📂Dynamics

Leslie age structure Model

Definition ¹

Suppose there are $m \in \mathbb{N}$ stages of age in a population, and let’s represent the population size at stage $a \in \left\{ 1 , \cdots , m \right\}$ at time $t$ as $x_{a}(t)$. If we denote the average number of offspring produced by individuals at age stage $a$ as $b_{a}$, and the survival rate of individuals moving from age stage $a$ to $a+1$ as $s_{a}$, then it can be formulated as follows: $$ \begin{align*} x_{1} \left( t+1 \right) =& \sum_{a=1}^{m} b_{a} x_{a} (t) \\ x_{a+1} \left( t+1 \right) =& s_{a} x_{a} (t) \end{align*} $$

This dynamics system, when expressed through a map, is called the Leslie age structured Model, and it can be represented in matrix form as follows: $$ X (t+1) := \begin{bmatrix} x_{1} (t+1) \\ x_{2} (t+1) \\ x_{3} (t+1) \\ \vdots \\ x_{m} (t+1) \end{bmatrix} = \begin{bmatrix} b_{1} & b_{2} & \cdots & b_{m-1} & b_{m} \\ s_{1} & 0 & \cdots & 0 & 0 \\ 0 & s_{2} & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & s_{m-1} & 0 \end{bmatrix} \begin{bmatrix} x_{1} (t) \\ x_{2} (t) \\ x_{3} (t) \\ \vdots \\ x_{m} (t) \end{bmatrix} =: L X (t) $$

In this case, the matrix $m \times m$ $L$ is referred to as the Leslie Matrix.

Explanation

The introduced definition only includes the simplest assumptions. Since most organisms have a specific reproductive prime, models vary significantly, and while it is common to only consider the number of females, depending on the species, both sexes might be considered, as well as symbiotic relationships with other species, or even food availability, leading to more complex models. Although we are discussing age, it doesn’t necessarily have to refer to ‘age’ precisely. For example, insects undergoing complete metamorphosis can be divided into eggs, larvae, pupae, and adults.

If $L$ is independent of time $t$, then given the initial condition $X(0)$, $L$ can be simply represented by multiplying $L$ by $t$ times as follows: $$ X (t) = L^{t} X(0) $$

The Leslie Model focuses primarily on the birth, death, and growth of a population considering age, which falls within Vital Dynamics. However, thinking about the age structure of a population is a widely used concept not limited to this alone. For instance, in the SIR model, differential disease transmission based on age-specific contact frequency is an example.

Continuous age structure Model

Models applying the von Foerster equation are known.