Question

Using a Logistic Differential Equation In Exercises 53 and \(54,\) the logistic differential equation models the growth rate of a population. Use the equation to (a) find the value of \(k,\) (b) find the carrying capacity, (c) graph a slope field using a computer algebra system, and (d) determine the value of \(P\) at which the population growth rate is the greatest. $$ \frac{d P}{d t}=0.1 P-0.0004 P^{2} $$

Short answer

The value of \(k\) is 0.1, the carrying capacity \(M\) is 250, and the population growth rate is greatest when \(P\) is 125.

Step-by-step solution

Solve for k

The logistic differential equation is given as \(\frac{dP}{dt} = kP(1 - \frac{P}{M})\), where \(k\) is the rate of growth and \(M\) is the carrying capacity. The exercise provides the equation in the form of \(\frac{dP}{dt} = 0.1P - 0.0004P^2\). By comparing coefficients, we find that \(k\) is equal to 0.1.

Find the Carrying Capacity \(M\)

The carrying capacity \(M\) is the value of \(P\) for which the function \(\frac{dP}{dt}\) becomes 0. This translates to \(kP = 0.0004P^2\). Solving for \(P\) when \(k = 0.1\), the carrying capacity \(M\) is equal to 250.

Graph a Slope Field

Using a computer algebra system such as Desmos or Geogebra, graph the slope field for the logistic differential equation \(\frac{dP}{dt}\).

Compute the Value of \(P\) at which Growth Rate is Greatest

To determine the value of \(P\) for which the population's growth rate is the greatest, take the derivative of \(\frac{dP}{dt}\) and equate it to 0. This gives us \(P = \frac{M}{2}\). Substituting \(M = 250\) from Step 2, we find that the population growth rate is greatest when \(P\) equals 125.

Key concepts

Differential Equations

Differential equations are mathematical expressions that describe the relationship between a function and its derivatives. They are an essential part of calculus and are used extensively in various scientific disciplines such as physics, engineering, and biology. In essence, they allow us to model how a certain quantity changes over time or space. A logistic differential equation is a specific type of differential equation commonly used to model situations where growth is limited by resource availability or other environmental factors, such as population growth.

Population Growth Model

In the field of ecology, population growth models are used to predict changes in population sizes over time. The simplest model is exponential growth, which assumes unlimited resources and space. However, in real-world scenarios, there are limits to how much a population can grow - this is where the logistic growth model comes into play. It describes a more realistic scenario where the growth rate decreases as the population size approaches a maximum capacity—the carrying capacity of the environment.

Carrying Capacity

Carrying capacity, often denoted by the symbol 'M' in logistic models, represents the maximum population size of a species that an environment can sustain indefinitely. Factors determining carrying capacity include resource availability, competition between individuals, predation, disease, and habitat space. In the logistic differential equation, the carrying capacity plays a crucial role, as it provides a ceiling to the population size, beyond which growth is no longer feasible.

Slope Field

A slope field, also known as a direction field, is a graphical representation of the solutions to a first-order differential equation without finding an explicit formula. By plotting small line segments with the slope given by the differential equation at various points, a slope field visually demonstrates the behavior of the solution curves. Using a computer algebra system to graph a slope field helps to predict and understand the patterns and directions of the population growth over time.

Growth Rate

The growth rate in the context of the logistic population model refers to how quickly the population size changes with time. Initially, when the population size is low compared to the carrying capacity, the growth rate is high, similar to exponential growth. As the population increases, the growth rate decreases until it reaches zero at the carrying capacity. In mathematical terms, the growth rate is represented by the first derivative of the population size with respect to time, \(\frac{dP}{dt}\).

Calculus

Calculus, the branch of mathematics that deals with continuous change, is the foundation for understanding and solving differential equations. In calculus, we study the concepts of limits, derivatives, and integrals, which are tools essential for analyzing the rate of change (in our case, population growth). The logistic differential equation combines principles of calculus with biological insights to model population dynamics mathematically. Through techniques such as differentiation, we can analyze changes in growth rate and identify the values at which the growth rate is at its peak.