs.

They have seven distinct life stages to the life of a loggerhead turtle (see Fig. 2), as follows: eggs and hatchlings, small juveniles, large juveniles, sub-adults, novice breeders, 1st year remigrants, and mature breeders. Females lay their eggs on shore and when the eggs hatch the baby turtles make their way to the water. Once the turtles in the water (small juvenile), they immediately start swimming and exploring. Young turtles (large juvenile) drift out to sea, where they inhabit floating debris in the ocean and scavenge for food. When they are at 12 years of age (sub adult), loggerhead turtles move into coastal regions. Loggerhead turtle will sexually mature only after 35 years (novice breeders).

Figure 2. Different stages of logger head turtle

Humans started hunting loggerhead turtles as a source of meat, oil and shells were also used in various crafts. From 2010 onwards loggerhead was considered as an endangered species. International Union for Conservation of Nature (IUCN) has taken several conservative measures to protect this animal. Loggerhead turtles are still threatened by habitat destruction they mainly rely on healthy waters and beaches to survive. These turtles can live to quite old; they will sexually mature around 35 years old. People enjoy by watching them in some parts of the world, IUCN hopes their efforts will preserve loggerhead turtle populations on the survival of globe earth.