| 研究実績の概要 |
I have studied stochastic parabolic evolution equations and some mathematical models in biology and ecology.
A. Stochastic parabolic evolution equations. The equations are of the form: dX=[AX+F(t)+G(X)]dt + K(t)dW(t). I have obtained results on existence, uniqueness, maximal regularity and regular dependence on initial data of solutions to stochastic linear and semilinear evolution equations by presenting new techniques in the semigroup approach.
B. Mathematical models. I have investigated three models: Stochastic Forest Ecosystem Model, Lotka-Volterra Model and Swarm Behaviour Model. For the first model, we achieved results on existence, uniqueness and boundedness of global positive solutions as well as existence of an invariant measure. We showed some sufficient conditions for sustainability of forest and proved that under the effect of large noise, the forest falls into decline. For the second one, we obtained results on existence of global solutions, periodic solutions and their stability. For the last one, we investigated mathematically the process of fish schooling by constructing and studying some stochastic ordinary differential equations based on the rules presented by Camazine-Deneubourg-Franks-Sneyd-Theraulaz-Bonabeau. Our models describe the movement of fish in environment with noise and obstacles. We obtained an insight of fish school cohesiveness by observing how obstacle-avoiding pattern changes as modelling parameters change.
The results have been submitted to some academic journals for publications.
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| 今後の研究の推進方策 |
I am currently studying non-autonomous evolution equations and semilinear evolution equation with additive noise. This is motivated by internal development of the theory of stochastic processes on one side, and by a need to study some our models (for example, Stochastic Diffusion Coat Pattern and Stochastic Forest Ecosystem) on the other side (to the best of our knowledge, the previous results do not seem strong enough to be applied to such models). The framework includes the following problems:
1. Existence and uniqueness of (global and local, strict and mild) solutions.
2. Regularity and regular dependence on initial data of solutions.
3. Long-time behaviour of solutions.
Once the task if fulfilled, it can be applied to Stochastic Diffusion Coat Pattern and Stochastic Forest Ecosystem to obtain more information on these systems.
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