Problem 26 - Calculus for Biology and Medicine

Question

Suppose the size of a population at time \(t\) is \(N(t)\), and the growth rate of the population is given by the logistic growth function $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right), \quad t \geq 0 $$ where \(r\) and \(K\) are positive constants. (a) Graph the growth rate \(\frac{d N}{d t}\) as a function of \(N\) for \(r=3\) and \(K=10\) (b) The function \(f(N)=r N(1-N / K), N \geq 0\), is differentiable for \(N>0\). Compute \(f^{\prime}(N)\), and determine where the function \(f(N)\) is increasing and where it is decreasing.

Step-by-Step Solution

Answer

(a) The growth rate graph is a downward parabola peaking at \( N = 5 \), zero at \( N = 0 \) and \( N = 10 \). (b) \( f(N) \) is increasing for \( 0 < N < 5 \) and decreasing for \( N > 5 \).

Step 1: Understand the Logistic Growth Rate Function

The problem provides the logistic growth rate function: \( \frac{dN}{dt} = rN\left(1-\frac{N}{K}\right) \). This represents the rate of change of the population \( N \) with respect to time \( t \). The function is characterized by the rate \( r \) and the carrying capacity \( K \).

Step 2: Graph the Logistic Growth Rate Function for Part (a)

Substitute the given values \( r=3 \) and \( K=10 \) into the function: \( \frac{dN}{dt} = 3N\left(1-\frac{N}{10}\right) \). The graph of this function as a function of \( N \) will have \( N \) on the x-axis and \( \frac{dN}{dt} \) on the y-axis. It is a downward-facing parabola starting at the origin, peaking at \( N = 5 \), and returning to zero at \( N = 10 \).

Step 3: Express and Differentiate the Logistic Function for Part (b)

The function given is \( f(N) = 3N(1 - \frac{N}{10}) \). To find \( f'(N) \), expand the function: \( f(N) = 3N - \frac{3N^2}{10} \). Differentiate this expression: \( f'(N) = 3 - \frac{3}{5}N \).

Step 4: Determine Increasing and Decreasing Intervals

To find where the function \( f(N) \) is increasing or decreasing, analyze \( f'(N) = 3 - \frac{3}{5}N \). Set \( f'(N) = 0 \) to find critical points: \( 3 - \frac{3}{5}N = 0 \Rightarrow N = 5 \). For \( 0 < N < 5 \), \( f'(N) > 0 \) indicating \( f(N) \) is increasing. For \( N > 5 \), \( f'(N) < 0 \), indicating \( f(N) \) is decreasing.

Key Concepts

Differential Equations

Differential equations are mathematical tools used to model how quantities change with respect to one another.
In the context of population dynamics, they help us understand how populations grow over time.
The logistic growth model, represented by the differential equation \( \frac{dN}{dt} = rN\left(1-\frac{N}{K}\right) \), describes the rate of change of a population, \( N \), over time, \( t \).
This equation takes into account both an intrinsic growth rate \( r \) and a limiting factor \( K \), known as the carrying capacity.
Understanding the solutions of a differential equation helps us predict future population sizes based on current data.

• The term \(\frac{dN}{dt}\) represents the rate at which the population changes.
• The expression \( rN\left(1-\frac{N}{K}\right) \) balances the population's potential growth with environmental constraints.
• Simplifying differential equations often involves finding a particular solution that suits a given set of initial conditions.

Population Dynamics

Population dynamics is the study of how populations of living organisms change over time due to births, deaths, and migration patterns.
In the logistic growth model, we analyze how the size of a population, \( N(t) \), changes as it approaches the carrying capacity \( K \).
The logistic equation suggests that populations grow rapidly at first, slow down as they near the carrying capacity, and eventually stabilize at \( K \).

• Initially, when \( N \) is small, the growth rate \( rN(1-\frac{N}{K}) \) is close to exponential, as the effect of the carrying capacity is minimal.
• As \( N \) grows larger, the term \( 1-\frac{N}{K} \) becomes smaller, slowing the growth rate.
• At \( N = K \), the growth rate reaches zero, indicating the population has stabilized at its carrying capacity.

Studying population dynamics helps identify how factors like resource availability and environmental conditions impact growth.

Carrying Capacity

The concept of carrying capacity is crucial in understanding how populations interact with their environment.
Carrying capacity, \( K \), refers to the maximum number of individuals an environment can sustainably support based on available resources like food, water, and space.

• In the logistic growth model, \( K \) is a limiting factor. As the population approaches \( K \), growth slows and eventually stops.
• The value of \( K \) can change based on environmental conditions. For example, a drought may reduce \( K \), while a sudden increase in food supply may raise it.
• Understanding \( K \) helps in predicting population behavior and planning for resource conservation.

By incorporating the concept of carrying capacity into their models, ecologists and environmental planners can make informed decisions to maintain ecological balance.