A Model for the Effectiveness of Hibernate on Complex Ecology System

Corresponding Author: Jiangdong Liao School of Mathematics and Statistics, Yangtze Normal University, Fuling 408100, Chongqing, P.R. China Email: ljd88073@163.com Abstract: There are significant differences between plant activities and animal activities in ecological networks. On this basis, this article aims to evaluate the effectiveness of the hibernate (forgetting and remembering mechanism). For this purpose, the hibernation interaction is derived model (linear UAH model) of continuous-time individual-level activity. This model is an effective approach to understand the effectiveness of the construction of the communication network on the complex ecology system. These standards cover the influence of the fundamental parameters and network structure on network effectiveness of hibernate. Theoretical analysis shows that the supreme eigenvalue of the related model matrix is determined whether hibernators tend to become extinct or continue. Moreover, the simulation experiments demonstrate that dynamics of the linear UAH model is very consistent with the actual situation activityhibernate interacting process. and so, the linear UAH model provide appropriate basis for evaluating effectiveness of hibernate.


Introduction
As a means of communication, activities have individual life plants. The dynamics of plant activities is aimed at modeling and research the activity process of plants, to understand the effect of different factors on plant prevalence of plants, so as to formulate costeffective restriction strategy activities. Van Mieghem et al. (2009) suggested an individual-level mode in 2009, (the accurate SIS model), which an accurate description of the average dynamics of the SIS epidemic. The continuous-time individual-level models are particularly useful in investigating the effective of the network topology and suitable for the study of ecological network. Thus this article aims to evaluate the effectiveness of hibernate (forgetting and remembering mechanism). For this purpose, the individual-level activity-hibernate interacting model (the linear UAH model) is derived. Then put forward a set of standards for extinction activities. These standards capture the combined effects of main parameters and network structure on the effectiveness of hibernation. The simulation experiments demonstrate that the dynamics of the linear UAH model is very suitable for the actual activity-hibernate interacting process. Therefore, the linear UAH model provides an appropriate basis for evaluating the effectiveness of hibernate.
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The materials are arranged in this way. Section 2 derives the exact UAH model and the linear UAH model respectively. Section 3 discusses the dynamics of the linear UAH model. At last, Section 4 generalizes this study.

Formation of the Linear UAH Model
This section is dedicated to establish a individuallevel model, which is capturing the effectiveness of hibernate for plant activity.

Notions, Notations and Fundamental Hypotheses
Consider an ecological network consisting of N plants labelled 1,2,···,N and let V (G) = {1,2,…,N} an ecological activity through a ecological network G = (V (G),E(G)), (i,j) E(G) only if j V (G) can effect iV (G). Let A = [aij]NN indicate adjacency matrix of G. Hereafter, the network is always assumed to be a strong connection.
After the appearance of the activity, at any time, assume that each plant in the ecological network is in one of the following three possible states: Uncertain, acting and hibernating. Depending on the individual difference, every plant may choose to be uncertain, or to be activity, or to be hibernator. At time t, let Pi(t) = 0, 1 and 2 mean that, plant I is uncertain, activity and hibernator, separately. At time t, the state of ecological network is indicated by the vector:

The Linear UAH Model
Based on equivalent models and these independent assumptions, the following model can be derived: We name this model as linear UAH because of the acting rate are linear in the arguments. Figure 1 shows these state transition rates of plant under the linear UAH model.

Dynamics of the Linear UAH Model
The generic UAH model is considered. Let A(t) mean the fraction of plant-activity at time t. So: The main purpose of this study is to define the tendency of A(t). For fundamental knowledge on matrix, from. In the following, only real square matrices are considered. Given the matrix M, let s(M) denote the maximum eigen value of M. If the off-diagonal entries of A are all non-negative, it is Metzler.
The linear UAH model might have two different equilibria, which are defined as below.

Definition 1
Let E = (U1,…,UN,A1,…,AN,H1,…,HN) T be an equilibrium of the linear UAH model (3): represents for a steady ecological network there is certainly no active plant (b) E is uncertain-free if U= 0. A uncertain-free equilibrium represents for a steady ecological network there is certainly no uncertain plant Obviously, the linear UAH model always has a unique activity-free equilibrium E1 =(1,…,1,0,…,0,0,…,0) T and a unique uncertain-free equilibrium   2 1 1 0, ,0, , , , . For the aim of checking hibernate equilibria of the model, we define a matrix: Since G is strongly connected andQ1is irreducible. On this basis, define an auxiliary matrix as: We have the comparison system: Let T = max{T1,…,TN}.For all t ≥ T, from the model (2), it follows that: Let w(t) = (w1(t),…,wn(t)) T and define a positive definite function as: Through calculation, we have: Since Q2 has the negative spectrum, then choose a ε, so that matrix: has the negative spectrum. Let u1,u2,…,uN mean the eigen values of Q ′ 2 and assume u1 is the maximum eigen value. As Q ′ 2 is symmetric, we have the orthogonal matrix T so that: It follows from the Lemma (Theorem 31.4 in) and Lemma (Corollary 3.3 in (Strauss and Yorke, 1967)) that limtw(t) = 0, which implies that, 1 ≤ i ≤ N. Then, for any ε >0, have T >0 so that t ≥ T, there is It suffices to enough to prove that H recognizes a unique fixed point and we must to prove two affirmers.

Experiment
Scale-free and Small-world networks are a type of networks with a wide range of applications (Albert and Barabási, 2002). Take a stochastic generated scale-free network and Small-world networks with 100 nodes and the experiments on the networks are shown in Figs. 2 and 3 separately.

Fig. 3: The time plots of Linear A(t), Linear H(t),Exact A(t), Exact H(t)
The following outcomes are from the previous above experiments: (a) If A(t) approaches a nonzero value, then the linear UAH model can truly capture the average evolutionary process of the plant (b) If H(t) becomes a nonzero value, after that the linear UAH model can truly capture the average evolutionary process of the plant

Concluding Remarks
This article has discusses the effects that of hibernate on plant activity and the linear UAH model has been exported. Under this model, a group of criteria for extinction of a activities is given. The extensive simulations result that, the dynamics of the linear UAH model fits well with the actual of the hibernate for activity process of hibernate. The following completions are drawn from the above demonstrates. In this case, the linear UAH model works fine; it can be utilized to quickly predict this average dynamics of activity in the ecological network. For this purpose, one individuallevel activity-hibernate interacting model (the linear UAH model) is derived. Simulation experiments show that dynamics of the linear UAH model are very consistent with the actual activity-hibernate interacting process. So, the linear UAH models provide an appropriate basis for evaluating the effectiveness of hibernate.
Moving in the direction, there are many works under study. With the generic UAH model criteria, the existence/activity of coexistence balance should be found. Thus, it is valuable exploit a new UAH model that brings the restraining efficacy into account. Because of individuallevel model, it is practical importance to know the influence of many factors on the complex ecology system.

Author's Contributions
Lingli Pei: Contributed to the conception of the study; performed the experiment; performed the data analyses and wrote the manuscript.
Jiangdong Liao: Contributed significantly to analysis and manuscript preparation.
Hongjun Wang: Helped perform the analysis with constructive discussions.

Ethics
Authors should be able to submit, upon request, a statement from the research ethics committee or institutional review board indicating approval of the research.