18.3: Two-species blending
- Page ID
- 25536
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Blending two-species systems is a similar process, but has more options in the parameters. Equation 18.3.1 is an example with limited options that produced the phase spaces in Figures 10.1.3 through 10.1.5.
\[\frac{1}{N_1}\frac{dN_1}{dt}\,=\,r_1(N_1)\,+\,s_{1,1}N_1\,+\,s_{1,2}(N_1)N_2\\\frac{1}{N_2}\frac{dN_2}{dt}\,=\,r_2(N_2)\,+\,s_{2,2}N_2\,+\,s_{2,1}(N_2)N_1\]
Changing the parameters uniformly from \(a\,b\) value when the corresponding \(N\) value is 0 to an \(a\,+\,b\) value when the corresponding \(N\) value is 1 is analogous to the blending that produced Figure 4.4.1. The parameters would vary as follows, using four distinct \(a\) values (\(a_1,\,a_2,\,a_{1,2},\,a_{2,1}\)), plus four distinct \(b\) values with matching subscripts (\(b_1,\,b_2,\,b_{1,2},\,b_{2,1}\)).
\(r_1(N_1)\,=\,a_1N_1\,+\,b_1,\qquad\,s_{1,2}(N_1)\,=\,a_{1,2}N_1\,+\,b_{1,2}\)
\(r_2(N_2)\,=\,a_2N_2\,+\,b_2,\qquad\,s_{2,1}(N_2)\,=\,a_{2,1}N_2\,+\,b_{2,1}\)
Substituting the above into Equation 18.3.1 and collecting terms gives an equation having all the RSN terms present, but now with a cross-product in terms of \(N_1N_2\) added at the end:
\[\frac{1}{N_1}\frac{dN_1}{dt}\,=\,b_1\,+\,(a_1\,+\,s_{1,1})N_1\,+\,b_{1,2}N_2\,+\,a_{1,2}N_1N_2\\\frac{1}{N_2}\frac{dN_2}{dt}\,=\,b_2\,+\,(a_2\,+\,s_{2,2})N_2\,+\,b_{2,1}N_1\,+\,a_{2,1}N_1N_2\]
In the specific case of Figures 10.1.3 through 10.1.5, we used \(s_{1,1}\,=\,s_{2,2}\,=\,−0.98\) and
\(r_1(N_1)\,=\,0.75N_1\,−0.5\qquad\,s_{1,2}(N_1)\,=\,−1.15N_1\,+\,2.5\)
\(r_2(N_2)\,=\,0.75N_2\,−0.5\qquad\,s_{2,1}(N_2)\,=\,−0.45N_2\,+\,1.3\)
which gave
\(\frac{1}{N_1}\frac{dN_1}{dt}\,=\,-0.5\,-\,0.23N_1\,+\,2.50N_2\,-\,1.15N_1N_2\)
\(\frac{1}{N_2}\frac{dN_2}{dt}\,=\,-0.5\,-\,0.98N_2\,+\,2.50N_1\,-\,0.45N_1N_2\)
for the flow in the figures.