Study On Fractional Nonlinear Population Diffusion Model

Fractional calculus operator is a powerful tool to study fractal dynamics,and has been successfully applied to natural science technology and some otherfields. More and more studies show that some diffusion phenomenon in notpure, and the media cannot described by the standard diffusion due to thecomplexity of its diffusion. we introduce the fractional calculus operator intothe anomalous diffusion, thus makes the research problem more universal.On the basis of the fractional calculus theory combined with the studyof population diffusion model, fractional nonlinear single population diffusionmodel, fractional nonlinear two populations interaction diffusion model andfractional nonlinear Fokker-Planck equation are introduced in this disserta-tion, And thus some studies of these model are also given. The study of thefractional population diffusion model is still in the initial stage. And this dis-sertation provides a new perspective to study the fractional population model.The main content of this dissertation is as follows:Firstly, single time population fractional Fisher type diffusion model andspatial fractional single population Fisher type diffusion model are established,and the approximate solution of these model are obtained by using the homo-topy perturbation method and the variational iteration method. comparisonbetween the integer single population Fisher diffusion are also given.Secondly, two population time fractional interaction diffusion model isestablished, and the approximate solution of this model are obtained by usingvariational iteration method in different initial conditions. The tendency oftwo populations of the prey-predator system, mutual competition system andmutual coexistence system between two populations are studied under thedensity dependence and density independence.Thirdly, numerical simulation of the fractional nonlinear populationmodel are studied, and the time evolution rule and state space distribution of the population are shown in the simulation results. And the solutions of frac-tional population model is continuous dependent on the change of fractionalorder.Finally, a generalize nonlinear Fokker-Planck diffusion equation with ex-ternal force and absorption are established. The solution of the integer non-linear anomalous diffusion with the diffusion coeffcient are obtained by qffexponential function. And the solutions of the multi-fractional nonlinear dif-fusion are also studied in detail. The solutions can have a compact behavioror a long tailed behavior.