# inordinatum

Physics and Mathematics of Disordered Systems

## Stochastic Logistic Growth

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A classic model for the growth of a population with a finite carrying capacity is logistic growth. It is usually formulated as a differential equation,

$\displaystyle \partial_t n(t)=r\, n(t)\left[1-\frac{n(t)}{K}\right]$. (1)

Here $n(t)$ is the size of the population at time $t$, $r$ is the growth rate and $K$ is the carrying capacity.

The dynamics of eq. (1) is deterministic: The initial population $n(0)=n_0 > 0$ grows (or decays) towards the constant carrying capacity $n(t\to\infty) = K$, which is a fixed point of eq. (1). This is seen in the solid trajectories in the figure below:

Deterministic vs. stochastic logistic growth. Solid lines: deterministic model in eq. (1). Transparent lines: Realizations of stochastic logistic growth model in eq. (2) below. The colors blue, orange and green correspond to $K=50, 30, 10$. In all cases $r=1$.

To make this model more realistic, let’s see how we can extend it to include stochastic fluctuations (semi-transparent trajectories in the figure above). I’ll look in the following at a simple stochastic logistic growth model (motivated by some discussions with a friend), where the steady state can be calculated exactly. The effect of stochasticity on the steady state is twofold:

• The population size for long times is not fixed, but fluctuates on a scale $\sigma \approx \sqrt{K}$ around the carrying capacity.
• The average population size $\left\langle n \right\rangle$ is increased above the carrying capacity $K$, but the shift goes to 0 as $K$ increases (i.e. the deterministic model is recovered for large $K$).

Now let’s look at the calculation in detail…

## A stochastic, individual-based logistic growth model

In eq. (1), the population is described by a real-valued function $N(t)$. In reality, populations consist of discrete individuals and the population size doesn’t change continuously. So, a more realistic approach is to describe the population dynamics as a birth-death process with the following reactions:

$\begin{array}{rl} A & \stackrel{r}{\longrightarrow} A+A \\ A+A & \stackrel{d}{\longrightarrow} A\end{array}$ (2)

In other words, we assume that during a time interval $dt$ two events may happen:

• Birth: With a probability $r \, dt$, any individual may give birth to offspring, thus increasing the population size by 1.
• Death due to competition: With a probability $d\, dt$, out of any two individuals one may die due to the competition for common resources. Thus the population size decreases by 1. Note that $d$ is related to the carrying capacity $K$ in eq. (1) by $d = r/K$.

Note that the stochasticity in this model is not due to random external influences but due to the discreteness of the population (demographic noise).

## Solution of the stochastic model

The system of reaction equations (2) translates into the following master equation for the probabilities $p_n(t)$ of the population size at time $t$ being equal to $n$:

$\begin{array}{rl} \displaystyle \partial_t p_n(t) = & \displaystyle n(n+1)d\,p_{n+1}(t) - n(n-1)d\, p_n(t) \\ & \displaystyle + (n-1)r\,p_{n-1}(t) - nr\, p_n(t)\end{array}$ (3)

This looks daunting. However, it can be simplified a lot by introducing the generating function

$\displaystyle G(\lambda,t) := \sum_{n=0}^{\infty} \lambda^n p_n(t)$

After some algebra we obtain

$\displaystyle \partial_t G(\lambda,t) = \lambda(1-\lambda) \left[d\, \partial_\lambda^2 G(\lambda,t) - r \,\partial_\lambda G(\lambda,t)\right]$.

This looks simpler than eq. (3) but still finding the full time-dependent solution $G(\lambda,t)$ does not seem feasible. Let’s focus on the steady state where $\partial_t G(\lambda,t)=0$. Together with the boundary conditions $G(\lambda=1,t)=1$ and $G(\lambda=0,t)=0$, we obtain the steady-state solution $G_s(\lambda)$

$\displaystyle G_s(\lambda) = \frac{e^{K\lambda}-1}{e^{K}-1}$. (4)

Here we set $d=r/K$ to connect to the notation of eq. (1). Correspondingly, the steady-state probabilities for population size $n$ are

$\displaystyle p_n = \frac{K^n}{n!}\frac{1}{e^K-1}$. (5)

Of course, these can equivalently also be obtained by solving the recursion relation (3) with $\partial_t p_n(t) = 0$. This result can be easily checked against simulations, see figure below.

Stationary probability distribution for the stochastic logistic growth model in eq. (2), for $r=1, K=5, d=r/K=0.2$. Blue line: exact solution in eq. (5). Yellow bars: Histogram over $10^6$ samples.

## Steady state with stochasticity

Let’s try to understand what the steady state of our stochastic model looks like. From eq. (4) we easily obtain the first moments of the distribution of the population size. The mean population size is:

$\displaystyle \left\langle n \right\rangle = \partial_\lambda\big|_{\lambda=1}G_s(\lambda) = \frac{K}{1-e^{-K}}$

So for large carrying capacities $K \gg 1$, we recover the same result as in the deterministic model (which is reasonable since for large population sizes demographic noise is less pronounced!). However, for smaller $K$ fluctuations increase the average population size. E.g. for $K=2$, the average population size in our stochastic model is $2/\left(1-e^{-2}\right) \approx 2.31$.

Now let us look at the variance,

$\begin{array}{rl} \sigma^2 := &\displaystyle \left\langle n^2\right\rangle - \left\langle n \right\rangle^2\\=&\displaystyle\partial_\lambda\big|_{\lambda=1}\lambda \partial_\lambda G_s(\lambda)-\left(\frac{K}{1-e^{-K}}\right)^2\\=& \displaystyle\frac{K\left(1-e^{-K}-K\,e^{-K}\right)}{\left(1-e^{-K}\right)^2}\end{array}$.

For large $K \gg 1$, we have $\sigma^2 \approx K$. So we see that stochastic fluctuations spread out the steady state to a width $\sigma \approx \sqrt{K}$ around the carrying capacity.