Problem 77 - Calculus Single Variable

Question

Suppose that the size \(P(t)\) of a population at time \(t\) is $$ P(t)=\frac{P_{0} M}{P_{0}+\left(M-P_{0}\right) e^{-k M t}} $$ for some positive constants \(K, P_{0},\) and \(M>P_{0} .\) (This growth model is known as logistic growth. a. What is the initial population size? What is the limiting size of the population as \(t \rightarrow \infty\) ? b. Verify that \(P^{\prime}(t)=k P(t)(M-P(t))\).

Step-by-Step Solution

Answer

The initial population size is \(P_0\), and the limiting size as \(t \to \infty\) is \(M\). The derivative \(P'(t)\) matches \(kP(t)(M-P(t))\).

Step 1: Determine Initial Population Size

To find the initial population size, substitute \(t = 0\) into the population equation: \[P(0) = \frac{P_0 M}{P_0 + (M-P_0) e^{-k M \cdot 0}} = \frac{P_0 M}{P_0 + (M-P_0) \cdot 1} = P_0.\] Therefore, the initial population size is \(P_0\).

Step 2: Determine Limiting Population Size as \(t \rightarrow \infty\)

As \(t\) approaches infinity, \(e^{-kMt}\) approaches zero because \(e^{-x} \rightarrow 0\) as \(x \rightarrow \infty\). Substituting into the equation gives: \[ P(t) = \frac{P_0 M}{P_0 + (M-P_0) \cdot 0} = \frac{P_0 M}{P_0} = M. \] Thus, the limiting size of the population is \(M\).

Step 3: Verify the Derivative Formula

Start by finding the derivative \(P'(t)\) using the quotient rule. Given: \[ P(t) = \frac{P_0 M}{P_0 + (M-P_0) e^{-k M t}}. \]Let \(f(t) = P_0 M\) and \(g(t) = P_0 + (M-P_0) e^{-k M t}\). Using the quotient rule \((\frac{u}{v})' = \frac{u'v - uv'}{v^2}\): 1. \(f'(t) = 0\) (since \(f(t)\) is a constant). 2. \(g'(t) = \frac{d}{dt}(P_0 + (M-P_0) e^{-k M t}) = -(M-P_0) (-kM) e^{-k Mt} = kM(M-P_0)e^{-kMt} \).Applying the derivatives to the quotient rule:\[ P'(t) = \frac{0 \cdot g(t) - f(t) \cdot g'(t)}{g(t)^2} = -\frac{P_0 M \cdot kM(M-P_0)e^{-kMt}}{(P_0 + (M-P_0)e^{-kMt})^2}. \]Plug \(P(t) = kP(t)(M - P(t))\) into the logistic equation:\[ k P(t) (M - P(t)) = k \frac{P_0 M}{P_0 + (M-P_0)e^{-kMt}} \left( M - \frac{P_0 M}{P_0 + (M-P_0)e^{-kMt}} \right). \]Simplify inside:\[ M - \frac{P_0 M}{P_0 + (M-P_0)e^{-kMt}} = \frac{M(P_0 + (M-P_0)e^{-kMt}) - P_0 M}{P_0 + (M-P_0)e^{-kMt}} = \frac{M(M-P_0)e^{-kMt}}{P_0 + (M-P_0)e^{-kMt}} \].Replace and simplify:\[ k P(t) (M-P(t)) = k \frac{P_0 M \cdot M(M-P_0)e^{-kMt}}{(P_0 + (M-P_0)e^{-kMt})^2}, \]which matches \( P'(t) = k P(t) (M - P(t)) \).

Step 4: Conclusion

We've identified that the initial population size is \(P_0\) and as \(t\) approaches infinity, the population size approaches \(M\). We have also confirmed that the expression for \(P'(t)\) satisfies \(P'(t) = k P(t)(M-P(t))\) by correctly applying the derivative steps.

Key Concepts

Population Dynamics

Population dynamics is the study of how populations change in size and composition over time. In the case of logistic growth, such as the one described in the exercise, the model considers factors that limit the growth of a population. This is a more realistic portrayal compared to exponential growth, where no such limits are considered.
The logistic growth model is represented by the equation \[ P(t)=\frac{P_{0} M}{P_{0}+(M-P_{0})e^{-k M t}} \]. This model integrates the initial conditions (starting population size) and assumes that as time passes, a limiting factor will prevent the population from growing indefinitely.

• **Initial Population Size**: This is designated as \(P_0\) at \(t=0\), representing the starting point of the population.
• **Carrying Capacity**: Defined as \(M\), it's the maximum population size that the environment can sustain indefinitely.
• **Growth Rate**: The constant \(k\) regulates how quickly the population approaches its carrying capacity.

Understanding these elements helps in predicting how real-world populations behave and adapt over time.

Differential Equations

Differential equations are equations that involve functions and their derivatives. They are fundamental in modeling the behavior of various phenomena. In the context of logistic growth, the differential equation is used to describe how the population size changes over time.
For logistic growth, the differential equation is given by \[ P'(t) = k P(t) (M - P(t)) \]. This equation reflects the rate of change of the population and illustrates the feedback mechanism:

• When \(P(t)\) is small, \(M - P(t)\) is large, meaning the population grows rapidly.
• When \(P(t)\) is close to \(M\), \(M - P(t)\) becomes very small, slowing growth as resources become limited.

This feedback mechanism is crucial to accurately modeling how populations grow under limited conditions, as seen in various species within an ecosystem.

Limit of a Function

The limit of a function is a fundamental concept in calculus, used to describe the behavior of a function as its input approaches a certain value. In the case of logistic growth, understanding the limit helps us determine the ultimate population size.
As time \(t\) approaches infinity, the exponential term \(e^{-k M t}\) in the logistic equation \( P(t)=\frac{P_{0} M}{P_{0}+ (M-P_{0}) e^{-k M t}} \) approaches zero. This simplifies the equation to \( \frac{P_0 M}{P_0} = M \), which indicates that the population will level off at its carrying capacity \(M\).
This concept of limiting population size is critical because it accounts for environmental constraints, allowing the population model to remain realistic even as time continues indefinitely. Thus, students learn how calculus helps predict long-term behavior in natural systems.

Quotient Rule

The quotient rule is a technique used in calculus to differentiate functions that are expressed as a quotient of two other functions. This rule is essential when solving problems involving rates of change, such as those seen in population dynamics.
For the logistic growth equation given as a quotient \( P(t) = \frac{P_0 M}{P_0 + (M - P_0) e^{-k M t}} \), we use the quotient rule to find the derivative, \( P'(t) \). The quotient rule states:

• If \( u(t) = \frac{f(t)}{g(t)} \), then \( u'(t) = \frac{f'(t)g(t) - f(t)g'(t)}{g(t)^2} \).

In the exercise, \( f(t) \) is a constant, \( P_0 M \), thus its derivative \( f'(t) \) is zero. This makes calculation easier but illustrates how the changing denominator, \( g(t) \), greatly affects the derivative. Applying the quotient rule correctly confirms the logistic growth rate \( P'(t) = k P(t)(M - P(t)) \), reinforcing how calculus tools are applied to real-world problems.