Designing a Resilient Allee-Ornstein-Uhlenbeck model

Abstract

In stochastic population dynamics, stochastic wandering can produce transition to an absorbing state. In particular, under Allee effects, low densities amplify the possibility of population collapse. We investigate this in an Allee-Ornstein-Uhlenbeck (Allee-OU) model, that couples a bistable Allee growth equation, with demographic noise, and environmental fluctuations modeled as an Ornstein-Uhlenbeck process. This process replaces the bifurcation parameter of the deterministic Allee effect equation. In the model, small noise may induce escape from the safe basin around the positive equilibrium toward extinction. We construct a stochastic control, altering the process to have a stationary distribution. We enable tractable control design, approximating the process by one with a stationary distribution. Two controlled models are developed, one acting directly on population size and another also modulating the environment. A threshold-based implementation minimizes the frequency of interventions while maximizing safe time. Simulations demonstrate that the control stabilizes fluctuations around the equilibrium.

Figures (8)

• Figure 1: (a) Phase portrait for the equation $d\mathbf{X}/dt=F(\mathbf{X})$, with $\lambda=0.097$ and $\rho_{\text{center}}=0.247$. Equilibria are marked with a black (stable) and open (unstable) circles. (b) Positive equilibria and isoclines of the components of $d\mathbf{X}/dt=F(\mathbf{X})$ (red parabola and grey continuous vertical line) for $\lambda=0.01$ and $\rho_{\text{center}}=0.247$. The basin of attraction of the stable equilibrium is determined by the separatrix emerging from the unstable equilibrium (solid blue).
• Figure 2: Sample paths of the process $\mathbf{X}_t$ starting at the stable equilibrium $\mathbf{X}_s^*$, with parameters $\lambda=0.01$, $\rho_{\text{center}}=0.247$, $\phi=0.04$, $\psi=0.1$ and $\epsilon=0.005$. Green dots along the paths are time markers at each 150 time steps, indicating where the path lingers most of the time.
• Figure 3: Histogram of times to extinction given a fixed computational duration of $4\times 10^{4}$ time steps. Extinction time is approximated by the first crossing of the level $x=0.4$. As illustrated in Figure \ref{['fig: path simulations']}, sample paths may cross the separatrix and subsequently be attracted to the equilibrium $[0\quad \rho_{\text{center}}]'$, corresponding to population extinction. The parameters used are the same as in Figure \ref{['fig: path simulations']}.
• Figure 4: Confidence ellipse for the approximate model with stationary law and level of confidence $\alpha=0.95$ plotted together with a sample path of the process $\mathbf{X}_t$, solution of equation (\ref{['eq: main model']}), with $\mathbf{X}_0=\mathbf{X}_s^*$ and time horizon $T$. Our impression from the plot is that the approximating stationary law is a reasonable proxi for the law of the process $\mathbf{X}_t$ within the chosen $T$. Parameters for the process are the same as in Figure 2.
• Figure 5: Sample paths of the system (\ref{['eq: controlled system']}) with control $U$ starting at $\mathbf{X}_s^*$ and plotted on top of Figure \ref{['fig: mean field']}(b) only for reference. Confidence ellipses under the approximate stationary law for the controlled process are the thick red curves. (a) $U$ acts on $x$ and $\rho$. (b) $U$ acts only on $x$. Contrasting with Figure \ref{['fig: confidence ellipse']}, the controlled paths are kept tightly close to the stable equilibrium $\mathbf{X}_s^*$ in both cases. All the parameters and the level of confidence, $\alpha$, are as in Figure \ref{['fig: confidence ellipse']}.