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Suppose that a population develops according to the logistic equation(dP/dt)=0.05P−0.00005P2 where t is measured in weeks.(a) What is the carriying capacity?(b) Is the solution increasing or decreasing when P is between 0 and the carriying capacity?(c) Is the solution increasing or decreasing when P is greater than the carriying capacity?

Respuesta :

LammettHash

a.

[tex]\dfrac{\mathrm dP}{\mathrm dt}=0.05P-0.00005P^2=0.00005P(1000-P)[/tex]

so that [tex]\frac{\mathrm dP}{\mathrm dt}=0[/tex] for [tex]P=0[/tex] or [tex]P=1000[/tex]. The derivative being equal to 0 gives you some information about the long-term stability of the population [tex]P[/tex]; namely, that the carrying capacity is 1000.

b. If [tex]0<P<1000[/tex], then [tex]0.00005P>0[/tex] and [tex]1000-P>0[/tex], so that [tex]\frac{\mathrm dP}{\mathrm dt}>0[/tex], which means the population would be increasing.

c. If [tex]P>1000[/tex], then [tex]0.00005P>0[/tex] but [tex]1000-P<0[/tex], so [tex]\frac{\mathrm dP}{\mathrm dt}<0[/tex] and the population would be decreasing.

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