Importance of Leslie Matrix and age distribution towards population projection

Importance of Leslie Matrix and age distribution towards population projection- this presentation explains population dynamics of insects describing how insect population increase with time as per Leslie matrix. It gives brief description of computational biology of population increase of organisms as per Leslie matrix.

1. Submitted to: Dr. SS Shaw Professor, Department of Entomology, IGKV Submitted by : Sonia Soni PhD. First year, Department of entomology, IGKV Importance of Leslie Matrix and age distribution towards population projection ENT -604 Advanced Insect Ecology

2. Leslie matrix B ritish ecologist Patrick Holt Leslie analyzed a matrix model for an age-structured populations of rodent in 1945 (Leslie, 1945) Leslie matrix is a discrete, age-structured model of population growth. It is an important method to describe the growth of populations (and their projected age distribution) in which a population is closed to migration growing in an unlimited environment and where only one sex i.e. the female is considered.

3. Leslie matrices are used in population ecology to determine how populations are affected by characteristics such as survival and fertility rates .

4. Properties of Leslie matrix A Leslie matrix is a square matrix with ‘n’ row and columns, where ‘n’ represents the number of age classes and is equal to the number of time intervals (i.e. time span of female). Only female population is modeled in the matrix. The male population and, thus, the total population, can be figured from knowing the ratio of female to male for the species. (Allen, 2007)

5. Continuation…. In Leslie matrix, there are entries on- the sub-diagonal directly below the main diagonal and in the first row. Every other element in this form of matrix is zero. Fig. The Leslie Matrix (Leslie, 1945)

6. Fig. The Leslie Matrix Sub-diagonal elements of Leslie matrix The entries on the sub-diagonal describe the survival probabilities for each age class (Allen, 2007). This survival probability P , is the chance that a female in age class i survives for one time interval to age i+1 (Caswell, 2001). Continuation…. First row elements of the Leslie matrix The entries in the 1 st row represent the fertilities F, for each age class (Allen, 2007). It is described by the number of individuals in age class 1 after one time interval (at time t+1) per individual in age class 1 during the original interval (or at time t; Caswell, 2001).

7. The initial state population vector : It is a vector where each entry represents the number of female individuals in each age group at time t (Caswell, 2001; Leslie, 1945). For example, suppose the female member of a population are divided into n stages or classes or age groups (where each age group having different fecundity rate and survival rate), then the number of female at different age group at time t o can be represented as:- Continuation…. = = Where, x represent number of females of different age groups (ranging from 1 to n) at time t o , This is initial state population vector.

8. Why do we need to go for Leslie matrix ? Lets understand this with help of following example: Suppose that the female members of a population are divided into two stages, each one year in length. Females in the first stage produce no offspring and have a 70% chance of surviving to the second stage. Females in the second stage produce an average of 3 female offspring per year, but are guaranteed to die after one year in stage 2. Let’s also suppose that initially there are 100 females in the first stage and 100 females in the second stage. Then what will the distribution of the female population look like in year 1? Or year 2?.....

9. Time (year) Number of female individuals Stage 1 Fecundity rate: produce no offspring Survival rate: 70% Stage 2 Fecundity rate: produce 3 female offspring per year Survival rate: 0% Initial population (at t ) 100 100 1 st year = (average number of offspring produced by stage 1 females x 100) + (average number of offspring produced by stage 2 females x 100) = (0 x 100) + (3 x 100) = 300 = number of stage 1 females reaching stage 2 + number of stage 2 females remaining in stage 2 = (probability of a stage 1 female reaching stage 2 x 100) + (probability of a stage 2 female remaining in stage 2 x 100) = (0.7 x 100) + (0 x 100) = 70 2 nd year = (average number of offspring produced by stage 1 females x 300) + (average number of offspring produced by stage 2 females x 70) = (0 x 300) + (3 x 70) = 210 = number of stage 1 females reaching stage 2 + number of stage 2 females remaining in stage 2 = (probability of a stage 1 female reaching stage 2 x 300) + (probability of a stage 2 female remaining in stage 2 x 70) = (.7 x 300) + (0 x 70) = 210 3 rd year So on…………………………………………… Continuation…. It seems easy with just 2 stages but what if there are n number of stages then the calculation become complex, thus to make it simple Leslie matrix is used.

10. Future population projection using Leslie matrix To project future population follow following steps:- First the initial population values are placed into a vector. This initial population vector is multiplied by the Leslie matrix that is raised to the power of the number of time intervals projected. The projection is expressed using the formula, Where X(t) is the projection matrix, L is Leslie matrix, t is the desired number of time intervals of projection and X(0) is the initial population vector. (Allen, 2007)

11. Matrix representation of the formula:- Where, n is the maximum age attainable in the population, t is the desired number of time intervals of projection, F is fertility and P is survival probability of respective age groups in the population. Projection matrix Initial population vector Leslie matrix raised to the power of number of time intervals projected (t)

12. Eigenvalues are one of the important properties of Leslie matrices as they give an idea of how the population will change over time. Eigenvalues can be calculated by subtracting lambda times the identity matrix I from the original matrix A, then find the determinant and set this equal to zero (0). That is, eigenvalues can be obtained by solving determinant of and set equal to 0

13. Continuation…. Suppose A is a Leslie matrix: Where a, b, c and d represent non-negative real numbers such that Identity matrix: A square matrix whose non-diagonal elements are zero and diagonal elements are unity i.e. one. I is the identity matrix:

14. Eigenvalues can be obtained by solving determinant of and set equal to 0 So by putting the value of a, b, c and d, lambda can be determined. Continuation….

15. For each Leslie matrix, there will then be a dominant eigenvalue that has a value greater than the absolute value of any other eigenvalue (Anton and Rorres , 1994). The value of the eigenvalue describes the changes in population in the future i.e. if- The population will decrease The population will increase Continuation….

16. Net reproduction rate:- Net reproduction rate is another statistic that can be found using the Leslie matrix. The net reproductive rate is simply the expected number of children a female will bear during her lifetime by taking the sum of each fertility rates multiplied by all previous survival probabilities . The net reproductive rate is represented by following expression:- The population has zero value population growth if its net reproductive rate is 1 (Anton and Rorres , 1994)

17. Stable age distribution:- Leslie matrix is based on stable age distribution i.e. the population projected should have a stable age distribution. Stable age distribution is the age distribution which would be approached by a population of stable age-schedule of birth-rate and death-rate (i.e., ‘ mx ’ and ‘ lx ’ are constant) when growing in unlimited space (Andrewartha and Birch, 1954).

18. Continuation…… When a population approaches stable age-distribution, its rate of increase becomes constant and is called intrinsic rate of increase or innate capacity for increase in numbers ( rm ). Such population can maintain its ‘ rm ’ over an indefinite period under a given set of environmental conditions. Therefore, the stable age-distribution is the only sound basis on which to make comparisons between values of rates of increase whether between different species or one species under different physical conditions.

19. Numerical Let’s look at a simple example. In 1941, H. Bernadelli explored a beetle population that consists of three age-classes. One-half of the females survive from year 1 to year 2, one-third of the females survive from year 2 to year 3. The females reproduce in their third year, producing an average of six new females. After they reproduce, the females die.

20. Age classes of female One year old Two year old Three year old Fertility rate 6 Survival probability Then the Leslie matrix for our beetle population is given by:

21. Suppose that in a given year there are 60 beetles of age 1 year, 60 beetles of age 2 years and 60 beetles of age 3 years. In other words, the population of beetles at time 0 is given by the vector Question 1 : What will the age distribution of the beetles look like in the following year? How about 5 years from now? How about 10 years from now? Continuation….

22. Solution: The age distribution of the beetles in year one is given by:- So after one year the age distribution of beetles population will be: 360 beetles of age 1 year, 30 beetles of age 2 years and 20 beetle of age 3 years.

23. The age distribution of the beetles five years from now is given by:- Let us solve first:-

24. So now putting the value of So after five years the age distribution of beetles population will be: 120 beetles of age 1 year, 180 beetles of age 2 years and 10 beetle of age 3 years.

25. The age distribution of the beetles ten years from now is given by:- So after ten years the age distribution of beetles population will be: 360 beetles of age 1 year, 30 beetles of age 2 years and 20 beetle of age 3 years.

26. Question 2 : What will happen to the population of beetles in the long run? Will it die out? Will it grow? Will the population get younger? Older? Let’s see if we can determine what will happen. Beginning with Calculate What will happen if we calculate Can we now describe the long-term behavior of the beetle population?

27. Solution :The age distribution of the beetles after one year, two years, three years, four years, five years and six years respectively from now is given by:-

28. Question 3 : Find the eigenvalues for the matrix L. Does this help explain the behavior you observed in the previous problem? For this example Solution: For the Leslie matrix with order 3 we can use the following equation for calculating lambda - The eigenvalue is 1, thus the population is stable and we observe the cyclic nature of the population.

29. References: Allen, L.J., 2007. An introduction to mathematical biology.  Upper Saddle River, New Jersey . Andrewartha, H.G. and Birch, L.C., 1954.  The distribution and abundance of animals  (No. Edn 1). University of Chicago press. Anton, H., and Rorres , C. 1994. Elementary Linear Algebra – applications version. 7th edition. New York: Wiley. Bernadelli , H., 1941. Population waves. J. Burma Res. Soc., 31: 1-18. Caswell, H., 2001. Matrix Population Models: Construction, Analysis, and Interpretation, 2nd edn . Sinauer Associates, Sunderland. Leslie, P.H., 1945. On the use of matrices in certain population mathematics. Biometrika . 33:183-212. doi : 10.1093/ biomet /33.3.183. PMID: 21006835.

30. Thank you…..