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A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential
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 |
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.
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. |
[2] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972. |
[3] |
L. H. Chuan and A. Yagi, Dynamical system for forest kinematic model, Adv. Math. Sci. Appl., 16 (2006), 393-409. |
[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. |
[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. |
[6] |
S. R. Foguel, The ergodic theory of positive operators on continuous functions, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 19-51. |
[7] |
A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1976.
![]() ![]() |
[8] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, NorthHolland, Tokyo, 1981. |
[9] |
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-1-4612-0949-2. |
[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. |
[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. |
[12] |
X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, 2008.
doi: 10.1533/9780857099402. |
[13] |
L. Michael, Conservative Markov processes on a topological space, Isr. J. Math., 8 (1970), 165-186.
doi: 10.1007/BF02771312. |
[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. |
[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. |
[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. |
[17] |
A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010. |
show all references
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. |
[2] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972. |
[3] |
L. H. Chuan and A. Yagi, Dynamical system for forest kinematic model, Adv. Math. Sci. Appl., 16 (2006), 393-409. |
[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. |
[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. |
[6] |
S. R. Foguel, The ergodic theory of positive operators on continuous functions, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 19-51. |
[7] |
A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 1976.
![]() ![]() |
[8] |
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, NorthHolland, Tokyo, 1981. |
[9] |
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-1-4612-0949-2. |
[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. |
[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. |
[12] |
X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, 2008.
doi: 10.1533/9780857099402. |
[13] |
L. Michael, Conservative Markov processes on a topological space, Isr. J. Math., 8 (1970), 165-186.
doi: 10.1007/BF02771312. |
[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. |
[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. |
[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. |
[17] |
A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer, Berlin, 2010. |




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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