具扩散的两种群互惠模型解的若干性质
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摘要
随着现代科学技术与社会的发展,无论是在工程、医学、生态等自然科学领域,还是在经济、金融等社会科学领域,数学都起着重要的作用。特别地,数学在生态学中的作用日益重要。对于生态学中产生的有趣问题,数学可以为其提供模型和方法去帮助理解。反过来再由生态学去检验数学模型的正确性。目前已有大量的工作研究了生态学中复杂的模型,其中的许多模型可以用非线性抛物和椭圆型偏微分方程组来描述。
本文将讨论来自生态领域中的反应扩散方程组,为使本文更具可读性和系统性。本文将紧紧围绕描述两种群互惠关系的模型,对有关的数学问题进行深入系统地研究。全文由五个部分组成。
第一部分,我们简要地介绍与本文所研究的问题有关的背景知识和研究工作的发展概况。
第二部分,我们考虑了互惠模型中的一个具有自由边界的拟线性抛物方程组。通过采用拉直边界的方法,结合Schauder不动点理论给出了古典解的局部存在唯一性。然后利用先验估计给出了全局解的存在性。同时还对自由边界的渐近性态进行了细致的刻划。结论表明,在种群内部竞争强时,该自由边界问题有全局的慢解,而当种群内部竞争弱时,存在爆破解和全局快解。
第三部分,考察具有自扩散和交错扩散的强耦合椭圆问题。为克服强耦合扩散项带来的困难,我们通过变换将强耦合椭圆问题转化为弱耦合方程组,然后利用上下解和单调迭代的方法给出其共存解。结果表明,当种群的出生率大而且自扩散和种群内部竞争强时,至少存在一个共存解。最后,通过一个简单数值算例对理论结果进行模拟。
由于受到季节更替的影响。生物的出生率、死亡率、种群的相互影响以及环境的容纳量呈现出周期性变化。所以,可以考虑用周期的非线性扩散方程组来描述生态模型。因此,在第四部分,我们重点讨论系数为周期函数的两种群Lotka-Voltcrra互惠模型.考察了Robin边界条件下该反应扩散系统的T周期解的存在性。给出了该问题的一个极大T周期解和极小T周期解。同样地,在这一部分我们也给出了数值算例来描绘模型解的周期性。
非线性扩散方程组的爆破理论也是偏微分方程的重要内容之一。第五部分研究了含自扩散的两种群抛物模型。首先给出了全局解的存在性。为了方便阅读,我们先回顾了不含自扩散模型的爆破结论,在此基础上,给出了相应的含自扩散的抛物模型的解的爆破条件。通过复杂的计算。最终给出了解在有限时刻爆破的充分条件,并且也给出了一个爆破解的数值模拟。
最后,我们在总结上述结论的基础上,对今后的研究工作做进一步的思考。
With the development of modern science and technology,mathematics has been extensively applied both in natural sciences such as engineering,medicine, ecology,and in social sciences such as economics,finance.Especially,mathematics is playing an increasing important role in ecology.Ecology produces interesting problems,mathematics provides models and ways to understand them. and ecology returns to verify the mathematical models.A great deal of research has been done to sophisticated models in ecology,most models can be described mathematically by nonlinear parabolic and elliptic partial differential equations.
This presentation is devoted to reaction diffusion systems describing ecological models.To make it more readable and systematic,we concerns primarily with two-species mutualistic models and the qualitative properties of these models are extensively studied.It consists of five parts.
In the first part,we briefly introduce the background and history about the related work.
The second part is concerned with a system of semilinear parabolic equations with a free boundary,which arises in a mutualistic ecological model.The local existence and uniqueness of a classical solution are obtained by straightening the free boundary and using Schauder fixed point theorem.The global existence of the solution is given by establishing a priori estimates.The asymptotic behavior of the free boundary problem is studied.Our results show that the free boundary problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.
The third part deals with strongly-coupled elliptic systems with self-diffusion and cross-diffusion.To overcome the difficulty caused by cross-diffusion,we will change the strongly-coupled problem into a weakly-coupled problem by using a translation.The existence of coexistence follows from the upper and lower solutions and corresponding iterations.Our results show that this strongly coupled mutualistic model possesses at least one coexistence state if the birth rate is big and self-diffusions and intra-specific competitions are strong.Finally,a numerical simulation is presented to illustrate the main results.
Because of the periodicity of the birth and death rates:rates of interactions and environmental carrying capacities on seasonal scale.nonlinear periodic diffusion equations arise naturally in ecological models.In the forth part,the cooperating two-species Lotka-Volterra model with periodic coefficients is discussed. The existence and asymptotic behavior of T-periodic solutions for the periodic reaction diffusion system under homogeneous Dirichlet boundary conditions are investigated.We show that the problem admits a maximal T-periodic solution and a minimal T-periodic solution.An numerical simulation is also given to illustrate the periodicity of the model.
The blowup theory of nonlinear diffusion equations is also one of important contents.The fifth part studies a parabolic system with diffusion and selfdiffusion in a two species mutualistic model.The global existence of the solution is given first.For the sake of convenience,we present the blowup result results for the parabolic system without self-diffusion.Then we give the blowup result of the corresponding parabolic equation with self-diffusion and finally the sufficient conditions are given for the parabolic system with self-diffusion to blow up in finite time by complex calculations.An numerical simulation is also given to illustrate the blowup results.
Finally,by summing up the given conclusions:we try to make further consideration for future research.
本文将讨论来自生态领域中的反应扩散方程组,为使本文更具可读性和系统性。本文将紧紧围绕描述两种群互惠关系的模型,对有关的数学问题进行深入系统地研究。全文由五个部分组成。
第一部分,我们简要地介绍与本文所研究的问题有关的背景知识和研究工作的发展概况。
第二部分,我们考虑了互惠模型中的一个具有自由边界的拟线性抛物方程组。通过采用拉直边界的方法,结合Schauder不动点理论给出了古典解的局部存在唯一性。然后利用先验估计给出了全局解的存在性。同时还对自由边界的渐近性态进行了细致的刻划。结论表明,在种群内部竞争强时,该自由边界问题有全局的慢解,而当种群内部竞争弱时,存在爆破解和全局快解。
第三部分,考察具有自扩散和交错扩散的强耦合椭圆问题。为克服强耦合扩散项带来的困难,我们通过变换将强耦合椭圆问题转化为弱耦合方程组,然后利用上下解和单调迭代的方法给出其共存解。结果表明,当种群的出生率大而且自扩散和种群内部竞争强时,至少存在一个共存解。最后,通过一个简单数值算例对理论结果进行模拟。
由于受到季节更替的影响。生物的出生率、死亡率、种群的相互影响以及环境的容纳量呈现出周期性变化。所以,可以考虑用周期的非线性扩散方程组来描述生态模型。因此,在第四部分,我们重点讨论系数为周期函数的两种群Lotka-Voltcrra互惠模型.考察了Robin边界条件下该反应扩散系统的T周期解的存在性。给出了该问题的一个极大T周期解和极小T周期解。同样地,在这一部分我们也给出了数值算例来描绘模型解的周期性。
非线性扩散方程组的爆破理论也是偏微分方程的重要内容之一。第五部分研究了含自扩散的两种群抛物模型。首先给出了全局解的存在性。为了方便阅读,我们先回顾了不含自扩散模型的爆破结论,在此基础上,给出了相应的含自扩散的抛物模型的解的爆破条件。通过复杂的计算。最终给出了解在有限时刻爆破的充分条件,并且也给出了一个爆破解的数值模拟。
最后,我们在总结上述结论的基础上,对今后的研究工作做进一步的思考。
With the development of modern science and technology,mathematics has been extensively applied both in natural sciences such as engineering,medicine, ecology,and in social sciences such as economics,finance.Especially,mathematics is playing an increasing important role in ecology.Ecology produces interesting problems,mathematics provides models and ways to understand them. and ecology returns to verify the mathematical models.A great deal of research has been done to sophisticated models in ecology,most models can be described mathematically by nonlinear parabolic and elliptic partial differential equations.
This presentation is devoted to reaction diffusion systems describing ecological models.To make it more readable and systematic,we concerns primarily with two-species mutualistic models and the qualitative properties of these models are extensively studied.It consists of five parts.
In the first part,we briefly introduce the background and history about the related work.
The second part is concerned with a system of semilinear parabolic equations with a free boundary,which arises in a mutualistic ecological model.The local existence and uniqueness of a classical solution are obtained by straightening the free boundary and using Schauder fixed point theorem.The global existence of the solution is given by establishing a priori estimates.The asymptotic behavior of the free boundary problem is studied.Our results show that the free boundary problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.
The third part deals with strongly-coupled elliptic systems with self-diffusion and cross-diffusion.To overcome the difficulty caused by cross-diffusion,we will change the strongly-coupled problem into a weakly-coupled problem by using a translation.The existence of coexistence follows from the upper and lower solutions and corresponding iterations.Our results show that this strongly coupled mutualistic model possesses at least one coexistence state if the birth rate is big and self-diffusions and intra-specific competitions are strong.Finally,a numerical simulation is presented to illustrate the main results.
Because of the periodicity of the birth and death rates:rates of interactions and environmental carrying capacities on seasonal scale.nonlinear periodic diffusion equations arise naturally in ecological models.In the forth part,the cooperating two-species Lotka-Volterra model with periodic coefficients is discussed. The existence and asymptotic behavior of T-periodic solutions for the periodic reaction diffusion system under homogeneous Dirichlet boundary conditions are investigated.We show that the problem admits a maximal T-periodic solution and a minimal T-periodic solution.An numerical simulation is also given to illustrate the periodicity of the model.
The blowup theory of nonlinear diffusion equations is also one of important contents.The fifth part studies a parabolic system with diffusion and selfdiffusion in a two species mutualistic model.The global existence of the solution is given first.For the sake of convenience,we present the blowup result results for the parabolic system without self-diffusion.Then we give the blowup result of the corresponding parabolic equation with self-diffusion and finally the sufficient conditions are given for the parabolic system with self-diffusion to blow up in finite time by complex calculations.An numerical simulation is also given to illustrate the blowup results.
Finally,by summing up the given conclusions:we try to make further consideration for future research.
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