Reducing the extinction risk of stochastic populations via nondemographic noise

We consider nondemographic noise in the form of uncertainty in the reaction step size and reveal a dramatic effect this noise may have on the stability of self-regulating populations. Employing the reaction scheme $mA\to kA$ but allowing, e.g., the product number $k$ to be a priori unknown and sampled from a given distribution, we show that such nondemographic noise can greatly reduce the population's extinction risk compared to the fixed $k$ case. Our analysis is tested against numerical simulations, and by using empirical data of different species, we argue that certain distributions may be more evolutionary beneficial than others.

Figure 1

Frequency of appearance of different litter sizes. The bars correspond to empirical data for (a) a laboratory mouse [30], (b) a guinea pig [31], (c) a spotted hyena [32], and (d) a wild boar [33]. The lines correspond to binomial distribution fits (see the text) with best fit parameters of (a) $m=15$ and $p=0.775$, (b) $m=5$ and $p=0.75$, (c) $m=3$ and $p=0.575$, (d) and $m=9$ and $p=0.6$. The insets correspond to the ratio ${\tau }^{\mathrm{emp}}/{\tau }^{\mathrm{ref}}$ as a function of $N$ with $B=2$ (see the text).

Figure 2

MTE as a function of $\gamma$. The lines are obtained by numerically evaluating Eq. (3). The markers correspond to numerical Monte Carlo simulation results. Panels (a) and (b) correspond to the $K$-step and binomial cases, respectively. Parameters: $B=2,N=150$, and $m=10$.

Figure 3

Logarithm of the ratio between the binomial and reference MTEs versus $m$ and $K$. ($▴$) markers are obtained by numerically evaluating Eq. (3) with $B=2$ and $N=2000$. ($•$) markers correspond to Eq. (4) with $B=1.15$ and $N=25000$. Panel (a): $m=9$ (for which $K_{\mathrm{c}}=7$). Panel (b): $K=7$ (for which $m_{\mathrm{c}}=9$).

Figure 4

Three-dimensional plots of the logarithm of the ratio between the binomial and reference MTEs as a function of $m$ and $K$ in the (a) bifurcation and (b) nonbifurcation regimes. In (a) and (b), respectively, the results are obtained using Eq. (4), and by numerically evaluating Eq. (3). Along the solid lines in panels (a) and (b), ${\tau }^{\mathrm{bin}}={\tau }^{\mathrm{ref}}$. In panels (c) and (d) we plot sections of panels (a) and (b), putting $K=m-R$ for $R=0,1$. The maxima in panel (c), obtained at $m_{\mathrm{opt}}=2,8$, are accurately predicted by Eq. (6). In panel (d) the maxima at $m_{\mathrm{opt}}=2,7$ are still closely predicted by Eq. (6). Here $B=1.125$ and $N=15000$ for panels (a) and (c), and $B=2$ and $N=400$ for panels (b) and (d).