Since each creature interacts with each other through a food chain, an analysis of the food chain is a crucial problem in an ecosystem. In this paper, we consider the plankton-fish system consisting of fish, zooplankton and phytoplankton, and study the influence of the random noise on the stability of the plankton system under fish school by numerical simulations. The qualitative change of the solutions of differential equations by variation of a system parameter is generally called a bifurcation. This paper is concerned with the bifurcation analysis of a stochastic plankton-fish system. As major analytical methods for the stochastic bifurcation, the D-bifurcation (Dynamical bifurcation) and the P-bifurcation (Phenomenological bifurcation) approaches are cited. The D-bifurcation considers the stability of invariant measures, while the P-bifurcation is characterized with a qualitative change of the stationary probability distribution, e.g., a transition from unimodal to bimodal distribution. By the numerical simulations, we show that not the D-bifurcation but the P-bifurcation occurs in the stochastic plankton-fish system considered here.
In this paper, the author investigates the convergent properties of estimators concerned with expectations for a class of vague random phenomena, where the fuzzy random set is considered as a model of the capricious vague perception of a crisp phenomenon or a crisp random phenomenon.
First, fuzzy random sets as vague random perceptions of non-random or random phenomena are investigated, and their expectations are also considered. Secondly applying the standard Strong Law of Large Numbers(SLLN) for the random elements in a separable Banach space, the convergent properties of estimators for expectations of the proposed fuzzy random sets are examined, and finally they are confirmed numerically by simulation studies.
In this paper, we are concerned with a problem of optimization of the linear observations that are used in the stationary Kalman-Bucy filter. Especially, we consider the optimization of the gain matrix in the observation. In the previous works of the author, the corresponding problem for discrete-time systems was already considered and the condition of optimality was obtained. This paper is concerned with the case of the continuous-time systems and it is shown that the condition of optimality is given by the same form as the discrete-time case except for the accompanying Lyapunov equation, which is continuous-type whereas it was discrete-type in the discrete-time problem. We propose a method of solving the set of equations of the Riccati equation for the error covariance and the condition of optimality by a simple recursive algorithm. The results of numerical experiments show the efficiency of the proposed algorithm.
This paper deals with a consensus problem for directed two-layered networks in which dynamics of agents are driven by topologies on both cooperative and competitive graphs. A distributed algorithm for exchanging information between agents in the networks is introduced, where the communication noises are taken into account. A sufficient condition for the proposed algorithm to achieve the socalled ε-averaging consensus, i.e., system state gets arbitrarily close to the average value of agents' states with desired high probability, is shown. A rigorous stopping rule for the algorithm is also derived. It is shown that all the agents' states converge to the average value of the initial states when the modeling parameters are chosen appropriately.