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# The system of differential equations$\frac {dx}{dt} = 0.5x - 0.004x^2 - 0.001xy $$\frac {dy}{dt} = 0.4y - 0.001y^2 - 0.002xy is a model for the populations of two species.(a) Does the model describe cooperation, or competition, or a predator-prey relationship?(b) Find the equilibrium solutions and explain their significance. ### Answer ## (A).$$\begin{array}{l}d x / d t=0.5 x-0.004 x^{2}-0.001 x y=0.5 x(1-x / 125)-0.001 x y \\d y / d t=0.4 y-0.001 y^{2}-0.002 x y=0.4 y(1-y / 400)-0.002 x y\end{array}$$(B). There are four Equilibrium Solutions:(1)$$x=0$ and $y=0$$(2)$$x=0 \text { and } 400-y-2 x=0 \Rightarrow x=0 \text { and } y=400$$(3)$$500-4 x-y=0 \text { and } y=0 \Rightarrow x=125 \text { and } y=0$$(4)$500-4 x-y=0$and$400-y-2 x=0 \Rightarrow y=500-4 x$and$y=400-2 x \Rightarrow 500-4 x=400-2 x \Rightarrow100=2 x \Rightarrow x=50$and$y=300\$

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Differential Equations

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