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Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics
Authors:Dong-Wei Huang  Hong-Li Wang  Jian-Feng Feng  Zhi-Wen Zhu
Institution:1. School of Science, Tianjin Polytechnic University, No. 63, ChengLinZhuang Road, HeDong District, Tianjin 300160, PR China;2. School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China
Abstract:In this paper, we consider the growth of densities of two kinds of typical HAB algae: diatom and dianoflagellate on some coasts of China’s mainland. Since there exist many random factors that cause the change of the algae densities, we shall develop a new nonlinear dynamical model with stochastic excitations on the algae densities. Applying a stochastic averaging method on the model, we obtain a two-dimensional diffusion process of averaged amplitude and phase. Then we investigate the stability and the Hopf bifurcation of the stochastic system with FPK (Fokker Planck–Kolmogorov) theory and obtain the stationary transition probability density of the process. We obtain the critical values of parameters for the occurrences of Hopf bifurcation in terms of probability. We also investigate numerically the effects of various parameters on the stationary transition probability density of the occurrences of Hopf bifurcation. The numerical results are in good correlation with the analysis. We draw the conclusion that if the Hopf bifurcation occurs with a radius large enough, i.e., if the densities of the HAB algae reach a high value, the HAB will take place with comparatively high probability.
Keywords:HAB (harmful algae blooms)  Nonlinear stochastic dynamic system  FPK theory  Stochastic Hopf bifurcation  Stochastic averaging method
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