📂Dynamics

Pontryagin's Maximum Principle

Model ¹

$$ {{ \partial n } \over { \partial t }} + {{ \partial n } \over { \partial a }} = - \mu \left( a \right) n \qquad t, a \in (0, \infty) $$ The partial differential equation above is known as the Von Foester Equation, and it has the following two Dirichlet boundary conditions:

• Initial condition 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 example, if $a = 10$, it means 10 years old, and if $a = 54$ it means 54 years old.
• $n(t,a)$: The number of individuals of age $a$ at time $t$.

Parameters

• $\mu (a) \ge 0$: The death rate of the population at age $a$.
• $b (a) \ge 0$: The birth rate of the population at age $a$.

Explanation

The Von Foester Equation is the application of the uniform progressive wave partial differential equation in mathematical biology, where its solution, $n(t,a)$, models how many people of age $a$ there are at time point $t$. The motif is straightforward: as $t$ advances by 1, the entire population’s age $a$ also increases by 1, which can be seen as the movement of a wave. Moreover, the attenuation rate of the wave perfectly matches the death rate. Unlike most physical problems, the boundary $a = 0$ is not a constant $0$ but is given by $$ \int_{0}^{\infty} b(a) n \left( t, a \right) da $$ which is the inner product of population at each age and the birth rate over time. As time passes, it means the population members age, and a lower function value signifies deaths.

In this context, the population can be interpreted ecologically but can also be associated with diseases, such as in the context of disease modeling, where it’s also referred to as the McKendrick-Von Foerster Equation.

Discrete Age-Structured Models

The Leslie model is well-known.

Derivation

According to the Malthusian growth model, the population size $a$ can be represented as 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 $n (t,a)$ itself is $$ d n (t,a) = {{ \partial n } \over { \partial t }} dt + {{ \partial n } \over { \partial a }} da $$ Therefore, we obtain the following: $$ {{ \partial n } \over { \partial t }} dt + {{ \partial n } \over { \partial a }} da = - \mu (a) n (t,a) dt $$ Since the change in age is exactly equal to the change in time, $da / dt = 1$ and obviously $dt / dt = 1$, by canceling $dt$ from both sides, we 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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