Lotka-Volterra Ecological Systems

Mathematical ecology is to establish and study the mathematical models in ecology by using mathematical theories and methods. Lotka-Volterra models are important in mathematical ecology. In this thesis, we introduce the studies on Lotka-Volterra models briefly.In 1900, Italian mathematician Volterra made a lecture entitled "Tries of applied mathematics and the biological and social sciences" in Rome University, which marks a milestone in the development of biomathematics. He published the famous papers in 1926 to explainthe changes law of Finme's fish and put forward the well-known Lotka-Volterra model. Lotka also proposed this model in the study of chemical reaction in 1925. The proposal of the Lotka-Volterra model made a climax of the development of mathematical ecology.The model which was originally proposed by Lotka and Volterra is based on the predator-prey case of two species:where x and y denote the quantities of the prey and the predator respectively,a₁,a₂,b₁,b₂ > 0 are constants. The ecological phenomena represented by Lotka-Volterra models are often seen in nature. In addition to the case of predator-prey, parasite and the host parasitized of the parasitic phenomenon can also be described by the model. Generally speaking, in nature, neither one species is not aggrieved by other species, nor the existence of one species does not threaten the survival of other species. In a certain range, as long as there are two species in a domain, and they have such constraint relation which is dominant than other relations, we can use Lokta-Volterra model to describe the relationship of the two species.In the early years, Lotka-Volterra models are ordinary differential systems. With the development of models, as well as the needs of ecology, partial differential models are proposed and time delay was added to models. According to their ecological significance, Lotka-Volterra models can be divided into three types: competition models, predator-prey models and reciprocal and cooperative models.In this thesis, we introduce the studies on these three types of Lotka-Volterra models briefly.(â… ) Competition Lotka-Volterra ModelsCompetition is that the existence of a species inhibits the growth of another species. They can kill each other, and they can strive resources in most cases.For example, a group of tigers and a group of leopards that live in the same mountain are two competitive groups; and crops is competitive with the weeds in the same field. The general competition Lotka-Volterra model is as follows:Ahmad, Lazer, Zeaman, Li Biwen and others studied the ordinary differential competition models, and discussed the existence, the uniqueness and the asymptotic behavior of solutions, and obtained conditions for the extinction and coexistence of species.In 1979, Shigesada, Kawasaki and Taramoto [7] extended ordinary differential model to partial differential model and they put forward a competition Lotka-Volterra model for two-species in the n dimensional space, which is the following reaction-diffusion system (referred to as the SKT model):This is the first partical differential Lotka-Volterra model. M. Pozio, A.Tesei, Y. Lou, Y. WanLi, Deuring, Shim Seong, Kim, Yagi, G. Galiano, M. Garzon,A. Jungle and others studied the SKT model and discussed the existence, the uniqueness and properties of solutions. Amann, Pu Shengmao, Gao Haiyan, Cui Shangbin Cui and others studied the competition Lotka-Volterra model with cross-diffusion, and discussed the existence and uniqueness of local solutions and global solutions and the asymptotic behavior of global solutions.With the development of models, as well as the needs of ecology, the following model with time delay was proposedThe studies on this model and other competition Lotka-Volterra models with time delay were referred to the works by Teng, Yu, Qiu Jianlong, Cao Jinde, Fang Hui, Xiao Yongfeng, Xu Rui, Chen Lansun, etc.(â…¡) Predator-Prey Lotka-Volterra ModelPredator-prey is a species prey on another species. The general predator-prey Lotka-Volterra model is as follows:where x is the prey and y is the predator.Parasitism and the host system is a special case of this model, where parasitism is the predator and host is the prey. The general two-species predator-prey model with cross-diffusion is as follows:Sun Wujun, Teng Zhidong, Yu Yuanhong, Lu Zhonghua, Chen Lansun and others studied the ordinary differential predator-ptey model; Amann, Zeng Xianzhong, Fu Maosheng, Gao Haiyan, Kuto, Yamada, Wen Zijuan and others studied the predator-ptey model with cross-diffusion; Xu Rui, Chen Lansun, Li Biwen and others studied the predator-ptey model with time delay and functional response; Wang Yuanming, Meng Xinzhu,Jiao Jianjun, Chen Lansun and others studied the predator-prey model with discrete time delay; Song Xinyu, Chen Lansun, Li Biwen and others studied the predator-prey model with continuous time delay.(â…¢) Reciprocal and Cooperative Lotka-Volterra ModelReciprocal and cooperative model is that the existence of each of the species will play a positive role of the growth of other species. For example, bees and flowers are two species that are reciprocal and cooperative. The general reciprocal and cooperative Lotka-Volterra model is as follows:The general partial differential Lotka-Volterra reciprocal and coop- erative model is as follows:Lu Zhenyi, Li Biwen and others studied the ordinary differential reciprocal and cooperative model. Xie Yulong, Lou Yanling, Korman,Li, Mottoni, Kim, Lin, Wu and others studied the general partial differential Lotka-Volterra reciprocal and cooperative model and discussed the condition of the coexistence and the properties of the solutions. Fu Maosheng, Wen Zijuan, Yang Zhiguo, Li Shuyong and Wang Changyou studied the reciprocal and cooperative model with cross-diffusion. In 2008, Li Xueshi and Lin guo studied the following reciprocal and cooperative model with time delay...