摄动增量法在非线性生态动力系统中的应用

In this project, a Perturbation-Incremental (PI) method is proposed to obtain approximate analytical periodic solutions of nonlinear ecological dynamics systems. The essence of this method is that a periodic solution is approximated by the Chebyshev series with a nonlinear time s rather than the physical time t, so as to overcome the sudden jump in the phase portrait. Since the first derivative of an approximate limit cycle obtained from the present method is piecewise continuous which agrees qualitatively with the exact solution, it gives accurate analytical solutions for the nonlinear ecological dynamics systems. The nonlinear transformation in this method is also expressed by the Chebyshev series. This method is used for the stability and bifurcation analysis of periodic solutions. The approximate solution obtained from the present method is compared to other perturbation methods with good accuracy for many systems. Such as, cyclic interaction models, metapopulation models, biological population models and impulsive population models. Period solutions with large amplitude and those near to homoclinic or heteroclinic orbits are computed. Excellent agreement of the approximate presentations with the numerical simulation has been demonstrated and discussed.

本项目拟以非线性生态动力系统为研究目标，应用改进的摄动增量法，得到非线性生态系统的近似解析周期解，进而研究解的存在性、唯一性、稳定性、极限环、同/异宿轨、分岔以及解和参数的依赖关系。进一步丰富和发展非线性生态科学理论，促进相关学科分支的发展。方法的核心是：将原有摄动增量法中的非线性变换用分段Chebyshev级数表示，将原来用傅氏级数表示的周期解（极限环）也用分段Chebyshev级数表示。可以克服原有方法的某些缺点，保证得到的解的二阶导数在不同子区域间的临界点处是不连续的，在预测分析子区域间的临界点的行为时能保证其精准性,避免在这些点处出现一些不切实际的误差。将这个方法应用于若干非线性生态系统：循环交互模型、异质种群模型、生物种群模型和种群脉冲模型。最后利用数学、统计相关软件给出相应的数值模拟，证明将这种改进的摄动增量法用于研究非线性生态动力系统是行之有效的。

本项目研究了摄动增量法的应用问题，讨论了剪刀石头布海洋生态系统、捕食-食饵种群生态系统、异质种群生态系统、三维非线性自治系统的周期解、双参数Rayleigh方程系统的极限环、Duffing-VanderPol方程的极限环、二次等时系统、非线性传送带系统等系统。通过对微分方程理论、生态学、强非线性系统的定量分析方法、瞬时动力系统、时滞动力系统、脉冲动力系统的定性理论、泛函分析、复变函数和数值计算等进行系统学习，掌握了基本的数学与统计软件的使用。 通过对傅里叶级数应用，摄动增量法得到很好的推广，开展了对非线性系统周期解的综合分析，得到了参数控制的关键要素，稳定性条件、以及关键参数控制的设计和改进策略。 通过引入合理的非线性变换, 用摄动增量法得到非线性动力系统的具有高精度的周期解析解表达式。进而研究解的基本性质及稳定性，讨论周期解的同宿异宿轨道出现情况。揭示系统的运动规律, 阐明系统运动特性与参数之间的原理, 建立新的非线性动力系统的定量分析方法。