Growth models

Logistic model

A resource consumption view

Consider a resource consumption model that follows the density of a single microbial species through time $N(t)$ and the density of that species' limiting resource $R(t)$:

$$ \frac{dR}{dt} = -a R N \ \frac{dN}{dt} = \epsilon a R N $$

where $a$ is the uptake rate of the microbes in question, in units of $(\rho T)^{-1}$ (where $\rho$ is density and $T$ is time), and $\epsilon$ is the dimensionless conversion rate of resource to biomass.

The following treatment follows (Arditi et al. 2015). By definition:

$$ \frac{d(\epsilon R + N)}{dt} = \ \epsilon \frac{dR}{dt} + \frac{dN}{dt} = \ -\epsilon a R N + \epsilon a R N \equiv 0 $$

Therefore, $\epsilon R + N$ is constant:

$$ M \equiv \epsilon R(t) + N(t) = \ \epsilon R(0) + N(0) \Rightarrow \ M - N = \epsilon R \Rightarrow \ \frac{dN}{dt} = a N (M-N) = aM N (1 - \frac{N}{M}) $$

By substituting $K=M$ and $r=aK$ we get the logistic model:

$$ \frac{dN}{dt} = r N \Big(1 - \frac{N}{K}\Big) $$

This derivation gives an interesting interpretation of the model parameters: We usually refer to $K$ as the carrying capacity, maximum population size, yield or density. Under this interpretation, $K=M=N(0) + \epsilon R(0)$ is the initial population density plus whatever population density there is to make from converting all the resource. This is in line with the standard interpretation.

We usually refer to $r$ as the proportional increase of the population density in one unit of time. Under this interpretation, $r=aK=a \epsilon R + a N$ is in $T^{-1}$; since $a$ is the rate of resource uptake per resource density per population density per time unit, if $N \approx 0$ then $r \approx a \epsilon R$ is the rate at which each population density unit uptakes and converts the resources at hand ($R$), which is in line with the standard interpretation.

Another derivation in (Arditi et al. 2015) assumes that the resource is biotic (prey) that has a logistic growth of its own; McArthur showed that the predator growth is logistic if the conversion rate is slow enough to allow separation of time scales.

Intraspecific interference

Yet another derivations in (Arditi et al. 2015) assumes direct intraspecific interference. Assume the population grows exponentialy:

$$ \frac{dN}{dt} = r N $$

but that encounters between individuals can lead to mortality. Assuming perfect mixing, the number of individulas dying due to interferece is $\lambda N^2$ and we get:

$$ \frac{dN}{dt} = r N - \lambda N^2 = r N \Big(1 - \frac{N}{K}\Big) $$

where $K=r/\lambda$.

Generalized logistic model

The generalized logistic model is an extension of the logistic model, introducting the parameter $\nu$:

$$ \frac{dN}{dt} = r N \Big(1 - \Big( \frac{N}{K})^{\nu} \Big) $$

This model is also called the Richards model (Richards, 1959) or, in its discrete time version, the $\theta$-logistic model (Gilpin & Ayala, 1973).

When $\nu=1$, this is the logistic model; when $\nu=0$ this is the Gompertz model.

According to (Richards, 1959), one interpretation of $\nu$ is that $(1+\nu)^{-1/\nu}$ states explicitly the proportion of the final size ($K$) at which the growth rate $\Big(\frac{dN}{dt}\Big)$ is maximal; i.e., this is the value of $N/K$ at which the inflexion point of the growth curve $\Big( \frac{d^2N}{dt^2}=0\Big)$ occurs (note that Richards uses the symbol $m=\nu+1$ and therefore the inflexion point occurs as $N/K=m^{1/(1-m)}$). When $\nu=0$, this occurs at $N/K=e^{-1}$; when $\nu=1$, this occurs at 1/2.

Solution

To solve this model (Skiadas, 2010), we define $y=N/K$ and $z=y^{-\nu}$ to get:

$$ \frac{dz}{dt} = -r \nu (z - 1) \ z(0) = \Big( \frac{N_0}{K} \Big)^{-\nu} $$

which is solved to $$ z(t) = 1 + e^{-r \nu t} \cdot C \ z(0) = 1 + C \Rightarrow \ C = \Big(\frac{N_0}{K}\Big)^{-\nu} - 1 \Rightarrow \ z(t) = 1 + e^{-r \nu t} \Big(\frac{N_0}{K}\Big)^{-\nu} \Rightarrow \ N(t) = \frac{K}{\Big[1 - \Big( 1- \Big( \frac{K}{N_0}\Big)^{\nu}\Big) e^{-r \nu t} \Big]^{1/\nu}} $$

Derivation

Following (Schnute, 1981), we define the population size $N$ and the per capita growth rate $Z$:

$$ \frac{dN}{dt} = N Z \Rightarrow Z = \frac{1}{N} \frac{dN}{dt} \ \frac{dZ}{dt} = \nu Z ( Z-r) = \nu Z^2 -\nu r Z $$

Integrating this system with the initial conditions $N(0)=N0, \lim{t \to \infty}{N(t)} = K$ gives the same solution as above.

According to (Schnute, 1981), this differential equation system can also be written as a second-order differential equation:

$$ \frac{d^2N}{dt^2} = \ \frac{dN}{dt} \Big( \frac{1 + \nu}{N} \cdot \frac{dN}{dt} -r\nu \Big) $$

This derivation allows a new interpretation of the model parameters:

  • $r$ is the maxmimum per capita growth rate, and
  • $\nu$ is the per capita decelaration rate of the per capita growth rate.

References