随机环境下生物资源保护及最优捕获问题
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摘要
二十世纪以来，生物资源的持续利用问题倍受世人关注。本文用数学模型从理论上研究了生物资源的保护及最优捕获问题。考虑到不同类型的噪音干扰，本文建立了三个随机保护区模型，并研究其一系列渐近性质。本文还研究了两类具有多种噪音干扰的随机最优捕获问题。本文的研究内容包括以下几个方面：
（1）由白噪音和有色噪音共同驱动的具有单参数干扰的随机保护区模型的动力学性质。利用随机分析技巧研究了模型的一些渐近性质，包括各种持久性与灭绝性。分析了保护区的正面效果并给出提高保护区效果的多种措施和划分保护区的方法。
（2）由白噪音驱动的具有双参数干扰的随机保护区模型的动力学性质。这个模型考虑到了环境白噪音对生物种群生长模型中两个参数的干扰。通过研究模型的一系列渐近性质说明了种群的内禀增长率是该模型中对环境干扰反应最为灵敏的一个系数。
（3）由白噪音和有色噪音共同驱动的具有双参数干扰的随机保护区模型的动力学性质。将（1）和（2）综合起来进行了全面的分析，得到随机环境下保护区的建立能够很好的起到保护生物种群的作用，当环境条件非常恶劣时，保护区的建立也能够减缓生物种群灭绝的速度。
（4）随机环境下Gompertz模型的最优捕获问题。分别研究了由布朗运动驱动和由Le′vy过程驱动的随机Gompertz模型的最优捕获问题。用遍历性的方法证明了期望和平稳概率密度意义下的目标函数和最优捕获问题的等价性。本文利用遍历方法来解决随机最优捕获问题。
（5）随机环境下Logistic模型的最优捕获问题。分别研究了由Le′vy过程驱动的以及由Le′vy过程和Markov过程共同驱动的Logistic模型的最优捕获问题。用遍历性方法证明了关于时间平均和平稳概率密度目标函数在最优捕获问题上的等价性。为随机模型的最优捕获问题提供了一种新的思路，避免了求解Fokker-Planck方程精确解的难题。
Utilization of renewable biological resources has received great attention from theentire world since twentieth century. Based on mathematical models, this paper studiesthe protection and the optimal harvesting problems of biological resources theoretical-ly. Considering diferent types of environmental noise, this paper builds three stochasticmodels with protection zone, and studies the corresponding asymptotic properties. Thispaper also studies the optimal harvesting problems for two types of models with severalkinds of noise. The content of this paper includes the following several aspects:
(1) The dynamical properties for a stochastic protection zone model with interfer-ence on single parameter which driven by both white noise and color noise. Using s-tochastic analysis techniques, some asymptotic properties are studied, such as persistenceand extinction. We analyze the positive efects of protection zone, give several measuresto improve the efects and get a method of dividing protection zone.
(2) The dynamical properties for a stochastic protection zone model with interfer-ence on two parameters which driven by white noise. This model considers that twoparameters in the population growth model are disturbed by white noise. Through theresearch of a series of asymptotic properties for this model, we illustrate that the intrinsicgrowth rate of biological population is the most sensitive coefcient under environmentaldisturbance.
(3) The dynamical properties for a stochastic protection zone model with interfer-ence on two parameters which driven by both white noise and color noise. We give anoverall analysis by synthesizing both (1) and (2). Results say establishing protection zonecan protect the biological population nicely under the random environment, and it alsocan slow down the speed of extinction under terrible circumstance.
(4) Optimal harvesting problem for Gompertz models in random environment. Westudy the optimal harvesting problems for stochastic Gompertz models driven by Brownmotion and Le′vy process respectively. The equivalency of optimal harvesting problem-s under expectation and stationary probability density meanings is obtained by ergodicmethod. This paper applies ergodic method on the optimal harvesting problem.
(5) Optimal harvesting problem for stochastic Logistic models. The optimal harvest-ing problems for stochastic Logistic models driven by Le′vy process and driven by both Le′vy process and Markov process are studied respectively. We show the equivalency ofobjective function on optimal harvesting problem between time average and stationaryprobability density meanings by ergodic method. This provide us a new idea for stochas-tic harvesting problem which can avoid the puzzle of solving Fokker-Planck equation.
（1）由白噪音和有色噪音共同驱动的具有单参数干扰的随机保护区模型的动力学性质。利用随机分析技巧研究了模型的一些渐近性质，包括各种持久性与灭绝性。分析了保护区的正面效果并给出提高保护区效果的多种措施和划分保护区的方法。
（2）由白噪音驱动的具有双参数干扰的随机保护区模型的动力学性质。这个模型考虑到了环境白噪音对生物种群生长模型中两个参数的干扰。通过研究模型的一系列渐近性质说明了种群的内禀增长率是该模型中对环境干扰反应最为灵敏的一个系数。
（3）由白噪音和有色噪音共同驱动的具有双参数干扰的随机保护区模型的动力学性质。将（1）和（2）综合起来进行了全面的分析，得到随机环境下保护区的建立能够很好的起到保护生物种群的作用，当环境条件非常恶劣时，保护区的建立也能够减缓生物种群灭绝的速度。
（4）随机环境下Gompertz模型的最优捕获问题。分别研究了由布朗运动驱动和由Le′vy过程驱动的随机Gompertz模型的最优捕获问题。用遍历性的方法证明了期望和平稳概率密度意义下的目标函数和最优捕获问题的等价性。本文利用遍历方法来解决随机最优捕获问题。
（5）随机环境下Logistic模型的最优捕获问题。分别研究了由Le′vy过程驱动的以及由Le′vy过程和Markov过程共同驱动的Logistic模型的最优捕获问题。用遍历性方法证明了关于时间平均和平稳概率密度目标函数在最优捕获问题上的等价性。为随机模型的最优捕获问题提供了一种新的思路，避免了求解Fokker-Planck方程精确解的难题。
Utilization of renewable biological resources has received great attention from theentire world since twentieth century. Based on mathematical models, this paper studiesthe protection and the optimal harvesting problems of biological resources theoretical-ly. Considering diferent types of environmental noise, this paper builds three stochasticmodels with protection zone, and studies the corresponding asymptotic properties. Thispaper also studies the optimal harvesting problems for two types of models with severalkinds of noise. The content of this paper includes the following several aspects:
(1) The dynamical properties for a stochastic protection zone model with interfer-ence on single parameter which driven by both white noise and color noise. Using s-tochastic analysis techniques, some asymptotic properties are studied, such as persistenceand extinction. We analyze the positive efects of protection zone, give several measuresto improve the efects and get a method of dividing protection zone.
(2) The dynamical properties for a stochastic protection zone model with interfer-ence on two parameters which driven by white noise. This model considers that twoparameters in the population growth model are disturbed by white noise. Through theresearch of a series of asymptotic properties for this model, we illustrate that the intrinsicgrowth rate of biological population is the most sensitive coefcient under environmentaldisturbance.
(3) The dynamical properties for a stochastic protection zone model with interfer-ence on two parameters which driven by both white noise and color noise. We give anoverall analysis by synthesizing both (1) and (2). Results say establishing protection zonecan protect the biological population nicely under the random environment, and it alsocan slow down the speed of extinction under terrible circumstance.
(4) Optimal harvesting problem for Gompertz models in random environment. Westudy the optimal harvesting problems for stochastic Gompertz models driven by Brownmotion and Le′vy process respectively. The equivalency of optimal harvesting problem-s under expectation and stationary probability density meanings is obtained by ergodicmethod. This paper applies ergodic method on the optimal harvesting problem.
(5) Optimal harvesting problem for stochastic Logistic models. The optimal harvest-ing problems for stochastic Logistic models driven by Le′vy process and driven by both Le′vy process and Markov process are studied respectively. We show the equivalency ofobjective function on optimal harvesting problem between time average and stationaryprobability density meanings by ergodic method. This provide us a new idea for stochas-tic harvesting problem which can avoid the puzzle of solving Fokker-Planck equation.
引文
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