A sustainability condition for stochastic forest model

Figure 1.  Sample trajectories of $u_t$ and $v_t$ of (2) with parameters: $a=2, b=1, c=2.5, f=4, h=1, \rho=5, \sigma=0.5$ and initial value $(u_0, v_0)=(2, 1).$ The left figure illustrates a sample trajectory of $(u_t, v_t)$ in the phase space; the right figure illustrates sample trajectories of $u_t$ and $v_t$ along $t\in [0,100]$

Figure 2.  Distribution of $(u_t, v_t)$ of (2) at $t=10^3$. The parameters and initial value are taken as in the legend of Fig. 1

Figure 3.  Graphs of $\mathbb Eu$ and $\mathbb Ev$ along $ t\in [0, 20]$. The parameters and initial value are taken as in the legend of Fig. 1

Figure 4.  Sample trajectory of two processes $I$ and $J$ defined by $I(t)=\frac{1}{t}\int_0^t u_sds$ and $J(t)=\frac{1}{t}\int_0^t v_sds$ along $ t\in [0,100].$ The parameters and initial value are taken as in the legend of Fig. 1

Figure 5.  Graph of probability functions $R$ and $S$ defined by $R(t)=\mathbb P\{(u_t, v_t)\in A; (u_0, v_0)=(2, 1)\}$ and $S(t)=\mathbb P\{(u_t, v_t)\in A; (u_0, v_0)=(3, 4)\}$ along $t\in [50,100], $ where $A=[0.5, 30]\times[0.1, 20]$ and the parameters of (2) are taken as in the legend of Fig. 1. These functions are calculated on the basis of 2000 sample trajectories of $(u_t, v_t)$ corresponding to each initial value

Figure 6.  Decline of forest under the effect of noise with large intensity $\sigma$. Here, $a=3, b=4, c=5, f=6, h=2, \rho=7, \sigma=4$ and initial value $(u_0, v_0)=(4, 3)$. The left figure is a sample trajectory of $(u_t, v_t)$ in the phase space; the right figure is a sample trajectory of $u$ and $v$ along $t\in [0, 1]$

Figure 7.  Decline of forest when the mortality $h$ of old trees is large. Here, $a=3, b=4, c=5, f=6, h=3.82, \rho=7, \sigma=0.25$ and initial value $(u_0, v_0)=(4, 3).$ The figure gives a graph of $\mathbb Eu$ and $\mathbb Ev$ along $t\in [0, 10]$