红松种群数学模型及其动力学行为研究
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摘要
红松是我国小兴安岭和东北东部山区的优势保护树种。红松材质优良,松籽富含油脂,经济价值很高。以红松为主形成的阔叶红松林是中国温带针阔混交林地区的顶级植被,也是中国木材和林副产品的重要产地。阔叶红松林具有复杂的群落结构和稳定的更替过程,对保护自然资源、维持生态平衡有重要作用。
红松的天然更新取决于种源和幼苗幼树的生长条件,而种源由母树产生的种子的传播条件和种子的消耗等因素决定。红松林中,种子是鼠类(主要指松鼠,也考虑花鼠、大林姬鼠和一些鸟类)的主要取食对象之一,幼苗的更新主要依赖鼠类的贮食行为,种子、鼠类和幼苗三种群互相影响和制约。鼠类和红松共同组成了一个典型的红松林生态系统。
数学模型已被广泛应用于经济、生态与环境科学中。本文依据数学生态学建模方法定量研究红松种群中种子、鼠类和幼苗三者之间的相互关系,构建了四类模拟红松种群的数学模型,并利用微分方程定性稳定性理论和时滞微分方程以及Hopf分支理论研究了模型的动力学行为,研究结果简述如下：
1根据红松种群的生态学特性,将其分为幼树(未达结实年龄个体)和母树(已达结实年龄个体),构建了两类模拟天然林内红松种群年龄更替的数学模型。通过定性分析,得出在一定条件下系统存在稳定的周期解和局部渐近稳定的正平衡点,并用Maple进行了数值模拟,结论表明天然林内红松幼树与母树会产生一个随时间变化的有规律、互为消长的特征,二者既相互协调又相互控制,保持稳定的动态平衡。
2红松种子、鼠类和幼苗是红松发生和演替的纽带,根据三者之间相互适应和相互调节的关系,构建了一类三维非线性红松种群模型,利用系统分析的方法研究了系统的稳定性,证明了系统在一定条件下存在唯一的周期解,讨论了其鞍结分支和Hopf分支,并对Hopf分支周期解进行了分析和计算,指出了红松种群中种子、鼠类和幼苗三种群数量具有周期波动的特征。
3探讨了一类三维的模拟红松种群中的种子、鼠类和幼苗的动态变化的时滞微分方程模型,利用系统分析的方法研究了模型的各种动力学行为,包括红松种群达到动态平衡的条件,平衡点的存在性和局部稳定性。结果表明时滞现象能导致稳定的平衡点转变为不稳定,得到了发生Hopf分支的条件。
研究结果揭示了红松种群系统的周期行为特征,为合理经营管理和保护红松林资源提供了理论依据。
Pinus koraiensis (Korean pine) is the dominant tree species in the eastern of Northeast China and Xiaoxing'an Mountains. The timber of Pinus koraiensis is excellent and its seeds are nutrient-rich which makes it an important species with highly economic values. The broad-leaved Pinus koraiensis forest is mainly formed by Pinus koraiensis and constitutes the zonal climax vegetation type in China's temperate mixed coniferous vegetation. It is considered as the key production base of timber and forest by-products in China. Furthermore, the complex community structure and stable regeneration process of the broad-leaved Pinus koraiensis forest makes it important to protect the natural resources and maintain ecological balance.
The natural regeneration of Pinus koraiensis depends on the seed sources and the growth conditions for seedlings. The seed sources are related to the dispersal and consumption of pine seeds etc. Since pine seeds are squirrels'(mainly referring to Sciurus vulgaris, also considering Eutamias sibiricus, Apodemus sylvticus and some birds) main food, and the regeneration of seedlings mainly relies on squirrels' evolutional hoarding behavior, interaction relationship exists among pine seeds, seedlings and squirrels. Pinus koraiensis forest and squirrels form a typical Pinus koraiensis forest ecosystem.
A wide range of mathematical models have been applied in economics, ecology and environmental sciences. The population of Pinus koraiensis including seeds, squirrels and seedlings is studied quantitatively by mathematical models. Four mathematical models were developed to describe the Pinus Koraiensis forest ecological system. Dynamics behavior of these models are investigated based on the stability theory and qualitative theory of ordinary differential equations, delay differential equations and Hopf bifurcation. The brief results are as follows:
1 Pinus koraiensis trees are classified to young trees (without seeds produce) and mother trees (with seeds produce) according to their ecological characteristics. Two mathematical models are developed to describe the interaction between young trees and mother trees. Sufficient conditions for the existence of periodic solution and globally asymptotic stability of the positive equilibrium are given based on qualitative analysis method. The results are validated by plotting with Maple, which indicate that there is a periodic fluctuation in the number of young trees and mother trees in natural forest. Young trees and mother trees interact and reach to a stable homeostasis.
2 Pine seeds, squirrels and Pinus koraiensis seedlings are the main factors influencing the occurrence and succession of the population of Pinus koraiensis forest. A nonlinear three-dimensional model of the population of Pinus koraiensis is developed based on the mutual accommodation and adjustment relationship of these factors. The stability of the model is studied using systematic analysis method and the results show that a unique and stable periodic solution exists under certain conditions. The saddle-node bifurcation and Hopf bifurcation are also discussed, and analysis for the periodic solution of the Hopf bifurcation is given. The results indicate that there is a periodic fluctuation among the number of seeds, squirrels and seedlings in the Pinus koraiensis forest ecosystem.
3 A three-dimensional model of the population of Pinus koraiensis including pine seeds, squirrels and seedlings with delay is considered. Dynamics behaviors, such as the condition for the ecosystem to reach dynamic balance, the existence and local stability of equilibria, are investigated by systematic analysis method. It is proved that the time delay may destabilize the positive equilibrium, and Hopf bifurcation occurs under certain conditions.
The results indicate that the evolution of the population of Pinus koraiensis exists periodic characteristics. Some theoretical basis for rational management and protection of Pinus koraiensis in China are provided.
红松的天然更新取决于种源和幼苗幼树的生长条件,而种源由母树产生的种子的传播条件和种子的消耗等因素决定。红松林中,种子是鼠类(主要指松鼠,也考虑花鼠、大林姬鼠和一些鸟类)的主要取食对象之一,幼苗的更新主要依赖鼠类的贮食行为,种子、鼠类和幼苗三种群互相影响和制约。鼠类和红松共同组成了一个典型的红松林生态系统。
数学模型已被广泛应用于经济、生态与环境科学中。本文依据数学生态学建模方法定量研究红松种群中种子、鼠类和幼苗三者之间的相互关系,构建了四类模拟红松种群的数学模型,并利用微分方程定性稳定性理论和时滞微分方程以及Hopf分支理论研究了模型的动力学行为,研究结果简述如下：
1根据红松种群的生态学特性,将其分为幼树(未达结实年龄个体)和母树(已达结实年龄个体),构建了两类模拟天然林内红松种群年龄更替的数学模型。通过定性分析,得出在一定条件下系统存在稳定的周期解和局部渐近稳定的正平衡点,并用Maple进行了数值模拟,结论表明天然林内红松幼树与母树会产生一个随时间变化的有规律、互为消长的特征,二者既相互协调又相互控制,保持稳定的动态平衡。
2红松种子、鼠类和幼苗是红松发生和演替的纽带,根据三者之间相互适应和相互调节的关系,构建了一类三维非线性红松种群模型,利用系统分析的方法研究了系统的稳定性,证明了系统在一定条件下存在唯一的周期解,讨论了其鞍结分支和Hopf分支,并对Hopf分支周期解进行了分析和计算,指出了红松种群中种子、鼠类和幼苗三种群数量具有周期波动的特征。
3探讨了一类三维的模拟红松种群中的种子、鼠类和幼苗的动态变化的时滞微分方程模型,利用系统分析的方法研究了模型的各种动力学行为,包括红松种群达到动态平衡的条件,平衡点的存在性和局部稳定性。结果表明时滞现象能导致稳定的平衡点转变为不稳定,得到了发生Hopf分支的条件。
研究结果揭示了红松种群系统的周期行为特征,为合理经营管理和保护红松林资源提供了理论依据。
Pinus koraiensis (Korean pine) is the dominant tree species in the eastern of Northeast China and Xiaoxing'an Mountains. The timber of Pinus koraiensis is excellent and its seeds are nutrient-rich which makes it an important species with highly economic values. The broad-leaved Pinus koraiensis forest is mainly formed by Pinus koraiensis and constitutes the zonal climax vegetation type in China's temperate mixed coniferous vegetation. It is considered as the key production base of timber and forest by-products in China. Furthermore, the complex community structure and stable regeneration process of the broad-leaved Pinus koraiensis forest makes it important to protect the natural resources and maintain ecological balance.
The natural regeneration of Pinus koraiensis depends on the seed sources and the growth conditions for seedlings. The seed sources are related to the dispersal and consumption of pine seeds etc. Since pine seeds are squirrels'(mainly referring to Sciurus vulgaris, also considering Eutamias sibiricus, Apodemus sylvticus and some birds) main food, and the regeneration of seedlings mainly relies on squirrels' evolutional hoarding behavior, interaction relationship exists among pine seeds, seedlings and squirrels. Pinus koraiensis forest and squirrels form a typical Pinus koraiensis forest ecosystem.
A wide range of mathematical models have been applied in economics, ecology and environmental sciences. The population of Pinus koraiensis including seeds, squirrels and seedlings is studied quantitatively by mathematical models. Four mathematical models were developed to describe the Pinus Koraiensis forest ecological system. Dynamics behavior of these models are investigated based on the stability theory and qualitative theory of ordinary differential equations, delay differential equations and Hopf bifurcation. The brief results are as follows:
1 Pinus koraiensis trees are classified to young trees (without seeds produce) and mother trees (with seeds produce) according to their ecological characteristics. Two mathematical models are developed to describe the interaction between young trees and mother trees. Sufficient conditions for the existence of periodic solution and globally asymptotic stability of the positive equilibrium are given based on qualitative analysis method. The results are validated by plotting with Maple, which indicate that there is a periodic fluctuation in the number of young trees and mother trees in natural forest. Young trees and mother trees interact and reach to a stable homeostasis.
2 Pine seeds, squirrels and Pinus koraiensis seedlings are the main factors influencing the occurrence and succession of the population of Pinus koraiensis forest. A nonlinear three-dimensional model of the population of Pinus koraiensis is developed based on the mutual accommodation and adjustment relationship of these factors. The stability of the model is studied using systematic analysis method and the results show that a unique and stable periodic solution exists under certain conditions. The saddle-node bifurcation and Hopf bifurcation are also discussed, and analysis for the periodic solution of the Hopf bifurcation is given. The results indicate that there is a periodic fluctuation among the number of seeds, squirrels and seedlings in the Pinus koraiensis forest ecosystem.
3 A three-dimensional model of the population of Pinus koraiensis including pine seeds, squirrels and seedlings with delay is considered. Dynamics behaviors, such as the condition for the ecosystem to reach dynamic balance, the existence and local stability of equilibria, are investigated by systematic analysis method. It is proved that the time delay may destabilize the positive equilibrium, and Hopf bifurcation occurs under certain conditions.
The results indicate that the evolution of the population of Pinus koraiensis exists periodic characteristics. Some theoretical basis for rational management and protection of Pinus koraiensis in China are provided.
引文
参考文献
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