一类分数阶非线性种群扩散模型的研究
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摘要
分数阶微积分算子作为研究分形动力学的有力工具,已经成功地应用到了自然科学和技术科学的很多领域。越来越多的研究也表明某些不纯介质中的扩散现象由于本身扩散的复杂性不能用标准的扩散方程来描述。我们将分数阶微积分算子引入到反常扩散中,从而使问题的刻画更具广泛性。
在国内外相关研究的基础上,本文运用分数阶微积分算子的理论和方法结合种群扩散模型的研究,构建了分数阶非线性单种群扩散模型和分数阶非线性两种群相互作用扩散模型,最后对种群模型的扩展模型Fokker-Planck方程进行了研究。关于分数阶种群扩散模型的研究尚处于初步阶段。本文为分数阶种群模型的分析研究提供了一种新的研究视角。本文完成的工作和主要创新点如下:
1.建立了时间分数阶单种群Fisher型扩散模型和空间分数阶单种群Fisher型扩散模型,应用同伦扰动法和变分迭代法求得问题的近似解,与整数阶单种群Fisher扩散方程进行了比较。
2.建立了时间分数阶两种群相互作用模型,利用变分迭代法求解了不同的种群初始值情况下的近似解。并分别讨论了密度制约条件下的两种群捕食与被捕食、相互竞争、互惠共存情况下种群的变化情况,另外也讨论了非密度制约条件下的两种群捕食与被捕食、相互竞争、互惠共存情况下种群的变化情况。
3.运用数值模拟的方法研究分数阶种群非线性扩散模型,从模拟的结果了解种群数量随时间推移的演化规律以及种群的状态空间分布,并与整数阶模型进行了简单地比较,分数阶种群扩散模型的解是连续依赖于分数阶导数的变化。
4.建立了含有外力和吸附项推广的非线性Fokker-Planck扩散方程。我们详细讨论了整数阶非线性对流-扩散方程和含外力和吸附项的多分数阶非线性对流-扩散方程,利用q?指数函数和q?对数函数的特性,以及在参量满足某种关系下求得了解析解,研究了模型解的特性。
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.
在国内外相关研究的基础上,本文运用分数阶微积分算子的理论和方法结合种群扩散模型的研究,构建了分数阶非线性单种群扩散模型和分数阶非线性两种群相互作用扩散模型,最后对种群模型的扩展模型Fokker-Planck方程进行了研究。关于分数阶种群扩散模型的研究尚处于初步阶段。本文为分数阶种群模型的分析研究提供了一种新的研究视角。本文完成的工作和主要创新点如下:
1.建立了时间分数阶单种群Fisher型扩散模型和空间分数阶单种群Fisher型扩散模型,应用同伦扰动法和变分迭代法求得问题的近似解,与整数阶单种群Fisher扩散方程进行了比较。
2.建立了时间分数阶两种群相互作用模型,利用变分迭代法求解了不同的种群初始值情况下的近似解。并分别讨论了密度制约条件下的两种群捕食与被捕食、相互竞争、互惠共存情况下种群的变化情况,另外也讨论了非密度制约条件下的两种群捕食与被捕食、相互竞争、互惠共存情况下种群的变化情况。
3.运用数值模拟的方法研究分数阶种群非线性扩散模型,从模拟的结果了解种群数量随时间推移的演化规律以及种群的状态空间分布,并与整数阶模型进行了简单地比较,分数阶种群扩散模型的解是连续依赖于分数阶导数的变化。
4.建立了含有外力和吸附项推广的非线性Fokker-Planck扩散方程。我们详细讨论了整数阶非线性对流-扩散方程和含外力和吸附项的多分数阶非线性对流-扩散方程,利用q?指数函数和q?对数函数的特性,以及在参量满足某种关系下求得了解析解,研究了模型解的特性。
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.
引文
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