Question

Using a Logistic Equation In Exercises 51 and 52 , the logistic equation models the growth of a population. Use the equation to (a) find the value of \(k,\) (b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach 50\(\%\) of its carrying capacity, and (e) write a logistic differential equation that has the solution \(P(t)\) . $$ P(t)=\frac{2100}{1+29 e^{-0.75 t}} $$

Short answer

The constant \(k = 0.75\), the carrying capacity \(L = 2100\), the initial population \(P(0) = 70\), and the population reaches half its carrying capacity at about \(t = 1.72\) years. The logistic differential equation is \( \frac{dP}{dt} = 0.75 P(1 - \frac{P}{2100}) \).

Step-by-step solution

Identify the constant of proportionality \(k\)

From the logistic equation \( P(t) = \frac{2100}{1+29 e^{-0.75 t}} \), comparing with standard logistic function, it can be seen that \(k = 0.75\).

Identify the Carrying Capacity

The Carrying Capacity is the maximum sustainable population, represented by \(L\). In this logistic equation, \(L = 2100\).

Identify the Initial Population

The initial population is \( P(0) \). Substituting \( t = 0 \) into the logistic equation gives \( P(0) = \frac{2100}{1+29 e^{0}} = \frac{2100}{1+29} = 70 \).

Find when the population reaches 50% of its carrying capacity

We need to solve for \( t \) when \( P(t) = 0.5L = 0.5*2100 = 1050 \). Substituting \( P(t) = 1050 \) into the equation gives \( 1050 = \frac{2100}{1+29 e^{-0.75 t}} \). Solving for \( t \) gives \( t \approx 1.72 \) years.

Represent As A Logistic Differential Equation

The logistic differential equation is given by \( \frac{dP}{dt} = k P(1 - \frac{P}{L}) \). Substituting \( k = 0.75 \) and \( L = 2100 \), we obtain \( \frac{dP}{dt} = 0.75 P(1 - \frac{P}{2100}) \).

Key concepts

Carrying Capacity

The carrying capacity is a term used in population ecology to describe the maximum number of individuals in a population that an environment can sustain indefinitely without being degraded. In our context, the logistic growth model we're looking at defines this using the variable 'L'. From the equation \(P(t)=\frac{2100}{1+29 e^{-0.75 t}}\), we can determine that the carrying capacity, \(L\), is 2100. This means that no matter how much time passes, the population will approach but never exceed 2100 individuals.

Understanding the concept of carrying capacity is crucial because it sets a boundary for population growth. In real-world scenarios, carrying capacities can be influenced by food availability, habitat space, water supply, and other environmental factors. It represents an equilibrium state where the population size remains stable, barring any changes to environmental conditions or resource availability.

Initial Population

The initial population refers to the size of the population at the beginning of the observation or at time \(t=0\). In the logistic growth model, this is indicated by the equation's value when \(t\) is set to zero. For the given logistic function \(P(0)=\frac{2100}{1+29}\), the initial population is found to be 70. This is a critical starting point when analyzing population dynamics because it sets the stage for future growth.

To understand the importance of the initial population, it's helpful to know that different initial population sizes can affect how quickly a population grows and how long it takes to reach carrying capacity. In general, a smaller initial population may take longer to reach the carrying capacity, while a larger initial population may approach the limit more rapidly.

Logistic Differential Equation

The logistic differential equation is at the heart of modeling population growth in environments with limited resources. It is given by the formula \( \frac{dP}{dt} = kP(1 - \frac{P}{L}) \), where \(P\) represents the population at time \(t\), \(k\) is the constant of proportionality, and \(L\) is the carrying capacity.

Within this context, the equation accommodates the accelerating growth when a population is small and the decelerating growth as the population approaches the carrying capacity. The usage of this equation allows us to predict the growth pattern of the population over time, considering the limitations imposed by the carrying capacity.

The logistic differential equation also focuses on the rate of change of the population, rather than the population size itself, highlighting how growth rates are not constant but change in response to the population size relative to the carrying capacity.

Population Growth

Population growth in the logistic model that we're discussing is depicted as starting off exponentially when the initial population is much smaller than the carrying capacity. However, as it grows and starts to reach the limit set by the carrying capacity, the growth rate slows down and eventually stabilizes.

By using the logistic growth model, we can examine not just the quantity of growth but also the rate at which it occurs. For instance, at half the carrying capacity, we can expect the population growth rate to be at its maximum. Beyond this point, the resources start getting scarcer, leading to a reduction in the growth rate until population size stabilizes around the carrying capacity.

Application in Exercises

In the example provided, we are tasked with determining when the population will reach 50% of its carrying capacity. Through calculations, we determine that this milestone is reached approximately 1.72 years after the initial population size was recorded.

Constant of Proportionality

The constant of proportionality, denoted as \(k\) in population dynamics, is a factor that influences the rate at which the population grows or declines within the logistic growth model. It can be thought of as the inherent per capita rate of growth when conditions are ideal and resources are abundant.

The constant of proportionality reflects both the ability of the population to reproduce and the intensity of the competition for resources. In the given logistic equation, \(k\) is determined by solving the equation and comparing it with the standard logistic function, resulting in \(k = 0.75\). This tells us that, initially, each individual in the population could contribute to the growth rate by a factor of 0.75 per unit of time, in this case years, before considering the effects of reaching the proximity of the carrying capacity.

Population Regulation

A higher value of \(k\) suggests a population that can grow rapidly under ideal conditions, while a lower value indicates a population with a slower growth potential. It plays a significant role in regulating population size and is essential for understanding the dynamics of population growth and stabilization.