A Free Boundary Problem Describing Plant Invasion

In environmental issues, the problem of biological invasions has aroused extensive attention all around the world over the recent years. The main reason is that the negative biological invasions always disequilibrate the local ecological balance and cause the intrinsic ecological structure to change. It is regarded as the enemy number one for biodiversity. According to the statistics, there are more than 500 kinds of alien species in China, and most of them are plants. With the fast expansion of survival region, the invasive plants have destroyed the stability of agricultural ecosystem and the environment of habitats for raising livestock and fish, which even affect people’s health. If valid prevention and control work not be done on time, then the situation will be even worse.The key in the theoretical studies of this subject is to be clear of the spreading process and internal mechanism of invasive plants. Thus, the study of mathematical model and space structure is necessary. At present, many kinds of mathematical models for investigating kinetic process of plant invasions have been proposed. However, the majority of them only consider the problem in the fixed region, ignoring the variation of moving fronts of the survival domain when the plants are spreading. Based on these, we consider the problem with free boundaries. Combined with the ASCS (age-structured continuous-space) model which was first proposed by Kuznetsov et al. in 1994, we discuss the conditions of failing and successful plant invasion and the long behavior of the moving fronts.The paper is organized as follows.In the first section, we introduce the background of the plant invasions, the research status and trend of this subject and the ASCS model.In the part of Section 2, we first apply the contraction mapping theorem to prove the global existence and uniqueness of the solution of the model, then give good properties of the free boundaries by strong maximum principle and Hopf lemma.Section 3 of this paper is devoted to define "failing invasion" and "successful invasion" by using mathematical language and discuss the long behavior of the positive solution of the plant invasions model. For the sake of this, a threshold parameter JF0 (t) is introduced. And by using comparison principals, sufficient conditions are given to determine when the plants invade successfully or fail to expand. Our results show that if J0≤1, then the plants fail to expand, if J0F(t0)≥ 1 for some t0≥ 0, the invasive plants will successfully establish themselves in the new environment, while if J0F(0)<1, the invasive plants expand with small initial size, if J0F (0)< 1< J0, with large initial values.In Section 4, a more detailed study of the invasions of the plants is displayed. That is, we find the sharp threshold depending on initial values between the cases:"failing invasion" and "successful invasion". More precisely, the plants will invade successfully when the value is bigger than the sharp threshold, otherwise, they will fail to expand.Finally, we give a brief summary of this paper and prospects in Section 5.