生物型体竞争模型的高分辨率高精度方法
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摘要
生物型体竞争模型(Hierarchical size-structured population model)是生物数学中一类非常重要的竞争模型，这一类模型通过生物以型体大小为基础的相互竞争关系描述了生物总数量随时间演变的规律，具有重要而广泛的应用价值，比如森林中各种植物对阳光的竞争模型、动物之间争夺食物与生殖优势的竞争模型等等。计算此类模型的主要难点在于方程的一些系数与边界条件中包含生物密度函数的全局积分，以及方程中包含的非线性的生长率、死亡率和繁殖率等函数。
本文主要对此类型体竞争模型进行了细致的研究分析，构造发展了一系列便于计算的数值计算格式，包括一阶显式迎风有限差分格式、二阶显式高分辨率有限差分格式和五阶显式高精度有限差分WENO(weighted essentially non-oscillatory)格式，并通过理论分析与大量数值算例证明了这些格式在数值计算这类模型方程中的良好性质与优越性。
对于一阶迎风格式和二阶高分辨率格式，我们证明了其具有总变差有界即TVB(Total Variation Bounded)性质，进而证明了这两种数值格式的稳定性与收敛性。同时我们分别给出了光滑解和间断解的数值算例验证了这两种数值格式的良好性质。
针对生物型体竞争模型的具体特点，我们又构造了相应的高阶精度的WENO差分格式，并通过大量数值算例验证了该格式的优异性质。对比一阶迎风格式、二阶高分辨率格式和其他已有的一阶与二阶差分格式，我们的高阶WENO格式展现了其显著的卓越性，在所有计算模型方程的光滑解和间断解的数值算例中，高阶WENO格式都可以使用少得多的格点数来得到更为优异精确的结果。我们又将其应用于食蚊鱼的型体竞争模型(Gambussia affinis)，进一步展现了高阶WENO格式的计算优越性。
Hierarchical size-structured population model is an important structured population model in mathematical biology. This model mainly describes the evolution of hierarchically size-structured population at a given time. Hierarchical size-structured population model has been used in modeling many biology problems such as modeling the competition for sunlight in a forest and modeling the competition for food and the advantage of reproduction among some kind of animals. The main technical complication is the existence of global terms in the coefficient and boundary condition for this model with nonlinear growth, mortality and reproduction rates.
In this paper we develop and discuss three explicit finite difference schemes, namely a first order upwind scheme, a second order high resolution scheme and a fifth order weighted essentially non-oscillatory (WENO) scheme for solving the hierarchical sizestructured population model with nonlinear growth, mortality and reproduction rates.
For the first order upwind scheme and the second order high resolution scheme, we prove their TVB (Total Variation bounded) property. Then we prove stability and convergence for both schemes and provide numerical examples to demonstrate their capability in solving smooth and discontinuous solutions.
Secondly we develop a high order explicit finite difference WENO scheme for solving the model. We carefully design approximations to these global terms and boundary conditions to ensure high order accuracy. Comparing with the first order monotone and second order total variation bounded schemes for the same model, the high order WENO scheme is more efficient and can produce accurate results with far fewer grid points. Numerical examples including one in computational biology for the evolution of the population of Gambussia affinis, are presented to illustrate the good performance of the high order WENO scheme.
本文主要对此类型体竞争模型进行了细致的研究分析，构造发展了一系列便于计算的数值计算格式，包括一阶显式迎风有限差分格式、二阶显式高分辨率有限差分格式和五阶显式高精度有限差分WENO(weighted essentially non-oscillatory)格式，并通过理论分析与大量数值算例证明了这些格式在数值计算这类模型方程中的良好性质与优越性。
对于一阶迎风格式和二阶高分辨率格式，我们证明了其具有总变差有界即TVB(Total Variation Bounded)性质，进而证明了这两种数值格式的稳定性与收敛性。同时我们分别给出了光滑解和间断解的数值算例验证了这两种数值格式的良好性质。
针对生物型体竞争模型的具体特点，我们又构造了相应的高阶精度的WENO差分格式，并通过大量数值算例验证了该格式的优异性质。对比一阶迎风格式、二阶高分辨率格式和其他已有的一阶与二阶差分格式，我们的高阶WENO格式展现了其显著的卓越性，在所有计算模型方程的光滑解和间断解的数值算例中，高阶WENO格式都可以使用少得多的格点数来得到更为优异精确的结果。我们又将其应用于食蚊鱼的型体竞争模型(Gambussia affinis)，进一步展现了高阶WENO格式的计算优越性。
Hierarchical size-structured population model is an important structured population model in mathematical biology. This model mainly describes the evolution of hierarchically size-structured population at a given time. Hierarchical size-structured population model has been used in modeling many biology problems such as modeling the competition for sunlight in a forest and modeling the competition for food and the advantage of reproduction among some kind of animals. The main technical complication is the existence of global terms in the coefficient and boundary condition for this model with nonlinear growth, mortality and reproduction rates.
In this paper we develop and discuss three explicit finite difference schemes, namely a first order upwind scheme, a second order high resolution scheme and a fifth order weighted essentially non-oscillatory (WENO) scheme for solving the hierarchical sizestructured population model with nonlinear growth, mortality and reproduction rates.
For the first order upwind scheme and the second order high resolution scheme, we prove their TVB (Total Variation bounded) property. Then we prove stability and convergence for both schemes and provide numerical examples to demonstrate their capability in solving smooth and discontinuous solutions.
Secondly we develop a high order explicit finite difference WENO scheme for solving the model. We carefully design approximations to these global terms and boundary conditions to ensure high order accuracy. Comparing with the first order monotone and second order total variation bounded schemes for the same model, the high order WENO scheme is more efficient and can produce accurate results with far fewer grid points. Numerical examples including one in computational biology for the evolution of the population of Gambussia affinis, are presented to illustrate the good performance of the high order WENO scheme.
引文
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[18] A. Calsina and J. Saldana. Asymptotic behavior of a model of hierarchically structured population dynamics. Journal of Mathematical Biology, 35:967-987, 1997.
[19] J. Casper and H. Atkins. A finite-volume high-order ENO scheme for two dimensional hyperbolic systems. Journal of Computational Physics, 106:62-76, 1993.
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[21] P. Colella. A direct eulerian MUSCL scheme for gas dynamics. SIAM Journal on Scientific and Statistical Computing, 6:104-117, 1985.
[22] P. Colella and P. Woodward. The piecewise-parabolic method (PPM) for gas-dynamical simulations. Journal of Computational Physics, 54:174-201, 1984.
[23] M. G. Crandall and A. Majda. Monotone difference approximations for scalar conservation laws. Mathematics of Computation, 34:1-21, 1980.
[24] J. M. Cushing. The dynamics of hierarchical age-structured population. Journal of Mathematical Biology, 32:705-729, 1994.
[25] J. M. Cushing. An introduction to structured populations dynamics. In CMB-NSF Regional Conference Series in Applied Mathematics. SIAM, 1998.
[26] J. M. Cushing and J. Li. Juvenile versus adult competition. Journal of Mathematical Biology, 29:457-473, 1991.
[27] A. Dolezal and S. Wong. Relativistic hydrodynamics and essentially non-oscillatory shock capturing schemes. Journal of Computational Physics, 120:266-277, 1995.
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