Stochastic Sensitivity Analysis of Noise-Induced Extinction in the Ricker Model with Delay and Allee Effect

Abstract

A susceptibility of population systems to the random noise is studied on the base of the conceptual Ricker-type model taking into account the delay and Allee effect. This two-dimensional discrete model exhibits the persistence in the form of equilibria, discrete cycles, closed invariant curves, and chaotic attractors. It is shown how the Allee effect constrains the persistence zones with borders defined by crisis bifurcations. We study the role of random noise on the contraction and destruction of these zones. This phenomenon of the noise-induced extinction is investigated with the help of direct numerical simulations and semi-analytical approach based on the stochastic sensitivity functions. Stochastic transitions from the persistence regimes to the extinction are studied by the analysis of the mutual arrangement of the basins of attraction and confidence domains.

Appendix

As a mathematical model of many population systems, the following generalized discrete dynamic system is frequently used

$$\begin{aligned} x_{t+1}=f(x_{t}). \end{aligned}$$

(4)

Here, x is an n-vector and f(x) is a sufficiently smooth vector-function.

An influence of random factors can be studied on the base of the stochastic system

$$\begin{aligned} x_{t+1}=f(x_{t}) + \varepsilon \sigma (x_{t}) \xi _{t}, \end{aligned}$$

(5)

where $\sigma (x)$ is $n\times m$-matrix, $\xi _{t}$ is m-dimensional uncorrelated random process with parameters ${\mathrm E}\xi _{t} = 0, \; {\mathrm E}\xi _{t}\xi _{t}^\top =I,\; {\mathrm E}\xi _{t}\xi _{k}^\top =0 (t\ne k)$. Here, I is an identity $m\times m$-matrix. The scalar parameter $\varepsilon$ defines an intensity of noise.

Under the random disturbances, around attractors of deterministic system (4), a stationary probabilistic distribution $p(x,\varepsilon )$ of random states of system (5) is formed. For the approximation of this distribution in the presence of small noise, a method of stochastic sensitivity functions was proposed in Bashkirtseva et al. (2010) for equilibria and discrete cycles, and in Bashkirtseva and Ryashko (2014b) for closed invariant curves.

Stochastic sensitivity of the equilibrium

Let $\bar{x}=f(\bar{x})$ be an exponentially stable equilibrium of system (4). The stochastic sensitivity matrix W of the equilibrium $\bar{x}$ is governed by the following matrix equation (Bashkirtseva et al. 2010)

$$\begin{aligned} W = FWF^\top + Q,\quad F=\frac{\partial f}{\partial x}(\bar{x}),\quad Q=\sigma (\bar{x})\sigma ^\top (\bar{x}). \end{aligned}$$

(6)

Using the matrix W, one can get the Gaussian approximation

$$\begin{aligned} p(x,\varepsilon )\approx K\exp \left( -\frac{(x-\bar{x},W^{-1}(x-\bar{x}))}{\varepsilon ^2}\right) . \end{aligned}$$

The stochastic sensitivity matrix W characterizes the response of system (5) to small random disturbances near the equilibrium $\bar{x}$.

Using the stochastic sensitivity matrix W, one can get a simple geometric description of the dispersion of random states near the equilibrium in the form of confidence ellipses.

For $n=2$, the stochastic sensitivity matrix W has eigenvalues $\eta _{1}, \eta _{2} > 0$, and corresponding eigenvectors $u_{1}, u_{2}$. A confidence ellipse around the equilibrium $\bar{x}$ can be written as

$$\begin{aligned} (x - \bar{x}, W^{-1}(x-\bar{x})) = 2q^2 \varepsilon ^2, \end{aligned}$$

where $q^2 = -\ln (1 - P)$ and P is a fiducial probability. For coordinates $z_1 = (x - \bar{x}, u_{1}), z_{2} = (x - \bar{x}, u_{2})$, the equation of the confidence ellipse can be written in the canonic form:

$$\begin{aligned} \frac{z_{1}^2}{\eta _{1}} + \frac{z_{2}^2}{\eta _{2}} = 2q^2 \varepsilon ^2. \end{aligned}$$

Stochastic sensitivity of the discrete cycle

Let $\{\bar{x}_{1}, \ldots , \bar{x}_{k}\}$ be an exponentially stable k-cycle of deterministic system (4). Elements of the cycle are connected by equalities

$$\begin{aligned} f(\bar{x}_{i}) = \bar{x}_{i+1}\,(i = 1, \ldots , k-1),\quad f(\bar{x}_{k}) = \bar{x}_{1}. \end{aligned}$$

In this case, the stochastic sensitivity of the cycle is defined by the consequence of matrices $W_{1}, \ldots , W_{k}$, where $W_{i}$ is a stochastic sensitivity of the element $\bar{x}_{i}$.

The matrix $W_{1}$ is governed by the following Eq. (Bashkirtseva et al. 2010)

$$\begin{aligned} W_{1} = BW_{1}B^\top + Q. \end{aligned}$$

(7)

Other matrices $W_{2}, \ldots , W_{k}$ can be found recurrently:

$$\begin{aligned} W_{i+1} = F_{i} W_i F^\top _{i} + Q_{i},\quad i=1, \ldots , k-1. \end{aligned}$$

(8)

Here,

$$\begin{aligned} F_i= & {} \frac{\partial f}{\partial x}(\bar{x}_{i}),\quad Q_{i}=\sigma (\bar{x}_{i}) \sigma ^\top (\bar{x}_{i}),\quad B=F_{k}\cdot \ldots \cdot F_{2} F_{1},\\ Q= & {} Q_{k} + F_{k}Q_{k-1}F_{k}^\top + \cdots + F_{k}\cdot \ldots \cdot F_{2}Q_{1}F_{2}^\top \cdot \ldots \cdot F_{k}^\top . \end{aligned}$$

Stochastic sensitivity of the closed invariant curve

Let the closed invariant curve $\varGamma$ be an attractor of deterministic system (4).

At first consider a case when the closed curve $\varGamma$ consists of the family of k-cycles of deterministic system (4). This means that for any point $\bar{x} \in \varGamma$, the solution $\bar{x}_t$ of deterministic system (4) with the initial value $\bar{x}_1=\bar{x}$ is k-periodic. At the points $\{\bar{x}_{1} , \ldots , \bar{x}_{k}\}$ of the invariant curve $\varGamma$, the stochastic sensitivity is defined by the matrices $W_{1}, \ldots , W_{k}$, where $W_{i}$ is a stochastic sensitivity of the element $\bar{x}_{i}$.

The first matrix $W_1$ is a solution (Bashkirtseva and Ryashko 2014b) of the following equation:

$$\begin{aligned} W_1 =P_1[F W_1 F^\top +Q] P_1, \qquad F= F_kP_k F_{k-1}\cdot \dots \cdot P_2F_1,\qquad Q=Q^{(k)}. \end{aligned}$$

(9)

The matrix $Q^{(k)}$ can be found recurrently:

$$\begin{aligned} Q^{(0)}= & {} 0,\;\;Q^{(j)}=P_{j+1}\left[ F_j Q^{(j-1)} F_j^\top +G_j\right] P_{j+1}\;\;(j=1,\dots k-1),\\ Q^{(k)}= & {} F_k Q^{(k-1)} F_k^\top +G_k. \end{aligned}$$

Here,

$$\begin{aligned} F_t=\frac{\partial f}{\partial x}(\bar{x}_t),\;\;G_t=\sigma (\bar{x}_t) \sigma ^\top (\bar{x}_t). \end{aligned}$$

The matrix $P_t$ is a matrix of the projection on to the hyperplane $\varPi _t$ that is orthogonal to the curve $\varGamma$ at the point $\bar{x}_t$. Other matrices $W_2,\dots W_k$ can be found recurrently:

$$\begin{aligned} W_{t+1} =P_{t+1}[F_t W_t F_t^\top +G_t]P_{t+1}. \end{aligned}$$

(10)

For $n=2$, one can use the factorization $W_1=m_1 p_1 p_1^{\top }$, where $p_1$ is an orthonormal vector to $\varGamma$ at the point $\bar{x}_1$. The function $m_1$ is scalar, and the following explicit formula holds:

$$\begin{aligned} m_1=\frac{p_1^\top Q p_1}{1-\left( p_1^\top F p_1\right) ^2}. \end{aligned}$$

For small $\varepsilon$, the value $\varepsilon ^2 m_1$ approximates the dispersion of the random states of system (5) at the point $\bar{x}_1$ in the normal direction to the closed curve $\varGamma$.

Using the value $m(\bar{x})$ of the stochastic sensitivity at the point $\bar{x}$, one can find a confidence interval on the line that is orthogonal to $\varGamma$ at $\bar{x}$. Boundaries of this interval are $x_{1,2}= \bar{x} \pm \varepsilon q \sqrt{2m(\bar{x})}p(\bar{x}).$ Here, the parameter $q=\mathrm {erf}^{-1}(P)$ (P is a fiducial probability), and $\mathrm {erf}(x)=\frac{2}{\sqrt{\pi }}\int \limits _0^x e^{-t^2} \mathrm{d}t$ is the error function. Under the variation of $\bar{x}$ along the closed invariant curve $\varGamma$, the points $x_{1}, x_{2}$ describe a confidence band around $\varGamma$.

In the case when the closed curve $\varGamma$ consists of the family of quasiperiodic solutions of deterministic system (4), we can find the stochastic sensitivity using the approximation of $\varGamma$ by a periodic sequence (Bashkirtseva and Ryashko 2014b).