Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge
Juan Li Yongzhong Song Hui Wan Huaiping Zhu
Mathematical Biosciences & Engineering 2017, 14(2): 529-557 doi: 10.3934/mbe.2017032

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type Ⅱ response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.

keywords: Phytoplankton zooplankton refuge toxin algal blooms multiple equilibria stability Hopf bifurcation Bogdanov-Takens bifurcation
Licensing schemes in Stackelberg model under asymmetric information of product costs
Qing-you Yan Juan-bo Li Ju-liang Zhang
Journal of Industrial & Management Optimization 2007, 3(4): 763-774 doi: 10.3934/jimo.2007.3.763
Based on the conclusion that have been put forward by Sougata Poddar and Uday Bhanu Sinha (2002), in "The Role of Fixed Fee and Royalty in Patent Licensing", we construct a model to analyze the behavior of the patentee when the licensee have private information about its marginal product costs before licensing. In the paper, the patentee is not considered as an independent R&D institute anymore but an insider. Moreover, we extend the model to Homogeneous Stackelberg and present the patentee's optimal linear licensing schemes in the sense of maximizing its profits under the licensing contract. It is found that the expected profit of the patentee under the separating contract is higher than any pooling contract in this model.
keywords: Homogeneous Stackelberg Asymmetric Information Licensing.
Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs
Juan Li Wenqiang Li
Mathematical Control & Related Fields 2015, 5(3): 501-516 doi: 10.3934/mcrf.2015.5.501
In this paper, we consider a new type of reflected mean-field backward stochastic differential equations (reflected MFBSDEs, for short), namely, controlled reflected MFBSDEs involving their value function. The existence and the uniqueness of the solution of such equation are proved by using an approximation method. We also adapt this method to give a comparison theorem for our reflected MFBSDEs. The related dynamic programming principle is obtained by extending the approach of stochastic backward semigroups introduced by Peng [11] in 1997. Finally, we show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle.
keywords: Reflected mean-field backward stochastic differential equations (reflected MFBSDEs) dynamic programming principle viscosity solution. comparison theorem reflected MFBSDEs coupled with value function

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