Age-Structured Models with Density Dependence: The Leslie Matrix
by Mike Martin

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

The model helps to describe the dynamics between the different, age-based population classes within the population of a given species.   On this page we assume the fecundities depend on the size of the overall population or, better said, the density.   The vector x has components that represent the populations of each of the age-based classes of the females. The Leslie matrix has fertilities or fecundities on the first row of the matrix.   That is, the ith entry in the first row represent the average number of newborns produced by one couple in the ith age group that survive through the time interval in which they were born.   The entries in the first row and their respective parameters should all be non-negative.   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.     On this page we have arbitrarily set the number of population classes to three.

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

a =  

c1 =  

c2 =  

c3 =  


s1 =  

s2 =  


K =  

λ =  


x1 (0) =  

x2 (0) =  

x3 (0) =  


N =  



POPULATIONS PLOT
Class I in BLUE // Class II in PURPLE // Class III in BROWN




PERCENTAGE POPULATIONS PLOT
Class I in BLUE // Class II in PURPLE // Class III in BROWN




TOTAL POPULATION PLOT


POPULATION TRAJECTORY DATA


PROPORTION TRAJECTORY DATA


TOTAL POPULATION TRAJECTORY DATA


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