Stochastic dynamics and logistic population growth

The Verhulst model is probably the best known macroscopic rate equation in population ecology. It depends on two parameters, the intrinsic growth rate and the carrying capacity. These parameters can be estimated for different populations and are related to the reproductive fitness and the competition for limited resources, respectively. We investigate analytically and numerically the simplest possible microscopic scenarios that give rise to the logistic equation in the deterministic mean-field limit. We provide a definition of the two parameters of the Verhulst equation in terms of microscopic parameters. In addition, we derive the conditions for extinction or persistence of the population by employing either the momentum-space spectral theory or the real-space Wentzel-Kramers-Brillouin approximation to determine the probability distribution function and the mean time to extinction of the population. Our analytical results agree well with numerical simulations.

Figure 1

Stationary PDF for $X\stackrel{\lambda }{\to }2X$ and $2X\stackrel{\mu }{\to }X$ with (a) $N=100$, (b) $N=25$, (c) $N=200$, and (d) $N=50$. Simulation results (symbols) are based on 3000 realizations of the stochastic process up to time ${10}^6$.

Figure 2

Stationary PDF for $X\stackrel{\lambda }{\to }3X$ and $2X\stackrel{\mu }{\to }X$ with (a) $N=200$, (b) $N=50$, (c) $N=400$, and (d) $N=100$. Simulation (symbols) results are based on 3000 realizations of the stochastic process up to time ${10}^6$.

Figure 3

Coefficient of variation $c_v$ versus $N$ for $a=1$ and $a=2$. The log-log plot in the inset shows that $c_v$ decays like $N^{-1/2}$. Simulation results are based on 3000 realizations of the stochastic process up to time ${10}^6$. We set $\mu =2$ and $d=1$ and vary $\lambda$.

Figure 4

Stationary PDF for $X\stackrel{\lambda }{\to }3X$ and $2X\stackrel{\mu }{\to }\varnothing$ with (a) $N=200$, (b) $N=40$, (c) $N=400$, and (d) $N=100$. Simulation (symbols) results are based on 3000 realizations of the stochastic process up to time ${10}^6$.

Figure 5

Coefficient of variation versus $N$ for $a=2$ and $a=4$. The inset demonstrates that $c_v$ decays like $N^{-1/2}$. Simulation results are based on 3000 realizations of the stochastic process up to time ${10}^6$. We set $\mu =d=2$ and vary $\lambda$.

Figure 6

Mean time to extinction $\tau$ vs (a) $N$ and (b) $a$ for the reactions $X\stackrel{\lambda }{\to }(a+1)X$ and $2X\stackrel{\mu }{\to }\varnothing$. Solid curves are obtained from (3.45), while symbols correspond to numerical simulations. We set $\mu =2$ and vary $\lambda$. Simulations have been performed up to time ${10}^8$.

Figure 7

Mean time to extinction $\tau$ vs (a) $N$ and (b) $a$ for the reactions $X\stackrel{\lambda }{\to }(a+1)X$ and $2X\stackrel{\mu }{\to }\varnothing$. Solid curves are obtained from (3.50), while symbols correspond to numerical simulations. We set $\mu =2$ and vary $\lambda$. Simulations have been performed up to time ${10}^8$.

Figure 8

Mean time to extinction for birth-and-death reactions. In both cases we set $\mu =0.1$, $\lambda =10$, $d=1$, and $N=100$ and modify $\gamma$ to vary $R_0$. We compare different values for the MET for (a) different $d$ and the same $a$ and (b) different $a$ and the same $b$. Simulations (symbols) have been performed up to time ${10}^9$ and mean values are obtained by averaging over $4\times {10}^4$ realizations. Solid curves correspond to exact analytic results given by (4.33), (4.35), and (4.38).