Allee effects plus noise induce population dynamics resembling binary Markov highs and lows (Q2141300)

This paper proposes new flexible deterministic population dynamics models of the form \[ \frac{1}{x}\frac{\mathrm d x}{\mathrm d t} = \alpha x^q \left(1- \frac{x}{K}\right)- \mu\tag{1} \] and \[ \frac{1}{x}\frac{\mathrm d x}{\mathrm d t} = \hat \alpha x^q \left(1- \frac{x}{K}\right),\tag{2} \] where the parameters \(\alpha,\hat \alpha,\mu, K\) are positive and the nonnegative \(q\) expresses the intensity of the attenuated Allee effect. For \(q=1\) equations (1) and (2) reduce to the classical models of Volterra with strong and weak Allee effect, respectively. The authors study the consequences of adding environmental and demographic stochasticity to the dynamics. Numerical results suggest how these mechanisms generate a succession of sudden decrease, low phase, and increase in some populations. The transition times between the two types of states are observed to be approximately exponentially distributed, with different parameters, rendering the embedded hi-low process approximately Markov. The results also contain a reasonable caution for control programs that take advantage of Allee effects in trying to achieve pest eradication.