摘要
本文考虑一个既含有捕食被捕食关系又有竞争关系的环状模型,当不同种群具有不同稀释率,且消耗率中的参数δ_i(i=1,2,3)分别取常数和线性函数时,运用常微分方程的定性理论分析了系统的定性性质。
当消耗率中的参数δ_i(i=1,2,3)全部取为常数时,主要分析了半平凡平衡解的存在性和局部稳定性。通过构造Lyapunov函数,证明了系统在半平凡平衡解处的全局稳定性;在条件m_3=D_2和a_1m_2/a_2m_1>D_2/D_1的假设下,证明了正平衡点的存在性,并进一步证明了系统在正平衡点处的局部稳定性。最后分析了系统的一致持续生存性。
将微生物x_2对营养基s的消耗率中的常数取为一次函数,在这种情况下,主要分析了半平凡平衡解的存在性和局部稳定性,证明了系统在半平凡平衡解处的全局稳定性;给出了系统存在Hopf分歧的条件,在此条件下系统的平衡点为稳定一阶细焦点,并进一步证明了由此产生的周期解在一定条件下是稳定的。
考虑消耗率中的参数δ_i(i=1,2,3)部分取为线性函数的环状模型,在这种情况下,主要分析了半平凡平衡解的存在性和局部稳定性,证明了系统在半平凡平衡解处的全局稳定性。
对相关结果通过Matlab软件给出了相应的数值模拟图像。
In this paper,an annular model which included competition relation and predator-prey relation is discussed.Under the conditions that different populations with different dilution rates and the parametersδ_i(i = 1,2,3)of the consume rates are defined by constants or a linear function,the quality of the system is analyzed with the qualitative theorem of ODE.
When all the parametersδ_i(i = 1,2,3)of the consume rate are defined by constants, the existence and the local stability of the semi-trivial equilibrium are considered.By constructing a Lyapunov function the global stability of the the semi-trivial equilibrium is proved;On the assumption that m_3 = D_2 and a_1m_2//a_2m_1>D_2/D_1,the existence of the positive equilibrium and the local stability of the positive equilibrium are considered. In the end,the uniform persistence of the system is analyzed.
Under the conditions that one of the parametersδ_i(i = 1,2,3)of the consume rate is defined by a linear function.The existence and the local stability of the semi-trivial equilibrium are considered.The global stability of the the semi-trivial equilibrium is proved;the existence of the Hopf bifurcation is considered and the related equilibrium is proved to be one order's fine focus,further the stability of the periodic equilibrium is proved.
Considering the model of the parametersδ_i(i = 1,2,3)of the consume rate is defined by a linear function.The existence and the local stability of the semi-trivial equilibrium are considered.The global stability of the the semi-trivial equilibrium is proved.
Numerical simulation of the corresponding equlibria's existence and stability is presented by Matlab.
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