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

The Pacific halibut fishery has been modeled by the differen- tial equation $$\frac{d y}{d t}=k y\left(1-\frac{y}{K}\right)$$ where $y(t)$ is the biomass (the total mass of the members of the population) in kilograms at time $t($ measured in years), the carrying capacity is estimated to be $K=8 \times 10^{7} \mathrm{kg},$ and $k=0.71$ per year. (a) If $y(0)=2 \times 10^{7} \mathrm{kg},$ find the biomass a year later. (b) How long will it take for the biomass to reach $4 \times 10^{7} \mathrm{kg}$ ?

The Pacific halibut fishery has been modeled by the differen-
tial equation
$$\frac{d y}{d t}=k y\left(1-\frac{y}{K}\right)$$
where $y(t)$ is the biomass (the total mass of the members of
the population) in kilograms at time $t($ measured in years), the
carrying capacity is estimated to be $K=8 \times 10^{7} \mathrm{kg},$ and
$k=0.71$ per year.
(a) If $y(0)=2 \times 10^{7} \mathrm{kg},$ find the biomass a year later.
(b) How long will it take for the biomass to reach $4 \times 10^{7} \mathrm{kg}$ ?

Key Concepts

Logistic Differential Equation

This is a model for population dynamics that combines exponential growth with a decelerating effect as the population nears its environmental limit. The equation, typically written as dy/dt = k y (1 - y/K), illustrates how the growth rate decreases as the population approaches the carrying capacity, blending the concepts of unbounded growth and resource limitation.

Carrying Capacity

Carrying capacity represents the maximum population or biomass that an environment can sustain over time. It is a critical component of the logistic model, as it determines the point at which environmental constraints begin to significantly reduce the growth rate, ultimately leading the population to stabilize.

Growth Rate Constant

The growth rate constant is a parameter that indicates how rapidly a population grows when it is far below its carrying capacity. In the logistic model, it influences the initial speed of population increase, before slowing down as the impact of resource limitations becomes significant.

Initial Value Problem

An initial value problem involves solving a differential equation with a specified starting condition, which determines a unique solution among many possible ones. In population modeling, this allows the prediction of future states of the population based on its known state at a particular initial time.

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