Qualitative analysis of a korean pine forest model with impulsive thinning measure☆
Introduction
Korean pine (Pinus koraiensis Sieb.et Zucc.) is a precious and rare tree species and mainly distributes in Changbai Mountain and Xingan Mountain areas of China. A few of them distribute in some areas of Japan, Korea and Russia. To protect the korean pine forest and maintain its regeneration and succession, the management measures such as quantitative thinning, thinning and single tree selective cutting are taken. In those measures, the quantitative thinning is a main measure because it can not only protect the species but also make the managers obtain economic benefits.
Some references have studied the storage of fallen trees of korean pine mixed forest [1], population structure and regeneration mode [2]. A few of references have studied the dynamical behaviors of korean pine forest models. For examples, Ref. [3] developed succession and silviculture model for broad-leaved pinus koriensis in Changbai Mountain by combining the framework of ZELIG and characteristics of broad-leaved pinus koriensis forests in Changbai area. Ref. [4] studied the structure of food net, habitat conditions, nature regeneration, the species structure of young forest and mature forest, and gave the regeneration model of korean pine. Ref. [5] have studied the wave features of population changes of korean pine in natural forest. To consider the dynamical properties of the regeneration model, Ref. [6] presented and studied a kind of mathematical model of population age replace of korean pine in natural forest. In paper [6], the conditions for the existences and stabilities of the equilibrium and the limit cycle are given.
Considering the current level of korean pine, the strategies of tending and thinning was taken in Changbai Mountain and Xingan Mountain areas. The main purpose of tending and thinning is to implement the forest, adjust the stand density, improve stand condition, improving forest quality, enhancement and play a variety of beneficial function of forest. Besides, through the thinning activities, the production of a certain number of wood is to meet the needs of national construction and people’s life. But the time and the total yield of thinning depend on the state of the forest. Therefore, a model of korean pine with impulsive thinning is presented to describe the regeneration process of the forest under mankind’s management measure.
There are some papers investigating the biological and mathematical model with impulsive control. For example, Ref. [7], [8] discussed the pest models with impulsive control. Ref. [9], [10] discussed the periodic solution of two microorganism culture systems with impulsive state feedback control by the existence criteria of periodic solution of a general planar impulsive autonomous system. Ref. [11] considered the system with impulsive state feedback control as semi-continuous dynamical system, and gave the definitions and some methods to discuss the qualitative problems of the models. As the applications of semi-continuous dynamical system, papers [12], [13], [14], [15] etc. gave the preliminary results about the biomathematical model with impulsive state feedback control. In this paper, we will discuss a kind of korean pine model with impulsive thinning measure, which depends on the state of the species population, and show the periodicity of the regeneration of the forest.
Section snippets
Model formulation and preliminary
Paper [6] presented the following model to study the population age replace of korean pine forest.where x and y denote the population of sapling and seed trees (that is, young trees and mature trees),respectively. Parameters and d are positive. System (2.1) assumed that
- (H1)
The intrinsic rate of increase of the sapling trees satisfies the Logistic function , the restriction from the seed trees is similar to the Holling type II functional response, that
The existence of order-1 periodic solution for
From Lemma 2.2, we know that the positive equilibrium is globally stable if . In the following, we discuss the periodic solution for the cases of and , respectively.
Periodic solution for
When , we know from Lemma 2.4 that system (2.1) has a unique limit cycle. Denote the limit cycle by . The limit cycle intersects the isoclinic line at two point and (see Fig. 6). The saddle sparatrix intersects the isoclinic line at the point . Clearly, .
If , then we know from the proof of Theorem 3.1 that system (2.2) has a unique order-1 periodic solution.
If , then the trajectories of system (2.2) tend to the limit cycle
Numerical simulations
In order to verify the mathematical results given above, let , then . It is easily known from Lemma 2.3 that system (2.1) has no limit cycle and the equilibrium is a focus point, see the dot line in Fig. 8. Fig. 8(a) shows that there exists an order-1 periodic solution for and . The trajectory starting from the initial point (4, 1) tends to the order-1 periodic solution. Fig. 8(b) shows the change of position of the order-1
Conclusions and discussions
In this paper,we have discussed the existence of periodic solution of the korean pine forest model with impulsive thinning measure. The results show that the model has order-1 periodic solution for which is orbitally asymptotically stable if Theorem 3.2 holds, and the existence of the periodic solution for or needs some conditions to guarantee.
If , the existence of order periodic solution is complex. If , there exists a unique periodic solution. If
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This work is supported by the National Natural Science Foundation of China (No. 11171284, 11371306), the Natural Science Foundation of Henan Province (No. 122300410034, 132300410344) and the Education Department of Henan Province (No. 12A110019), the Universities Young Teachers Program of Henan Province (No. 2010GGJS-104).