随机种群模型若干性质的研究
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摘要
近些年来，确定性生物种群模型受到了广泛的重视并被广泛地研究。然而在现实世界中生物种群系统不可避免地受到环境噪声的影响，因此研究随机生物种群模型进而揭示环境噪声对于系统的影响就非常的重要。本文致力于研究几个重要的和基本的随机生物种群模型的动力学行为，主要包含以下内容：
1.随机非自治Logistic模型的持久性和灭绝性。由于其重要性，对于Logistic模型的研究已经是而且必将依然是生物数学里最有意思的方向之一。而该模型所表示的物种何时生存以及何时灭绝是非常有意思和重要的问题。本文研究了两个广泛应用的随机非自治Logistic模型，并分别给出了各个模型生存和灭绝的阈值。本文证明了如果阈值是正的，那么物种将会生存；如果阈值是负的，那么物种将会灭绝。
2.具有有色噪声扰动的推广的Logistic模型。本文建立了物种灭绝、非平均持久生存、弱持久生存和随机持久的充分条件，得到了模型生存和灭绝的阈值，并对其渐近性质进行了研究。结果表明不同类型的随机噪声对物种的生存和灭绝有不同的影响。
3.随机Lotka-Volterra互惠模型的持久性和灭绝性。对于每个物种，本文建立了其生存和灭绝的阈值，得到了系统随机持久的充分条件。从结果容易看出，随机噪声对于系统中所有物种的生存都不利。
4.具有Holling II型功能反应的随机捕食者-食饵模型的持久性和灭绝性。对于每一物种，本文建立了其生存和灭绝的充分条件，并在一定条件下得到了阈值。
5.提出了具有Beddington-DeAngelis型功能反应的随机捕食者-食饵模型。本文证明了虽然该系统的系数既不满足线性增长条件也不满足局部Lipschitz条件，该系统仍然具有全局的正解，并给出了系统正平衡点全局渐近稳定的充分条件。建立了具有Beddington-DeAngelis型功能反应和阶段结构的随机捕食者-食饵模型，并且给出了其正平衡点全局渐近稳定的充分条件。结果都揭示了如果随机噪声强度不大并且原来确定性系统的平衡点是全局渐近稳定的，那么随机系统将会保留这样的良好性质。
In recent years, deterministic population systems have received great attention andhave been studied extensively. However, population dynamics in the real world is in-evitably afected by environmental noises. Then it is important to study stochastic pop-ulation systems to reveal the efect of environmental noises on population systems. Thispaper devotes to investigating the dynamical properties of some important and basic s-tochastic population systems, the main results are as follows:
1. Persistence and extinction in stochastic non-autonomous Logistic systems. Theinvestigation of logistic system has long been and will continue to be one of the dominantthemes in mathematical ecology due to its importance. Persistence and extinction of thismodel is one of the most interesting and important topics. This paper studies two widelyused stochastic non-autonomous Logistic models and obtains the critical number betweenpersistence and extinction for each system. This paper proves that if the critical numberis positive, then the population is persistence; if the critical number is negative, then thepopulation goes to extinction.
2. Generalized stochastic Logistic model under regime switching. This paper estab-lishes sufcient condition for extinction, non-persistence in the mean, weak persistenceand stochastic permanence and obtains the critical number between persistence and ex-tinction. Asymptotic properties of this model are also studied. Results show that a difer-ent type of noise has a diferent efect on the persistence and extinction of the population.
3. Persistence and extinction of a stochastic Lotka-Volterra cooperation model. Foreach species, the critical number between persistence and extinction is obtained, sufcientcondition for stochastic permanence is established. From our results, it is easy to see thatthe stochastic noise is unfavorable for the persistence of all species.
4. Persistence and extinction of a stochastic predator-prey system with Holling IIfunctional response. Sufcient criteria for extinction and persistence for each species areestablished. The persistence-extinction threshold for each species is obtained in manycases.
5. A stochastic predator-prey system with Beddington-DeAngelis functional re-sponse is proposed. This paper shows that, although the coefcients of the system neithersatisfy the linear growth condition, nor local Lipschitz continuous, the model still has a globally positive solution, and this paper establishes sufcient conditions for stochas-tically asymptotic stability in the large of the positive equilibrium. A stage-structuredpredator-prey model with Beddington-DeAngelis functional response with stochastic per-turbation is considered, and this paper establishes the sufcient conditions for stochasti-cally asymptotic stability in the large of the positive equilibrium. These results revealthat if the positive equilibrium of the deterministic system is globally stable, then thestochastic model will preserve this nice property provided the noise is sufciently small.
1.随机非自治Logistic模型的持久性和灭绝性。由于其重要性，对于Logistic模型的研究已经是而且必将依然是生物数学里最有意思的方向之一。而该模型所表示的物种何时生存以及何时灭绝是非常有意思和重要的问题。本文研究了两个广泛应用的随机非自治Logistic模型，并分别给出了各个模型生存和灭绝的阈值。本文证明了如果阈值是正的，那么物种将会生存；如果阈值是负的，那么物种将会灭绝。
2.具有有色噪声扰动的推广的Logistic模型。本文建立了物种灭绝、非平均持久生存、弱持久生存和随机持久的充分条件，得到了模型生存和灭绝的阈值，并对其渐近性质进行了研究。结果表明不同类型的随机噪声对物种的生存和灭绝有不同的影响。
3.随机Lotka-Volterra互惠模型的持久性和灭绝性。对于每个物种，本文建立了其生存和灭绝的阈值，得到了系统随机持久的充分条件。从结果容易看出，随机噪声对于系统中所有物种的生存都不利。
4.具有Holling II型功能反应的随机捕食者-食饵模型的持久性和灭绝性。对于每一物种，本文建立了其生存和灭绝的充分条件，并在一定条件下得到了阈值。
5.提出了具有Beddington-DeAngelis型功能反应的随机捕食者-食饵模型。本文证明了虽然该系统的系数既不满足线性增长条件也不满足局部Lipschitz条件，该系统仍然具有全局的正解，并给出了系统正平衡点全局渐近稳定的充分条件。建立了具有Beddington-DeAngelis型功能反应和阶段结构的随机捕食者-食饵模型，并且给出了其正平衡点全局渐近稳定的充分条件。结果都揭示了如果随机噪声强度不大并且原来确定性系统的平衡点是全局渐近稳定的，那么随机系统将会保留这样的良好性质。
In recent years, deterministic population systems have received great attention andhave been studied extensively. However, population dynamics in the real world is in-evitably afected by environmental noises. Then it is important to study stochastic pop-ulation systems to reveal the efect of environmental noises on population systems. Thispaper devotes to investigating the dynamical properties of some important and basic s-tochastic population systems, the main results are as follows:
1. Persistence and extinction in stochastic non-autonomous Logistic systems. Theinvestigation of logistic system has long been and will continue to be one of the dominantthemes in mathematical ecology due to its importance. Persistence and extinction of thismodel is one of the most interesting and important topics. This paper studies two widelyused stochastic non-autonomous Logistic models and obtains the critical number betweenpersistence and extinction for each system. This paper proves that if the critical numberis positive, then the population is persistence; if the critical number is negative, then thepopulation goes to extinction.
2. Generalized stochastic Logistic model under regime switching. This paper estab-lishes sufcient condition for extinction, non-persistence in the mean, weak persistenceand stochastic permanence and obtains the critical number between persistence and ex-tinction. Asymptotic properties of this model are also studied. Results show that a difer-ent type of noise has a diferent efect on the persistence and extinction of the population.
3. Persistence and extinction of a stochastic Lotka-Volterra cooperation model. Foreach species, the critical number between persistence and extinction is obtained, sufcientcondition for stochastic permanence is established. From our results, it is easy to see thatthe stochastic noise is unfavorable for the persistence of all species.
4. Persistence and extinction of a stochastic predator-prey system with Holling IIfunctional response. Sufcient criteria for extinction and persistence for each species areestablished. The persistence-extinction threshold for each species is obtained in manycases.
5. A stochastic predator-prey system with Beddington-DeAngelis functional re-sponse is proposed. This paper shows that, although the coefcients of the system neithersatisfy the linear growth condition, nor local Lipschitz continuous, the model still has a globally positive solution, and this paper establishes sufcient conditions for stochas-tically asymptotic stability in the large of the positive equilibrium. A stage-structuredpredator-prey model with Beddington-DeAngelis functional response with stochastic per-turbation is considered, and this paper establishes the sufcient conditions for stochasti-cally asymptotic stability in the large of the positive equilibrium. These results revealthat if the positive equilibrium of the deterministic system is globally stable, then thestochastic model will preserve this nice property provided the noise is sufciently small.
引文
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