46105 以數學模型來詮釋「利他主義」的真諦

\begin{align} \frac{dN_1}{dt}=r_1 N_1 \Big(\frac{K_1-N_1-\alpha_{12} N_2}{K_1}\Big)=f(N_1,N_2 ),\label{1}\\ \frac{dN_2}{dt}=r_2 N_2 \Big(\frac{K_2-N_2-\alpha_{21} N_1}{K_2}\Big)=h(N_1,N_2 ),\label{2} \end{align}

\begin{align} \frac{dN_1}{dt}=r_1 N_1 \Big(\frac{K_1-N_1-\alpha_{12} N_2}{K_1}\Big)=f(N_1,N_2 ),\tag*{(1)}\\ \frac{dN_2}{dt}=r_2 N_2 \Big(\frac{K_2-N_2}{K_2}\Big)=g(N_1,N_2 ),\label{3} \end{align}

\begin{align} \frac d{dt} \Delta N_1=&%\frac{\p f}{\p N_{1_{(N_{10},N_{20})}}} f_{N_1}(N_{10},N_{20}) \Delta N_1+%\frac{\p f}{\p N_{2_{(N_{10},N_{20})}}} f_{N_2}(N_{10},N_{20}) \Delta N_2,\tag*{(4-1)}\\ \frac d{dt} \Delta N_2=&%\frac{\p g}{\p N_{1_{(N_{10},N_{20})}}} g_{N_1}(N_{10},N_{20}) \Delta N_1+%\frac{\p g}{\p N_{2_{(N_{10},N_{20})}}} g_{N_2}(N_{10},N_{20}) \Delta N_2,\tag*{(5-1)}\\ \hbox{ 亦即是 } \frac d{dt} \Delta N_1=&\Big\{r_1 \Big(\frac{K_1-N_{10}-\alpha_{12} N_{20}}{K_1} \Big)-\frac{r_1 N_{10}}{K_1}\Big\}\Delta N_1 -\frac{r_1 \alpha_{12} N_{10}}{K_1}\Delta N_2,\label{4}\\ \frac d{dt} \Delta N_2=&\Big\{r_2 \Big(\frac{K_2-N_{20}}{K_2} \Big)-\frac{r_2 N_{20}}{K_2} \Big\}\Delta N_2.\label{5} \end{align}

(a) 針對 $SS_1(0,0)$ (亦即兩物種全數滅絕狀況), 此時 $(N_{10},N_{20})=(0,0)$ \begin{align} \frac d{dt} \Delta N_1=&r_1 \Delta N_1,&\Delta N_1=\Delta N_{10} e^{r_1 t},~\qquad~\tag*{(4a)}\\ \frac d{dt} \Delta N_2=&r_2 \Delta N_2,&\Delta N_2=\Delta N_{20} e^{r_2 t}.~\qquad~\tag*{(5a)} \end{align}

(b) 針對 $SS_2(K_1,0)$ (亦即他種生物獨存, 而人類全數滅絕的狀況), 此時 $(N_{10},N_{20})=(K_1,0)$ \begin{align} \frac d{dt} \Delta N_1=&=-r_1 \Delta N_1-r_1 \alpha_{12} \Delta N_2,\tag*{(4b)}\\ \frac d{dt} \Delta N_2=&r_2 \Delta N_2\qquad \hskip 1cm\Delta N_2=\Delta N_{20} e^{r_2 t}.\hskip 1cm~\tag*{(5b)}\\ \hbox{依此推導可得} \Delta N_1=&\Delta N_{10} e^{-r_1 t}+\frac{r_1 \alpha_{12} \Delta N_{20}}{r_1+r_2}(e^{-r_1 t}-e^{r_2 t}).\tag*{(4b*)} \end{align}

(c) 針對 $SS_3(0, K_2)$ (亦即他種生物全數滅絕, 而人類獨活的窘境), 此時 $(N_{10},N_{20})=(0,K_2)$, 因此可得 \begin{align} \frac d{dt} \Delta N_1=&r_1 \Big(\frac{K_1-\alpha_{12} K_2}{K_1} \Big)\Delta N_1,&\Delta N_1 =&\Delta N_{10} e^{r_1 (1-\frac{\alpha_{12} K_2}{K_1})t},\hskip 1cm~\tag*{(4c)}\\ \frac d{dt} \Delta N_2=&-r_2 \Delta N_2,&\Delta N_2=&\Delta N_{20} e^{-r_2 t}.\tag*{(5c)} \end{align}

(d) 針對 $SS_4(K_1-\alpha_{12} K_2, K_2)$ (亦即兩物種共存的結局), 此時 $(N_{10},N_{20})=(K_1-\alpha_{12} K_2$, $K_2)$。 \begin{align} \frac d{dt} \Delta N_1=&\Big\{-\frac{r_1 (K_1-\alpha_{12} K_2)}{K_1}\Big\}\Delta N_1-\frac{r_1 \alpha_{12} (K_1-\alpha_{12} K_2)}{K_1} \Delta N_2,\hskip 1cm~\tag*{(4d)}\\ \frac d{dt} \Delta N_2=&-r_2 \Delta N_2\qquad~\hskip 2cm \Delta N_2=\Delta N_{20} e^{-r_2 t}.\tag*{(5d)}\\ \hbox{若設定 $\varnothing=K_1-\alpha_{12} K_2$, 則可推導得到} \Delta N_1=&\Delta N_{10} e^{-r_1\frac{\varnothing}{K_1}t}-\frac{r_1 \alpha_{12}\varnothing\Delta N_{20}} {\varnothing r_1-r_2 K_1}(e^{-r_2 t}-e^{-r_1 \frac{\varnothing}{K_1}t}). \tag*{(4d*)} \end{align}

"Only if we understand, will we care. Only if we care, will we help. Only if we help, shall all be saved."....... Jane Goodall

### 參考文獻

B. Y. Chen, Revealing characteristics of mixed consortia for azo dye decolorization: Lotka-Volterra model and game theory, Journal of Hazardous Materials 149, 508-514, 2007. N. J. Gotelli, Chapter 5 Competition, In: A Primer of Ecology, Sinauer Association Inc., Sunderland, MA, USA, pp. 111-138, 1995. P. J. Morin, Chapter 10 Competition: mechanisms, models, and niches, In: Community Ecology, Blackwell Science Inc., Malden, MA, USA, pp.29-66, 1999. G. F. Gause, Competitive Exclusion Principle. Science 131, 1292-1297, 1960.

---本文作者陳博彥任教於國立宜蘭大學化學工程與材料工程學系---