Consider the budworm population dynamics governed by the equation

where ,k the carrying capacity, and r, the birth rate of the budworm population, are positive parameters. Find out the steady states and use the perturbation to do the stability analysis of the equation for 0 < r < 1.

To find the steady states of the given equation for budworm population dynamics, we set the equation equal to zero:

\[0 = rP \left(1 - \frac{P}{k}\right).\]

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