This is an example of interfacing R and shiny to allow users to explore

a biological model often encountered in an introductory ecology class.

We are interested the growth of a population that is composed of multiple,

discrete stages or age classes. Patrick H. Leslie provides an in-depth

derivation of the model in his 1945 paper

“On the use of matrices in certain population mathematics”.

The population at time is represented by a vector ,

where each element of the vector represents the number of individulas

in each age class (e.g. if a population has age classes, then

has elements). Time is considered discrete and we

assume that the population is censused prior to breeding. We assume

that individuals within each age class are identical, and that each

has some probability of maturing to the next age class, surviving

(staying in the same age class), and reproduction. Changes in the

population from one timestep to another are represented as:

where is an x Leslie matrix (or more generally, a

projection matrix) that describes the contribution of each age class to

the population at time .

Suppose we are tasked with modeling the annual dynamics of a population

with four age classes, and $t$ represents years. For simplicity, we

model only females and assume that plenty of males are available for

breeding. Individuals in the first age class survive to class 2 with

probability 0.1, class 2 individuals survive to class 3 with probability

0.5, class 3 individuals survive to class 4 with probability 0.9, and

class four individuals survive each year with probability 0.7. Only the

fourth age class is reproductive, with individuals producing 100 class

1 individuals per year. We can represent this population graphically as:

Equivalently, as a Leslie matrix:

The long term population growth rate is related to the dominant

eigenvalue of . If the population

eventually declines to extinction, and if , the

population increases.

From a management perspective, it is often useful to know how limited

resources may be allocated to increase population growth or prevent

extinction. In other words, if an element such as fecundity or

survival could be manipulated by managers, how much would the long term

population growth rate change? To this end, one can calculate the

*sensitivity* of the dominant eigenvalue () to small

changes in :

where and are left and right

eigenvectors, respectively, associated with the dominant eigenvalue.

Because survival and fecundity are on different scales, sensitivity is

often scaled by a factor of for a measure

of *elasticity*.

#### Building the R shiny app

Files are accessible in this repository. Please feel free to clone for your own use and/or contribute.

Here is the resulting graphical user interface for the model.