How the Leslie Matrix Reveals Age‑Structured Population Dynamics

Leslie Matrix Model

The commonly used Malthus and Logistic models for population studies are simple but have two major drawbacks: they consider only total population size and ignore individual differences such as age‑specific fertility and mortality. By grouping the population by age, the Leslie matrix addresses these issues.

Age is a reasonable classification because individuals of the same age tend to have similar reproductive and survival rates. Thus the population can be divided into age classes (e.g., one‑year groups or five‑year groups). Time is also discretized into periods.

Let N_t be the vector of population numbers for each age class at time t . Let f_i denote the fertility rate of age class i and s_i the survival rate (probability of surviving to the next age class). The Leslie matrix L is constructed as:

<code>|  f_1  f_2  …  f_{k-1}  f_k |
|  s_1   0   …      0      0 |
|   0   s_2  …      0      0 |
|  …    …   …     …      … |
|   0    0  …   s_{k-1}  0 |
</code>

The population vector evolves according to N_{t+1}=L·N_t . When the population distribution at a given time step is known, the vector for the next step can be computed, and the total population size for each period follows.

Key Conclusions

Age‑structured models provide richer predictions than aggregate models.

The Leslie matrix allows calculation of future age distributions and total population size.

The same framework can be applied to other species by grouping according to relevant traits such as weight or size.

Reference

Python数学实验与建模 / 司守奎, 孙玺菁, 科学出版社