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# 18.3: Combinación de dos especies

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Mezclar sistemas de dos especies es un proceso similar, pero tiene más opciones en los parámetros. La Ecuación 18.3.1 es un ejemplo con opciones limitadas que produjeron los espacios de fase en las Figuras 10.1.3 a 10.1.5.

$\frac{1}{N_1}\frac{dN_1}{dt}\,=\,r_1(N_1)\,+\,s_{1,1}N_1\,+\,s_{1,2}(N_1)N_2\\\frac{1}{N_2}\frac{dN_2}{dt}\,=\,r_2(N_2)\,+\,s_{2,2}N_2\,+\,s_{2,1}(N_2)N_1$

Cambiar los parámetros uniformemente de$$a\,b$$ valor cuando el$$N$$ valor correspondiente es 0 a un$$a\,+\,b$$ valor cuando el$$N$$ valor correspondiente es 1 es análogo a la mezcla que produjo Figura 4.4.1. Los parámetros variarían de la siguiente manera, utilizando cuatro$$a$$ valores distintos ($$a_1,\,a_2,\,a_{1,2},\,a_{2,1}$$), más cuatro$$b$$ valores distintos con subíndices coincidentes ($$b_1,\,b_2,\,b_{1,2},\,b_{2,1}$$).

$$r_1(N_1)\,=\,a_1N_1\,+\,b_1,\qquad\,s_{1,2}(N_1)\,=\,a_{1,2}N_1\,+\,b_{1,2}$$

$$r_2(N_2)\,=\,a_2N_2\,+\,b_2,\qquad\,s_{2,1}(N_2)\,=\,a_{2,1}N_2\,+\,b_{2,1}$$

Sustituir lo anterior en la Ecuación 18.3.1 y recopilar términos da una ecuación que tiene todos los términos RSN presentes, pero ahora con un producto cruzado en términos de$$N_1N_2$$ agregado al final:

$\frac{1}{N_1}\frac{dN_1}{dt}\,=\,b_1\,+\,(a_1\,+\,s_{1,1})N_1\,+\,b_{1,2}N_2\,+\,a_{1,2}N_1N_2\\\frac{1}{N_2}\frac{dN_2}{dt}\,=\,b_2\,+\,(a_2\,+\,s_{2,2})N_2\,+\,b_{2,1}N_1\,+\,a_{2,1}N_1N_2$

En el caso específico de las Figuras 10.1.3 a 10.1.5, se utilizó$$s_{1,1}\,=\,s_{2,2}\,=\,−0.98$$ y

$$r_1(N_1)\,=\,0.75N_1\,−0.5\qquad\,s_{1,2}(N_1)\,=\,−1.15N_1\,+\,2.5$$

$$r_2(N_2)\,=\,0.75N_2\,−0.5\qquad\,s_{2,1}(N_2)\,=\,−0.45N_2\,+\,1.3$$

que dio

$$\frac{1}{N_1}\frac{dN_1}{dt}\,=\,-0.5\,-\,0.23N_1\,+\,2.50N_2\,-\,1.15N_1N_2$$

$$\frac{1}{N_2}\frac{dN_2}{dt}\,=\,-0.5\,-\,0.98N_2\,+\,2.50N_1\,-\,0.45N_1N_2$$

para el flujo en las figuras.

This page titled 18.3: Combinación de dos especies is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Clarence Lehman, Shelby Loberg, & Adam Clark (University of Minnesota Libraries Publishing) via source content that was edited to the style and standards of the LibreTexts platform.