Stochastic foundations in nonlinear density-regulation growth

In this work we construct individual-based models that give rise to the generalized logistic model at the mean-field deterministic level and that allow us to interpret the parameters of these models in terms of individual interactions. We also study the effect of internal fluctuations on the long-time dynamics for the different models that have been widely used in the literature, such as the theta-logistic and Savageau models. In particular, we determine the conditions for population extinction and calculate the mean time to extinction. If the population does not become extinct, we obtain analytical expressions for the population abundance distribution. Our theoretical results are based on WKB theory and the probability generating function formalism and are verified by numerical simulations.

Figure 1

MTE vs $\lambda$ for the processes given by $\text{A}\stackrel{\lambda }{\to }2\text{A}$ and $c\text{A}\stackrel{\mu }{\to }⌀$, for $\mu =3$. The value of $N$ is given by (15). Symbols correspond to numerical simulations performed up to time ${10}^8$ and averaging over ${10}^4$ realizations. Solid curves for $c=3,4,5$ are, respectively, obtained from Eqs. (32), (33), and Eq. (30) with $c=5$.

Figure 2

MTE for the processes given by $\text{A}\stackrel{\lambda }{\to }2\text{A}$ and $c\text{A}\stackrel{\mu }{\to }⌀$ vs $K$, for $\lambda =100$ and varying $\mu$. Symbols correspond to numerical simulations performed up to time ${10}^8$ and averaging over $1.5\times {10}^4$ realizations. Solid curves for $c=2,3,4$ are respectively obtained from Eqs. (31), (32), and (33).

Figure 3

Population abundance $P\left(n\right)$ as a function of $n$ for $K=100$. Theoretical results (lines) from Eqs. (40), (44), and (48) are compared with numerical simulations (symbols), where we have performed ${10}^4$ realizations from an initial population equal to $K$ up to ${10}^7$ time steps.

Figure 4

Mean population abundance $\langle n\rangle$ divided by $K$. Shown are theoretical results from Eqs. (41), (45), and (49). The mean population abundance approaches $K$ as $K\to \infty$.

Figure 5

Plot of the MTE for three different reaction schemes with different values of $d$. The parameters are $r=600$ [see Eq. (53)], $c=3$, and $K=20$. Symbols correspond to numerical simulations performed up to time ${10}^9$ and averaging over ${10}^4$ realizations. Solid curves are obtained from Eq. (61) for $d=1,2,3$.

Figure 6

Probability of extinction Q(n) starting from $n$ individuals. We have considered here $\mu =6,N=100$ and $b=2,3,4$. The symbols correspond to numerical simulations, and the solid curves are obtained from the exact solutions Eqs. (76), (77), and (78). The number of events considered in the simulations is ${10}^7$.