Fractal Analysis And Control Of The Competition Model

Studying the model of population growth is very important for the development of human beings and society.It has important applications in population control,alloca-tion of social resources,monitoring and improvement of the ecological environment,the protection of species and the development of aquaculture industry.With the continuous development of biological mathematics,people's understanding of mathematical theory has been deepened,and the mathematical model has been well developed and applied today.Among them,we are concerned about whether the population has a positive e-quilibrium and whether the equilibrium is stable(Which means,the initial size of the population changes because of the change of the environment or the influence of the out-side world,and whether it can be restored to the original equilibrium state as time goes on).In mathematics,this is about the stability of the solution.In this paper,several different population competition models are discussed.Lotka-Volterra population competition model plays an important role in mathematical models.Firstly,Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry.And feedback control is taken on the Julia set of the model.Synchronization of two different Julia sets of the model with different parameters is discussed,which makes one Julia set change to be another.Secondly,the continuous population competition model is analyzed.It is dispersed and its Julia set of the competition model is introduced similarly.Then feedback control and optimal function control are taken on the Julia set of the model.Moreover the fractal box dimensions of the controlled Julia set are calculated to depict the complexity of Julia set and the attractive domain.The above content is the use of the Julia set of ideas and methods in the scope of the real number system,but the Julia set itself is defined in the complex system.Thus the Julia set is discussed on the complex system.Extend the population competition model to the complex domain firstly.The continuous population competition model is dispersed.Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry.The stability of the system is proposed based on Jury test.Then feedback control and tracking control are taken on the Julia set of the model.Human impact on the population,which is conducive to the rational utilization of wildlife resources and protection,to provide theoretical guidance for the artificial breeding and planting industry,and provide scientific basis for the prediction and prevention of pests,maintain ecological balance,the formation of the sustainable development of harmony between man and nature of the state.For convenience,discussion of this paper is limited to the competitive relationship between the two groups.