Age-Structured Models: The Leslie Matrix
by Mike Martin

This page generates plots, information, and data for a Leslie-Matrix Model:

where
and
.

The model helps to describe the dynamics between the different, age-based population classes within the population of a given species.   Usually, the populations in the model are those of the females within a larger population.   The vector x has components that represent the populations of each of the age-based classes of the females. The Leslie matrix (or projection matrix) has fertilities or fecundities on the first row of the matrix.   That is, the parameters, bi, represent the average number of newborn females produced by one female in the ith age group that survive through the time interval in which they were born.   The bi's are all greater than or equal to zero. The other parameters, si's, represent the survival probabilities from one class to the next.   More specifically, si is the fraction of the ith age group that live to the (i+1)st age group.   Each si is positive and less than or equal to one.   A new population distribution, xt+1, is obtained by multiplying the Leslie matrix, L, by the old population distribution, xt.   On this page we have arbitrarily set the number of population classes to four (say, juveniles, young adults, adults, and old adults).

Enter in the values of the parameters (described above), the initial distribution, and the number of steps, then press the "Evaluate" button.

b1 =  

b2 =  

b3 =  

b4 =  


s1 =  

s2 =  

s3 =  


x1 =  

x2 =  

x3 =  

x4 =  


N =  



POPULATIONS PLOT
Class 1 (Juveniles) in BLUE // Class II (Young Adults) in PURPLE // Class III (Adults) in BROWN // Class IV (Old Adults) in Gray




PERCENTAGE POPULATIONS PLOT
Class 1 (Juveniles) in BLUE // Class II (Young Adults) in PURPLE // Class III (Adults) in BROWN // Class IV (Old Adults) in Gray


POPULATION TRAJECTORY DATA

PROPORTION TRAJECTORY DATA

LAST PERCENTAGE POPULATION DISTRIBUTION
has the following components ==>

If you multiply this vector by the scalar multiple of then you obtain a vector with components This last vector should be approaching the eigenvector, v1 (given below), as the number of steps grows large

As a consideration from the Perron-Frobenius Theory, the largest eigenvalue, λ1, is .
The eigenvector, v1, associated with λ1 has the following coordinates

The following number is the inherent net reproductive number and represents the expected number of offspring per individual per lifetime. For these parameter values,
the INHERENT NET REPRODUCTIVE NUMBER is ==>

References:

Perron-Frobenius Theory

The Leslie Matrix