📂Dynamics

폰 푀르스터 방정식

Model ¹

$$ {{ \partial n } \over { \partial t }} + {{ \partial n } \over { \partial a }} = - \mu \left( a \right) n \qquad t, a \in (0, \infty) $$ The above partial differential equation is referred to as the von Foerster equation and possesses the following two Dirichlet boundary conditions:

• Initial conditions of age structure $$ n \left( 0, a \right) = f(a) $$
• Number of births $$ n \left( t, 0 \right) = \int_{0}^{\infty} b(a) n \left( t, a \right) da $$

Variables

• $a$: Represents age. For instance, $a = 10$ signifies 10 years old, $a = 54$ means 54 years old.
• $n(t,a)$: Denotes the population of a group aged $a$ at time $t$.

Parameters

• $\mu (a) \ge 0$: The death rate of a group aged $a$.
• $b (a) \ge 0$: The birth rate of a group aged $a$.

Description

The von Foerster equation is an application of the uniform wave propagation partial differential equation to mathematical biology, with its solution, $n(t,a)$, modeling the population aged $a$ at time $t$. The motivation is straightforward: as time $t$ progresses by one unit, the age $a$ of the entire population increases by one unit, akin to the movement of a traveling wave. Furthermore, the attenuation of the wave can aptly represent the death rate. Contrariwise to typical physical problems, the boundary of $a = 0$ is not constant like $0$ but is given by $$ \int_{0}^{\infty} b(a) n \left( t, a \right) da $$ the functional inner product with the population and birth rate of each age. The passage of time signifies the aging of the population members, and a decrease in function value signifies that many have died.

Here, the term “population” can be interpreted ecologically or linked to the number of cells and infectious diseases. In the context of epidemic modeling, it is also called the McKendrick-Von Foerster equation.

Discrete Age Structure Model

The Leslie model is well known.

Derivation

According to the Malthusian growth model, the population $n(t,a)$ of a group aged $a$ at time $t$ can be described by the following ordinary differential equation. $$ {{ d n (t,a) } \over { d t }} = - \mu (a) n (t,a) \implies d n (t,a) = - \mu (a) n (t,a) dt $$ Meanwhile, the total derivative of $n (t,a)$ is $$ d n (t,a) = {{ \partial n } \over { \partial t }} dt + {{ \partial n } \over { \partial a }} da $$ thus yielding: $$ {{ \partial n } \over { \partial t }} dt + {{ \partial n } \over { \partial a }} da = - \mu (a) n (t,a) dt $$ The change in age precisely matches the change in time, so $da / dt = 1$, and trivially, $dt / dt = 1$, thus allowing for the cancellation of $dt$ on both sides to obtain the following equation. $$ {{ \partial n } \over { \partial t }} + {{ \partial n } \over { \partial a }} = - \mu \left( a \right) n (t, a) $$

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