1. CENTURION UNIVERSITY OF
TECHNOLOGY AND
MANAGEMENT,BBSR
NAME: ARYA PRATIK BEHERA
REGD NO.210301120240
B.TECH CSE , SECTION : F
SUBJECT: DIFFERENTIAL EQUATION AND LINEAR ALGEBRA
TOPIC: FIND THE GROWTH RATE IN LESLIE MODEL.
2. INTRODUCTION :
• In applied mathematics the 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 Patrick H. Leslie.
• Leslie matrices may be used to model the age distribution of a population as well as
population growth.
• The Leslie matrix is a square matrix with the same number of rows and columns
as the population vector has elements.
• The dominant eigen value tells us the long term population growth and the corresponding
eigen vector tells us the long term age distribution.
3. GROWTH RATE IN LESLIE MODEL:
• Generally, it is not possible to predict the rate of growth of a certain population of animal
based on the size of the population alone.
• A Leslie matrix models the growth of the female portion of a population, using survival
rates like the proportion that will survive until the next year or the next time period, and
fecundity rate the actual reproductive rate for females of different ages.
• A more accurate picture of survival rate can be obtained if we use survival rates specific to
the different age groups. So splitting a given population into smaller age categories allows a
much more accurate population model to be developed.
4. EXAMPLE :
• A wildlife officer is interested in the number of a certain species of rat in a
habitat and has compiled the following information on this species.
Population breakdown by age-:
AGE(x) AT YEAR END NUMBER(N) OF FEMALES
1 55
2 32
3 25
5. It assumed that rats do not live longer than 3 years and that there are the
same no. of males and females .
Fecundity rates ,𝑭𝒙(the no. of females offspring per female per year)
Age(X) Fecundity
1 0.6
2 2.4
3 0.5
The survival rates ,𝑺𝒙𝐨𝐟 each group have also been estimated
Age(X) Survival rate(𝑆𝑥)
1 0.5
2 0.6
6. 0.6 2.4 0.5
0.5 0 0
0 0.6 0
L= 𝑊1 = 55
32
25
Population at the 2nd year :
𝑊2 = L * 𝑊1
122.3
27.5
19.2
Number of 1 year old rats = number of 1 year old F₁+number of 2 year old F₂+ number of 3 year old
F3
The number of 2 year old rats in the next year=0.5x number of 1 year old.
The number of 3 year old rats in the next year = 0.6x number of 2 year old.
