arXiv 1301.62372013-01math.APq-bio.PE

具有突变的Lotka-Volterra竞争系统的平衡点收敛性

Convergence to the Equilibrium in a Lotka-Volterra Ode Competition System with Mutations

Jerome Coville, Frederic Fabre

备注：: Minor corrections

摘要：

本文研究一类Lotka-Volterra型突变竞争模型解的长时间行为。我们的主要动机来自于分析的Lotka-Volterra的竞争系统的突变，模拟人口遗传动力学的不同病毒在其主机：$$ \frac{dv_{i}(t)}{dt}=v_i\[r_i-\frac{1}{K}Ψ_i(v)\]+\sum_{j=1}^{N} μ_{ij}(v_j-v_i). $$在第一部分中，我们分析的情况下，竞争条款$Ψ_i$是独立的病毒类型$i$。在这种情况下，在对函数$Ψ_i$、系数$r_i$和变异矩阵$μ_{ij}$的一些相当一般的假设下，我们证明了存在唯一的正的全局稳定的平稳解，即该解吸引从任何非负初始数据开始的轨迹.此外，唯一的稳态$\bar v$是严格正的，在这个意义上，$\bar v_i>0$对所有$i$。这些结果与没有突变项的Lotka-Volterra方程的行为形成鲜明对比，在没有突变项的Lotka-Volterra方程中，已知存在多个非负稳态解，并且存在一个不相容原理（即对于所有的$i\neq i_0, \bar v_{i}=0$和$\bar v_{i_0}>0$）。然后，我们探讨了一个典型的例子，已经提出来解释一些实验数据。对于这种特殊的模型，我们证明了收敛到均衡的速度。在第二部分中，在一些额外的假设下，我们证明了存在一个积极的稳定状态的完整系统，我们分析了长期的动态。证明主要依赖于相对熵的构造，相对熵起着李雅普诺夫泛函的作用。

英文摘要：

In this paper we are investigating the long time behaviour of the solution of a mutation competition model of Lotka-Volterra's type. Our main motivation comes from the analysis of the Lotka-Volterra's competition system with mutation which simulates the demo-genetic dynamics of diverse virus in their host : $$ \frac{dv_{i}(t)}{dt}=v_i\[r_i-\frac{1}{K}Ψ_i(v)\]+\sum_{j=1}^{N} μ_{ij}(v_j-v_i). $$ In a first part we analyse the case where the competition terms $Ψ_i$ are independent of the virus type $i$. In this situation and under some rather general assumptions on the functions $Ψ_i$, the coefficients $r_i$ and the mutation matrix $μ_{ij}$ we prove the existence of a unique positive globally stable stationary solution i.e. the solution attracts the trajectory initiated from any nonnegative initial datum. Moreover the unique steady state $\bar v$ is strictly positive in the sense that $\bar v_i>0$ for all $i$. These results are in sharp contrast with the behaviour of Lotka-Volterra without mutation term where it is known that multiple non negative stationary solutions exist and an exclusion principle occurs (i.e For all $i\neq i_0, \bar v_{i}=0$ and $\bar v_{i_0}>0$). Then we explore a typical example that has been proposed to explain some experimental data. For such particular models we characterise the speed of convergence to the equilibrium. In a second part, under some additional assumption, we prove the existence of a positive steady state for the full system and we analyse the long term dynamics. The proofs mainly rely on the construction of a relative entropy which plays the role of a Lyapunov functional.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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