Problem 49 - Calculus for Biology and Medicine

Question

Suppose that the rate of change of the size of a population is given by $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right) $$ where \(N=N(t)\) denotes the size of the population at time \(t\) and \(r\) and \(K\) are positive constants. Find the equilibrium size of the population-that is, the size at which the rate of change is equal to \(0 .\) Use your answer to explain why \(K\) is called the carrying capacity.

Step-by-Step Solution

Answer

The equilibrium size of the population is \( N = K \). This is why \( K \) is called the carrying capacity, as it is the stable size that the environment can support.

Step 1: Understanding the Differential Equation

The given differential equation is \( \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) \). Here, \(N\) represents the population size at time \(t\), \(r\) is the growth rate, and \(K\) is the carrying capacity. We want to find the equilibrium size of the population when the rate of change, \( \frac{dN}{dt} \), is 0.

Step 2: Setting the Rate of Change to Zero

To find the equilibrium point, set the rate of change equal to zero: \( \frac{dN}{dt} = 0 = rN \left( 1 - \frac{N}{K} \right) \). This means the population size is neither increasing nor decreasing.

Step 3: Solving for the Equilibrium Population Size

Solve the equation \( 0 = rN \left( 1 - \frac{N}{K} \right) \). This results in two possibilities: either \( N = 0 \) or \( 1 - \frac{N}{K} = 0 \).

Step 4: Finding the Non-Trivial Solution

The trivial solution \( N = 0 \), implies no population. For meaningful equilibrium, solve \( 1 - \frac{N}{K} = 0 \) to get: \( 1 = \frac{N}{K} \). Multiply both sides by \( K \) to find \( N = K \).

Step 5: Explaining the Carrying Capacity

At \( N = K \), the equation \( \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) \) becomes 0, indicating that the population has reached a stable size that the environment can support without leading to growth or decline. This is why \( K \) is referred to as the carrying capacity.

Key Concepts

Logistic Growth Model

The logistic growth model is used to describe how populations grow in an environment with limited resources. At the heart of this model is the idea that as a population grows, resources such as food and space become more scarce, which in turn slows down the growth rate. This model is expressed using the equation:\[\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)\]where:

• \(N\) is the population size at time \(t\)
• \(r\) is the intrinsic rate of increase
• \(K\) is the carrying capacity of the environment

The logistic model goes beyond simple exponential growth by incorporating the concept of carrying capacity, highlighted through the term \(1 - \frac{N}{K}\). This term effectively reduces the growth rate as the population approaches the carrying capacity, \(K\). As a result, the population growth gradually slows and eventually stabilizes as it reaches a size that the environment can comfortably support.

Differential Equations

In mathematics and theoretical biology, differential equations are crucial in modeling how populations change over time. A differential equation expresses a relationship between a function and its derivatives. In the case of the population dynamics equation, it shows how the rate of change in population size \(\frac{dN}{dt}\) is dependent on the current population size \(N\).Understanding differential equations allows us to predict future scenarios. For instance, it provides insights into how quickly a population will grow, how environmental limitations impact growth, and when the growth will cease. In the logistic growth model:

• \(\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)\) is of the form \(\frac{dN}{dt} = f(N)\), showing how \(N\) changes over time.
• By setting \(\frac{dN}{dt} = 0\), we can solve for \(N\) at equilibrium.

Differential equations like these guide the exploration of dynamic systems in real-world applications.

Carrying Capacity

The carrying capacity, denoted by \(K\), is a central concept in ecology representing the maximum population size that an environment can sustainably support. In the logistic growth model equation:\[\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)\]\(K\) acts as a threshold. As the population size \(N\) approaches \(K\), the term \(1 - \frac{N}{K}\) approaches zero, reducing growth to zero. This behavior ensures that the population won't exceed the environmental limits, thus preventing depletion of resources.In practical terms:

• If \(N < K\), the population can still grow.
• If \(N = K\), the population size remains stable, as the birth rate balances with the death rate, and no growth occurs.
• If \(N > K\), the population will decrease since resources can't support the excess size.

Understanding carrying capacity helps in managing resources effectively and sustaining ecosystems.

Equilibrium Analysis

Equilibrium analysis involves finding the stable states of a system where the system is at rest (i.e., the rate of change is zero). In population dynamics, this requires setting the rate of change of the population, \(\frac{dN}{dt}\), to zero.By doing so in the logistic growth equation:\[0 = rN \left(1 - \frac{N}{K}\right)\]We identify the equilibrium states as:

• \(N = 0\): A trivial solution indicating no population.
• \(N = K\): A meaningful equilibrium where the population is stable and fully supported by the environment.

Equilibrium analysis confirms that the population will stabilize at the carrying capacity \(K\) when resources are optimally utilized. This analysis allows ecologists and environmentalists to comprehend how populations interact with their environments and predict how external changes might impact these systems.