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Determine whether the solutions x(t)=0 and x(t)=1 of the single scalar equation $dx/dt=-x(1-x)$ are stable or unstable.

So far in the book I have just done problems like this except with dx/dt=(some matrix)x and then i find the eigenvalues to determine stability, so I am not really sure how to do this problem.

$dx/dt=-x(1-x)$

$=-x+x^2$

Any help would be greatly appreciated

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Hint

Find the lines of stability or equilibrium. Then check to see what sign $dx/dt$ is above and below the line.

For example, one line of stability is $x=0$.

  • A little bit above this, say at $x=0.1$, we have a $dx/dt < 0$. A negative slope above $x=0$ indicates that the solution asymptotically approaches $x=0$ as $t\rightarrow\infty$. Stable from above.
  • A little bit below it, say at $x=-0.1$, we have $dx/dt >0$. A positive slope below $x=0$ indicates that the solution asymptotically approaches $x=0$ as $t\rightarrow\infty$. Stable from below.

Stable from both above and below $\implies$ $x=0$ is stable.

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Know very little about the topic, but it seems a lot like logistic function: http://en.wikipedia.org/wiki/Logistic_function#In_ecology:_modeling_population_growth

Hope it leads to a solution.

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