Study On Mathematical Models And Dynamics Behavior Of The Population Of Pinus Koraiensis

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.