# American Institute of Mathematical Sciences

March  2017, 16(2): 699-718. doi: 10.3934/cpaa.2017034

## A sustainability condition for stochastic forest model

 1 Promotive Center for International Education and Research of Agriculture, Faculty of Agriculture, Kyushu University, 6-10-1 Hakozaki, Nishi-ku, Fukuoka 812-8581, Japan 2 Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan 3 Department of Applied Physics, Graduate School of Engineering, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan

Received  May 2016 Revised  November 2016 Published  January 2017

Fund Project: 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.

Citation: TÔn Vı$\underset{.}{\overset{\hat{\ }}{\mathop{\text{E}}}}\,$T T$\mathop {\text{A}}\limits_.$, Linhthi hoai Nguyen, Atsushi Yagi. A sustainability condition for stochastic forest model. Communications on Pure & Applied Analysis, 2017, 16 (2) : 699-718. doi: 10.3934/cpaa.2017034
##### References:
 [1] M. Ya. Antonovsky, Impact of the factors of the environment on the dynamics of population (mathematical model), in Proc. Soviet-American Symp. Comprehensive Analysis of the Environment, Tbilisi 1974, Leningrad: Hydromet, (1975), 218-230. Google Scholar [2] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972.  Google Scholar [3] L. H. Chuan and A. Yagi, Dynamical system for forest kinematic model, Adv. Math. Sci. Appl., 16 (2006), 393-409.  Google Scholar [4] L. H. Chuan, T. Tsujikawa and A. Yagi, Asymptotic behavior of solutions for forest kinematic model, Funkcial. Ekvac., 49 (2006), 427-449. doi: 10.1619/fesi.49.427.  Google Scholar [5] L. H. Chuan, T. Tsujikawa and A. Yagi, Stationary solutions to forest kinematic model, Glasg. Math. J., 51 (2009), 1-17. doi: 10.1017/S0017089508004485.  Google Scholar [6] S. R. Foguel, The ergodic theory of positive operators on continuous functions, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 19-51.  Google Scholar [7] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1976.   Google Scholar [8] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, NorthHolland, Tokyo, 1981.  Google Scholar [9] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [10] P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE through Computer Experiments, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.  Google Scholar [11] Yu. A. Kuznetsov, M. Ya. Antonovsky, V. N. Biktashev and E. A. Aponina, A cross-diffusion model of forest boundary dynamics, J. Math. Biol., 32 (1994), 219-232. doi: 10.1007/BF00163879.  Google Scholar [12] X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [13] L. Michael, Conservative Markov processes on a topological space, Isr. J. Math., 8 (1970), 165-186. doi: 10.1007/BF02771312.  Google Scholar [14] L. T. H. Nguyen and T. V. Ta., Dynamics of a stochastic ratio-dependent predator-prey model, Anal. Appl. (Singap.), 9 (2011), 329-344. doi: 10.1142/S0219530511001868.  Google Scholar [15] T. Shirai, L. H. Chuan and A. Yagi, Asymptotic behavior of solutions for forest kinematic model under Dirichlet conditions, Sci. Math. Jpn., 66 (2007), 289-301.  Google Scholar [16] T. V. Ta., L. T. H. Nguyen and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl. (Singap.), 12 (2014), 63-73.  Google Scholar [17] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010.  Google Scholar

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##### References:
 [1] M. Ya. Antonovsky, Impact of the factors of the environment on the dynamics of population (mathematical model), in Proc. Soviet-American Symp. Comprehensive Analysis of the Environment, Tbilisi 1974, Leningrad: Hydromet, (1975), 218-230. Google Scholar [2] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972.  Google Scholar [3] L. H. Chuan and A. Yagi, Dynamical system for forest kinematic model, Adv. Math. Sci. Appl., 16 (2006), 393-409.  Google Scholar [4] L. H. Chuan, T. Tsujikawa and A. Yagi, Asymptotic behavior of solutions for forest kinematic model, Funkcial. Ekvac., 49 (2006), 427-449. doi: 10.1619/fesi.49.427.  Google Scholar [5] L. H. Chuan, T. Tsujikawa and A. Yagi, Stationary solutions to forest kinematic model, Glasg. Math. J., 51 (2009), 1-17. doi: 10.1017/S0017089508004485.  Google Scholar [6] S. R. Foguel, The ergodic theory of positive operators on continuous functions, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 19-51.  Google Scholar [7] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1976.   Google Scholar [8] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, NorthHolland, Tokyo, 1981.  Google Scholar [9] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar [10] P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE through Computer Experiments, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-57913-4.  Google Scholar [11] Yu. A. Kuznetsov, M. Ya. Antonovsky, V. N. Biktashev and E. A. Aponina, A cross-diffusion model of forest boundary dynamics, J. Math. Biol., 32 (1994), 219-232. doi: 10.1007/BF00163879.  Google Scholar [12] X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [13] L. Michael, Conservative Markov processes on a topological space, Isr. J. Math., 8 (1970), 165-186. doi: 10.1007/BF02771312.  Google Scholar [14] L. T. H. Nguyen and T. V. Ta., Dynamics of a stochastic ratio-dependent predator-prey model, Anal. Appl. (Singap.), 9 (2011), 329-344. doi: 10.1142/S0219530511001868.  Google Scholar [15] T. Shirai, L. H. Chuan and A. Yagi, Asymptotic behavior of solutions for forest kinematic model under Dirichlet conditions, Sci. Math. Jpn., 66 (2007), 289-301.  Google Scholar [16] T. V. Ta., L. T. H. Nguyen and A. Yagi, Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl. (Singap.), 12 (2014), 63-73.  Google Scholar [17] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010.  Google Scholar
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
. These functions are calculated on the basis of 2000 sample trajectories of $(u_t, v_t)$ corresponding to each initial value">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
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 $t\in [0, 1]$
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 $t\in [0, 10]$
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 −
 h (0, h*) (h*, h*) (h*, ∞) O unstable stable glob. asymp. stable P+ stable stable − P− − unstable −
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