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# A sustainability condition for stochastic forest model

This work was supported by JSPS KAKENHI Grant Number 20140047, This work was supported by JSPS Grant-in-Aid for Scientific Research (No. 26400166)
• A stochastic forest model of young and old age class trees is studied. First, we prove existence, uniqueness and boundedness of global nonnegative solutions. Second, we investigate asymptotic behavior of solutions by giving a sufficient condition for sustainability of the forest. Under this condition, we show existence of a Borel invariant measure. Third, we present several sufficient conditions for decline of the forest. Finally, we give some numerical examples.

Mathematics Subject Classification: Primary: 37H10; Secondary: 47D07.

 Citation: • • 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 $t\in [0, 10]$

Table 1.  Stability and instability of stationary solutions of (1)

 h (0, h*) (h*, h*) (h*, ∞) O unstable stable glob. asymp. stable P+ stable stable − P− − unstable −
• Figures(7)

Tables(1)

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