18.3: Two-species blending

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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.