The Pacific halibut fishery has been modeled by the equation B(t)=(8 ×10^7)/(1+3 e^-0.71 t) wher

step 1 Okay, so we're trying to take the limit of this as T goes to infinity So if we were to like esse

The Pacific halibut fishery has been modeled by the equation...

The Pacific halibut fishery has been modeled by the equation $$B(t)=\frac{8 \times 10^{7}}{1+3 e^{-0.71 t}}$$ where $B(t)$ is the biomass (the total mass of the members of the population) in kilograms at time $t$. What is $\lim _{t \rightarrow \infty} B(t)?$ What is the significance of this limit?

The Pacific halibut fishery has been modeled by the equation $$B(t)=\frac{8 \times 10^{7}}{1+3 e^{-0.71 t}}$$
where $B(t)$ is the biomass (the total mass of the members of the population) in kilograms at time $t$. What is $\lim _{t \rightarrow \infty} B(t)?$ What is the significance of this limit?

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Key Concepts

Logistic Growth Model

This model is used to describe how populations grow in environments with limited resources. It typically features an initial exponential growth phase that gradually slows down as the population (or biomass) approaches the carrying capacity. The model helps in predicting sustainable levels and long-term behavior of a population.

Carrying Capacity

In population dynamics, the carrying capacity represents the maximum population size (or biomass) that the environment can sustain indefinitely. It is a crucial concept in logistic growth models, marking the equilibrium point where growth levels off due to limited resources.

Limit of a Function

This concept involves determining the value that a function approaches as the independent variable approaches a certain point, often infinity. It is central in understanding the long-term behavior of models in mathematics, particularly in applied fields like biology and economics.

Exponential Decay

Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value, often seen in terms with a negative exponential argument. In long-term analysis, the decaying exponential term tends to zero, simplifying functions and revealing underlying steady state behaviors.

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