Stochastic Logistic Growth Process

Definition

The Stochastic Logistic Growth Process is a birth and death process where birth and death rates are quadratic functions of the population size:

${\lambda }_i=b_{1}i+b_2{i}^2>0,{\mu }_i=d_{1}i+d_2{i}^2>0$

where the coefficients satisfy:

$r=b_1-d_1>0\text{(intrinsic growth rate)}$ $K=\frac{b_1-d_1}{d_2-b_2}>0\text{(carrying capacity)}$

Interpretation

This is the stochastic analog of the deterministic logistic equation $\frac{dn}{dt}=rn(1-n/K)$. There are infinitely many choices of coefficients $(b_1,b_2,d_1,d_2)$ that yield the same $r$ and $K$, so there are infinitely many stochastic logistic models that correspond to the same deterministic model.

Properties

Extinction: Extinction occurs with probability 1 (${\lim }_{t\to \infty }{p}_0(t)=1$) and the expected time to extinction is finite. This follows from the ratio test applied to the extinction condition, since $d_2>b_2\ge 0$ implies death rates dominate at large population sizes.

Mean vs. Deterministic: The mean $m(t)$ of the stochastic process satisfies:

$\frac{dm}{dt}=rm(t)-\frac{r}{K}\mathbb{E}[X^2(t)]<rm(t)\left[1-\frac{m(t)}{K}\right]$

Thus $m(t)\le n(t)$ where $n(t)$ is the deterministic solution — the stochastic mean is always less than the deterministic prediction.

Density Dependence:

• If $b_2<0$: birth rate decreases with population size (crowding effect on births)
• If $d_2>0$: death rate increases with population size (crowding effect on deaths)

Finite vs Infinite State Space

For finite state space $\{0,1,\dots ,N\}$: ${\lambda }_i=0$ for $i\ge N$, extinction is certain.

For infinite state space: The condition $d_2>b_2$ ensures the ratio test confirms certain extinction.

Related

• Birth and Death Process
• Probability of Population Extinction
• Expected Time to Extinction
• Quasistationary Probability Distribution