Computing probabilistic solutions of the Bernoulli random differential equation

In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measu...

Mean exit time and escape probability for the stochastic logistic growth model with multiplicative α-stable Lévy noise

Establishment of cost-effective management strategy of aquaculture is one of the most important issues in fishery science, which can be addressed with bio-economic mathematical modeling. This paper deals with the aforementioned issue using a stochastic process model for aquacultured non-renewable fishery resources from the viewpoint of an optimal stopping (timing) problem. The goal of operating the model is to find the optimal criteria to start harvesting the resources under stochastic environment, which turns out to be determined from the Bellman equation (BE). The BE has a separation of variables type structure and can be simplified to a reduced BE with a fewer degrees of freedom. Dependence of solutions to the original and reduced BEs on parameters and independent variables is analyzed from both analytical and numerical standpoints. Implications of the analysis results to management of aquaculture systems are presented as well. Numerical simulation focusing on aquacultured Plecoglossus altivelis in Japan validates the mathematical analysis results. Copyright © 2017 John Wiley & Sons, Ltd.

Optimization model to start harvesting in stochastic aquaculture system

In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker-Planck equation for fish population X(t). In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness, and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker-Planck equation as growth rate r, carrying capacity K, the intensity of Gaussian noise ${\lambda}$, noise intensity ${\sigma}$ and stability index ${\alpha}$ vary. The MET from the interval (0,1) at the right boundary is finite if ${\lambda} {\sqrt2}$, the MET from (0,1) at this boundary is infinite. A larger stability index ${\alpha}$ is less likely to lead to the extinction of the fish population.

Mean Exit Time and Escape Probability for the Stochastic Logistic Growth Model with Multiplicative {\alpha}-Stable L\'evy Noise

Book reviewManagement system dynamics: R.G. COYLE, U.K., 1977, 463 pages, Hardback £16.95, $ 32.00, Paperback £6.95, $ 13.00

Two extensions of
stochastic logistic model for fish growth have been examined. The basic
features of a logistic growth rate are deeply influenced by the carrying
capacity of the system and the changes are periodical with time. Introduction
of a new parameter , enlarges the scope of investing the growthof different fish
species. For rapid growth  lying between 1 and 2 and for
slowly growing.

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Md. Asaduzzaman Shah

Papers

Stochastic Logistic Model for Fish Growth