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Equation  represents development and mortality, whereas equation  represents reproduction. Equation  specifies the number of individuals in the first age class and equation  specifies the number of individuals in all other age classes. In the equation , the number of individuals in age x+1 in time t+1 equals to the number of individuals in the previous age and previous time multiplied by age-specific survival rate sx. In the equation  the number of new-born organisms equals to the number of mothers (Nx,t) multiplied by the numbers of offspring produced (fflx). The number of offsprings is summed over all ages of mothers.

These two equations can be combined into one matrix equation:

where N is the vector of age distribution in the population at time t, and A is the transition matrix. When a matrix is multiplied by a vector, we take the 1st row of the matrix, multiply each number by the corresponding number in the vector-column and then sum all products. This sum is the value of the 1st element in the result vector. Then we take the 2nd row of the matrix, multiply it by the same vector and the sum becomes the 2nd element in the result vector. In the same way we can estimate all elements of the result vector.

The first element of the result vector corresponds to the equation , and all other elements correspond to the equation .

How to read matrix models?

Each column specifies the fate of organisms in specific state. The number in the intersection of column i and row j indicates how many organisms in state j are produced by one organism in state i. In the Leslie model, organisms' state is defined by age only. For example, the third column corresponds to age a=2. An organism in age 2 produces m offsprings of age 0 (first cell in the column), and goes to age class 3 with probability s (the cell under main diagonal).

Most simple stochastic process is defined by a matrix of transition probabilities between states: In this case, the sum of matrix elements in each column equals to 1 because it is assumed that each system passes through a series of states and neither die nor reproduce.

Leslie model is more complex because the sum of matrix elements in each column is not necessary equal to 1. This is a "branching process" because the life trajectory of a parent branches into life trajectories of its offsprings.

Matrix models are easy to iterate in time. In the next time step we again multiply the transition matrix by the vector of age distribution:

This equation can be used to simulate as many time steps as necessary.

7.2. Model Behavior

Two major characteristics of the behavior of Leslie model:

• A few damping oscillations are followed by an exponential growth

• Age distribution approaches a stable age distribution

These 2 features can be seen in the graphs that show simulations of sheep population dynamics:  In this simulation we started from a population of 100 new-born sheep:

The first graph shows exponential population growth (it becomes linear in a log scale) after several initial years. The second graph shows convergence of age distribution to a stable age distribution.

Here is an example of the model of Leslie implemented using Excel spreadsheet:

You can play with this model by changing model parameters (the matrix) and initial age-distribution.

You can simplify the analysis of matrix models using PopTools, which is a free Excel plugin developed by Greg Hood, CSIRO, Canberra, Australia.

7.3. Estimation of the intrinsic rate of population increase

Previously we discussed an approximate equation: