# 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.