The rate of change (dP)/(dt) of the number of deer on an island is modeled by the following differential equation: (dP)/(dt)=(4)/(8721)P(754-P) At t=0, the number of deer on the island is 108 and is increasing at a rate of 32 deer per day. Find lim_(t rarr oo)P^(')(t). Answer:

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The rate of change  (dP)/(dt) of the number of deer on an island is modeled by the following differential equation:  (dP)/(dt)=(4)/(8721)P(754-P) At  t=0, the number of deer on the island is 108 and is increasing at a rate of 32 deer per day. Find  lim_(t rarr oo)P^(')(t). Answer:

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Equilibrium Points: The given differential equation is (dP)/(dt) = (4/8721)P(754-P). To find the limit of P'(t) as t approaches infinity, we need to analyze the behavior of the function defined by the differential equation. The equilibrium points occur when (dP)/(dt) = 0, which happens when P = 0 or P = 754. Since the population cannot be negative, we discard P = 0 as a meaningful equilibrium in this context. Therefore, we consider P = 754 as the carrying capacity of the island, which is the maximum population that the environment can sustain indefinitely.Population Behavior: As t approaches infinity, the population P(t) will approach the carrying capacity, because the rate of change (dP)/(dt) will decrease as P gets closer to 754. This is due to the fact that (754 - P) becomes smaller, reducing the product P(754 - P), and thus reducing the rate of change (dP)/(dt).Approaching Carrying Capacity: Since the rate of change (dP)/(dt) depends on the product P(754 - P), as P approaches 754, the term (754 - P) approaches 0. This means that the rate of change of the population will approach 0 as the population approaches the carrying capacity.Limit of Rate of Change: Therefore, the limit of P'(t) as t approaches infinity is 0, because the population will stabilize at the carrying capacity and there will be no further change in the number of deer on the island.

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