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反应扩散模型和亚种群模型 (查看源代码)

2022年3月16日 (三) 09:37的版本

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The first term of the equation considers that only a fraction $d_k$ of particles moves out of the node per unit time. The second term instead accounts for the particles diffusing from the neighbors into the node of degree k. This term is proportional to the number of links k times the average number of particles coming from each neighbor. This is equal to the average over all possible degrees $k'$ of the fraction of particles moving on that edge, $d_{k'k}$, according to the conditional probability $P(k'|k)$ that an edge belonging to a node of degree k is pointing to a node of degree $k'$. Here the term $d_{k'k}$ is the diffusion rate along the edges connecting nodes of degree k and $k'$. In the simplest case of homogeneous diffusion each particle diffuses with rate $r$ from the node in which it is and thus the diffusion per link $d_{k'k}=r/k'$. On uncorrelated networks $P(k'|k)=k'P(k')/\left<k\right>$ and hence one easily gets in the stationary state $d\mathcal{N}/dt=0$ the solution <ref>Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.</ref><ref>Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>

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The first term of the equation considers that only a fraction $d_k$ of particles moves out of the node per unit time. The second term instead accounts for the particles diffusing from the neighbors into the node of degree k. This term is proportional to the number of links k times the average number of particles coming from each neighbor. This is equal to the average over all possible degrees $k'$ of the fraction of particles moving on that edge, $d_{k'k}$, according to the conditional probability $P(k'|k)$ that an edge belonging to a node of degree k is pointing to a node of degree $k'$. Here the term $d_{k'k}$ is the diffusion rate along the edges connecting nodes of degree k and $k'$. In the simplest case of homogeneous diffusion each particle diffuses with rate $r$ from the node in which it is and thus the diffusion per link $d_{k'k}=r/k'$. On uncorrelated networks $P(k'|k)=k'P(k')/\left<k\right>$ and hence one easily gets in the stationary state $d\mathcal{N}/dt=0$ the solution <ref name=":0">Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.</ref><ref name=":1">Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.</ref>

方程中的第一项表示在每个单位时间内有比例为<math> d_k </math>的粒子从度为<math> k </math>的节点中离开，第二项表示外部粒子从邻居节点扩散到度为<math> k </math>的节点中，该项与节点的连边数<math> k </math>成正比，也就是说一个节点与其邻居节点的连接越多，外部粒子扩散进入该节点的可能性越大。条件概率<math>P(k'|k)</math>表示的是一条边的一端连接度为<math>k</math>的节点时，其另一端指向度为<math>k'</math>的概率。<math>d_{k'k}</math>表示的是两端连接了度为<math>k</math>和<math>k'</math>节点的连边的扩散率。在均匀扩散的最简单情况下，每个粒子从其所在的节点以速率<math>r</math>扩散，因此沿着每条连边的扩散率为<math>d_{k'k}=r/k'</math>。在无关联的网络上，<math>P(k'|k)=k'P(k')/\left<k\right></math>，因此在稳态<math>d\mathcal{N}/dt=0</math>时，很容易得到解（Colizza等，2007b； Noh和Rieger，2004）：

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\end{equation} </nowiki>

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上面的公式显式地给出了系统中粒子扩散的描述，并指出网络拓扑结构在反应扩散过程中的重要性。从式中可以看到，如果一个节点的度越大，则其被扩散中的粒子访问到的可能性就越大。

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上面的公式显式地给出了系统中粒子扩散的描述，并指出网络拓扑结构在反应扩散过程中的重要性。从式中可以看到，如果一个节点的度越大，则其被扩散中的粒子访问到的可能性就越大。<ref name=":0" /><ref name=":1" />

==亚种群模型==

Bnustv

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