## 结果

$p_{2 \to 1}(k) = p_k (q_{2,k}-{s_{2,k} } )(1 -P)\frac{ {\sum \nolimits_j {jp_j q_{1,j}}}}{ {\sum \nolimits_j {jp_j }}}$

$p_{1 \to 2} (k) = p_k (q_{1,k} - s_{1,k} )\left[ {(1 - P) {\frac{ {\sum \nolimits_j {jp_j q_{2,j}} }}{ {\sum\nolimits_j {jp_j }} }} + P} \right]$

$p_{\bar i \to i} (k) = p_k (1 - q_{i,k} - \sum \nolimits_{j \ne i} {s_{j,k} } )(1 - P)\frac{ {\sum \nolimits_j {jp_j q_{i,j}} }}{ {\sum \nolimits_j {jp_j} }}$

$p_{i \to \bar i} (k) = p_k (q_{i,k} - s_{i,k} )\left[ {(1 - P)\left( {1 - \frac{ {\sum\nolimits_j {jp_j q_{i,j} } }}{ {\sum\nolimits_j {jp_j } }}} \right) + P} \right]$

$q_i^{w} = \frac{ {\sum\nolimits_j {jp_j q_{i,j} } }}{ {\sum\nolimits_j {jp_j } }} = \frac{ {\sum\nolimits_j {jn_j q_{i,j} } }}{ {N\left\langle k \right\rangle }}$

$p_{\bar i \to i} (k) = p_k (1 - q_{i,k} - \sum\nolimits_{j \ne i} {s_{j,k} } )(1 - P)q_i^{w} = p_k (q_{\bar i,k} - s_{\bar i,k} )(1 - P)q_i^{w} ,$ $p_{i \to \bar i} (k) = p_k (q_{i,k} - s_{i,k} )\left[ {(1 - P)(1 - q_i^{w} ) + P} \right] = p_k (q_{i,k} - s_{i,k} )\left[ {(1 - P)q_{\bar i}^{w} + P} \right]$

$q_{i,k}$的演化方程为

${d}q_{i,k} =\left[ (p_{\bar i \to i} (k)-p_{i \to \bar i} (k))/p_k \right] {d}t =\left[ {(1 - \sum\nolimits_j {s_{j,k} } )(1 - P)q_i^{w} - q_{i,k} + s_{i,k} } \right] {d}t$

$\hat q_i = s_i + \underbrace {(1 - s)(1 - P)\frac{ {s_i^{w} }}{ {P + (1 - P)\sum\nolimits_j {s_j^{w} } }}}_{ {\rm{Internal \; interactions}}}$

$p_{m \to \bar m} (k) = p_k (q_{m,k} - s_{m,k} )(1 - P)(1 - q_m^{w} ) = p_k (q_{m,k} - s_{m,k} )(1 - P)q_{\bar m}^{w} ,$ $p_{\bar m \to m} (k) = p_k (1 - q_{m,k} - \sum\nolimits_{j \ne m} {s_{j,k} } )\left[ {(1 - P)q_m^{w} + P} \right] = p_k (q_{\bar m,k} - s_{\bar m,k} )\left[ {(1 - P)q_m^{w} + P} \right].$

$q_{m,k}$的演化方程为

${d}q_{m,k} = \left[ {\left( { {\rm{1 - }}\sum\nolimits_j {s_{j,k} } } \right)\left( {(1 - P)q_m^{w} + P} \right) - q_{m,k} + s_{m,k} } \right]{d}t$

$\hat q_m^{w} = \frac{ {s_m^{w} + (1 - \sum\nolimits_j {s_j^{w} } )P}}{ {P + (1 - P)\sum\nolimits_j {s_j^{w} } }}$ $\hat q_m = s_m + \frac{ {(1 - s)(1 - P)s_m^{w} }}{ {P + (1 - P)\sum\nolimits_j {s_j^{w} } }} + (1 - s)P + \frac{ {(1 - s)(1 - P)(1 - \sum\nolimits_j {s_j^{w} } )P}}{ {P + (1 - P)\sum\nolimits_j {s_j^{w} } }}$

Updated: