Stationary Distribution, Extinction and Probability Density Function of a Stochastic Vegetation–Water Model in Arid Ecosystems

Abstract

In this paper, we study a three-dimensional stochastic vegetation–water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution $\varpi (\cdot )$. In a biological interpretation, the existence of the distribution $\varpi (\cdot )$ implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium $\overline{D}^*$ related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker–Planck equation, we obtain the approximate expression of the distribution $\varpi (\cdot )$ around the equilibrium $\overline{D}^*$. In addition, we obtain sufficient condition $\mathscr {R}_0^E<1$ for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics.

Appendix A. (Local stability of system (1.1))

In this section, we will focus on the local stability of the equilibria $D_0$ and $D^*$ of system (1.1).

Theorem A.1

If ${\mathscr {R}}_0<1$, the vegetation-free equilibrium $D_0$ of system (1.1) is locally asymptotically stable (LAS), but it is unstable when ${\mathscr {R}}_0>1$.

Proof

The Jacobi matrix of system (1.1) at the equilibrium $D_0$ is

$$\begin{aligned} J(D_0)=\left( \begin{array}{ccc} R_\mathrm{esp}({\mathscr {R}}_0-1) &{} \quad 0 &{} \quad 0 \\ \frac{R(1-w_0)}{k_2w_0}-\frac{q\gamma R_\mathrm{esp}{\mathscr {R}}_0}{c} &{} \quad -r_w &{} \quad \alpha w_0 \\ -\frac{R(1-w_0)}{k_2w_0} &{} \quad 0 &{} \quad -\alpha w_0 \end{array}\right) . \end{aligned}$$

By direct calculation, the characteristic polynomial of $D_0$ is

$$\begin{aligned} \phi _{J(D_0)}(y)=[y-R_\mathrm{esp}({\mathscr {R}}_0-1)](y+r_w)(y+\alpha w_0). \end{aligned}$$

Clearly, $J(D_0)$ has three real eigenvalues $y_1=R_\mathrm{esp}({\mathscr {R}}_0-1)$, $y_2=-r_w<0$ and $y_3=-\alpha w_0<0$. If ${\mathscr {R}}_0<1$, then $J(D_0)\in {\overline{RH}}(3)$. Combining Definition 2.1 and the Routh–Hurwitz criterion (Ma et al. 2015), we obtain that $E_0$ is LAS when ${\mathscr {R}}_0<1$. Conversely, if ${\mathscr {R}}_0>1$, we get that $y_1=R_\mathrm{esp}({\mathscr {R}}_0-1)>0$, implying that $D_0$ is unstable. This completes the proof of Theorem A.1. $\square$

Next, we define a critical value by

$$\begin{aligned} {\mathscr {R}}_1=\Bigl [\frac{c(R+r_\mathrm{w}k_1)({\mathscr {R}}_0-1)}{q\gamma (c\alpha _2 g_{\mathrm{co}_2}-R_\mathrm{esp})}+k_2w_0\Bigr ]^2-\frac{ck_2R(1-w_0)}{q\alpha \gamma }. \end{aligned}$$

Theorem A.2

If ${\mathscr {R}}_0>1$ and ${\mathscr {R}}_1\ge 0$, the vegetation-positive equilibrium $D^*$ is LAS.

Proof

Similar to Theorem A.1, the Jacobi matrix of system (1.1) at the equilibrium $D^*$ is

$$\begin{aligned} J(D^*)= & {} \left( \begin{array}{ccc} 0 &{} \quad \frac{c\alpha _2g_{\mathrm{co}_2}k_1P^*}{(W^*+k_1)^2} &{} \quad 0 \\ \frac{\alpha k_2(1-w_0)S^*}{(P^*+k_2)^2}-\frac{q\alpha _2\gamma g_{\mathrm{co}_2}W^*}{W^*+k_1} &{} \quad -\frac{q\alpha _2\gamma g_{\mathrm{co}_2}k_1P^*}{(W^*+k_1)^2}-r_w &{} \quad \frac{\alpha (P^*+k_2w_0)}{P^*+k_2} \\ -\frac{\alpha k_2(1-w_0)S^*}{(P^*+k_2)^2} &{} \quad 0 &{} \quad -\frac{\alpha (P^*+k_2w_0)}{P^*+k_2} \end{array}\right) \\:= & {} \left( \begin{array}{ccc} 0 &{} \quad a_{12} &{} \quad 0 \\ a_{21} &{} \quad -a_{22} &{} \quad a_{23} \\ a_{31} &{} \quad 0 &{} \quad -a_{23} \end{array} \right) , \end{aligned}$$

where $a_{12}=\frac{c\alpha _2g_{\mathrm{co}_2}k_1P^*}{(W^*+k_1)^2}>0$, $a_{21}=\frac{\alpha k_2(1-w_0)S^*}{(P^*+k_2)^2}-\frac{q\alpha _2\gamma g_{\mathrm{co}_2}W^*}{W^*+k_1}$, $a_{22}=\frac{q\alpha _2\gamma g_{\mathrm{co}_2}k_1P^*}{(W^*+k_1)^2}+r_w>0$, $a_{23}=\frac{\alpha (P^*+k_2w_0)}{P^*+k_2}>0$ and $a_{31}=\frac{\alpha k_2(1-w_0)S^*}{(P^*+k_2)^2}>0$. A direct calculation shows that

$$\begin{aligned} \phi _{J(D^*)}(y)=y^3+l_1y^2+l_2y+l_3, \end{aligned}$$

where $l_1=a_{22}+a_{23}>0$, $l_2=a_{22}a_{23}-a_{12}a_{21}$ and $l_3=a_{12}a_{23}(a_{31}-a_{21})$.

If ${\mathscr {R}}_0>1$, we determine that $P^*=\frac{c(R+r_\mathrm{w}k_1)({\mathscr {R}}_0-1)}{q\gamma (c\alpha _2g_{\mathrm{co}_2}-R_\mathrm{esp})}$ and $a_{31}-a_{21}=\frac{q\alpha _2\gamma g_{\mathrm{co}_2}W^*}{W^*+k_1}>0$, which means that $l_3>0$. Moreover, if ${\mathscr {R}}_1\ge 0$, by the equality $R=\frac{\alpha (P^*+k_2w_0)S^*}{P^*+k_2}$, we have

$$\begin{aligned} \begin{aligned} a_{22}a_{23}-a_{12}a_{31}=&\frac{\alpha (P^*+k_2w_0)}{P^*+k_2}\Bigl [\frac{q\alpha _2\gamma g_{\mathrm{co}_2}k_1P^*}{(W^*+k_1)^2}+r_w\Bigr ]\\&-\frac{c\alpha _2g_{\mathrm{co}_2}k_1k_2R(1-w_0)P^*}{(W^*+k_1)^2(P^*+k_2)(P^*+k_2w_0)}\\ \ge&\frac{q\alpha \alpha _2\gamma g_{\mathrm{co}_2}k_1P^*}{(W^*+k_1)^2(P^*+k_2)(P^*+k_2w_0)}\\&\Bigl [(P^*+k_2w_0)^2-\frac{ck_2R(1-w_0)}{q\alpha \gamma }\Bigr ]\\\ =&\frac{q\alpha \alpha _2\gamma g_{\mathrm{co}_2}k_1P^*{\mathscr {R}}_1}{(W^*+k_1)^2(P^*+k_2)(P^*+k_2w_0)}\ge 0. \end{aligned} \end{aligned}$$

Combined with $a_{21}<a_{31}$, we obtain that $l_2=a_{22}a_{23}-a_{12}a_{21}>a_{22}a_{23}-a_{12}a_{31}\ge 0$ and

$$\begin{aligned} l_1l_2-l_3=a_{22}l_2+a_{23}(a_{22}a_{23}-a_{12}a_{31})\ge a_{22}l_2>0. \end{aligned}$$

According to the Routh–Hurwitz criterion, we determine that $J(D^*)\in {\overline{RH}}(3)$. Thus, $D^*$ is LAS when ${\mathscr {R}}_0>1$ and ${\mathscr {R}}_1\ge 0$. This completes the proof of Theorem A.2. $\square$