What is a PSPM
Jaideep Joshi
11 March 2022
what_is_pspm.RmdA physiologically structured population model describes how the physiological structure of a population of individuals evolves through time. At its core is the mcKendrick-von Foerster Partial Differential Equation (PDE),
\[ \frac{\partial u(x,t)}{\partial t} = \frac{\partial }{\partial x}\left(g(x,t,E)u(x,t)\right) - \mu u(x,t,E), \] with the boundary condition \[ g(x_b)u(x_b) = s_e \int_{x_b}^{x_m} \beta(x,t,E)u(x,t)dt, \] and a specified initial distribution \[ u(x,0) = u_0(x), \]
where \(x\) is the physiological state variable, such as size, \(u(x)\) is the density of individuals of size \(x\) (such that the number of individuals within a small size range \([x, x+dx]\) is \(u(x)dx\)), \(g\), \(\mu\), and \(\beta\) are the demographic rates of individuals (growth rate \(g\), mortality rate \(\mu\), and fecundity rate \(\beta\)) as functions of their size \(x\), the environment \(E\), and time \(t\), \(x_b\) is the size of offspring at birth, \(x_m\) is the maximum size of individuals, and \(s_e\) is the establishment probability of offspring.
The PDE states that within a given size interval, the density of individuals increases because of smaller individuals entering the focal size-range via growth, and decreases by individuals growing out of the size range and by individuals dying. The boundary condition states that the density of individuals at the smallest size \(x_b\) is determined by the total number number of offspring that are produced by all individuals in the population, and the proportion of those that establish successfully.
A simple example of a PSPM
As a simple example, consider a population of trees in a 1 ha plot. The environment is the average light availability \(E\) within the canopy, which depends on the total crown area of trees in the plot. The physiological variable \(x\) represents the biomass of individual trees.
We assume that the growth rate of a tree is allometrically related to its biomass, so that
\[ g(x,t,E) = g_0 x^{\phi_g}. \]
We assume that the mortality rate of all individuals is constant and independent of biomass and the environment,
\[ \mu(x,t,E) = \mu_0. \] Next, we assume that the seed production rate of trees is proportional to their growth rate \(g\) and light availability \(L\),
\[ \beta(x,t,E) = \beta_0 g(x,t,E)\cdot E. \]
Further, let’s assume that the crown area \(A\) of a tree scales with its biomass as \[ A(x) = a_0 x^{\phi_a} \]
Thus light availability, which decreases with increasing total crown area, can be described by
\[ E = 1-E_0 \int_{x_b}^{x_m} A(x)u(x)dx \]
Some terminology used in libpspm
i-state
The variable \(x\) describes the individual physiological state, and is thus called “i-state”. libpspm currently supports only one state variable, but in principle, the i-state can be a vector of multiple “i-state variables”.
Cohort
All methods to solve the PDE described above rely on discretizing the i-state variable \(x\) into \(J\) classes. Let the i-states of individuals in the \(J\) classes be \(x_0, x_1, x_2,...,x_J\) and the densities of individuals in these classes be \(u_0, u_1, u_2,...,u_J\).
The set of all individuals with the same i-state is called a cohort. Thus, each cohort \(j\) is defined by its i-state \(x_j\) and density \(u_j\).