A population P obeys the logistic model. It satisfies the equation dP/dt=2/500*P

Question

A population P obeys the logistic model. It satisfies the equation
dP/dt=2/500*P(5-P) for P>0.

a) The population is increasing when ?<P<?
b) The population is decreaing when P>?
c) Assume that P(0)=3, find P(88)

P(88)=?

Explanation / Answer

dP/dt= 0.2P-0.002(P^2) as a differential equation must be solved dt = dP/(P/5 - P^2/500) = 5*dP/[P*(100-P)] = 5 * dP/P + 5 * dP/(100-P) so t = 5*LN(P) - 5*LN(P-100) + C t - C = 5*LN[P/(P-100)] = LN[P^5/(P-100)^5] or P = 100* e^[(t-C)/5]/{e^[(t-C)/5]-1} (a) the domain of P for the natural logarithm is P > 0 for the first log term and P - 100 > 0 for the second this means P > 100 the carrying capacity is 100 units (b) the solution undefined over the range between 0 and the carrying capacity of 100 units (c) the solution is decreasing when P is greater than the 100-unit carrying capacity

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