# 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, Lobry, and Sari 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, Lobry, and Sari 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, Lobry, and Sari 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 and 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 \(N(0)=N_0, \lim_{t \to \infty}{N(t)} = K\) 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

Arditi, Roger, Claude Lobry, and Tewfik Sari. 2015. “Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation.” *Theoretical Population Biology* 106 (December): 45–59. doi:10.1016/j.tpb.2015.10.001.

Gilpin, Michael E., and Francisco J. Ayala. 1973. “Global Models of Growth and Competition.” *Proceedings of the National Academy of Sciences of the United States of America* 70 (12 Pt 1-2): 3590–93. doi:10.1073/pnas.70.12.3590.

Richards, F. J. 1959. “A Flexible Growth Function for Empirical Use.” *Journal of Experimental Botany* 10 (2): 290–301. doi:10.1093/jxb/10.2.290.

Schnute, Jon. 1981. “A Versatile Growth Model with Statistically Stable Parameters.” *Canadian Journal of Fisheries and Aquatic Sciences* 38 (9): 1128–40. doi:10.1139/f81-153.

Skiadas, Christos H. 2010. “Exact Solutions of Stochastic Differential Equations : Gompertz , Generalized Logistic and Revised Exponential,” 261–70. doi:10.1007/s11009-009-9145-3.