Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Let \(m\left(p_{0}\right)\) be \(p^{\prime}(0)\) in the Gompertz model, where $$p^{\prime}(t)=-r p(t) \ln (p(t) / \beta), \quad t \geq 0, \quad p(0)=p_{0}$$ Then $$m\left(p_{0}\right)=-r p_{0} \ln \left(p_{0} / \beta\right)$$ For \(p_{0} / \beta \ll 1\) it follows from Explore 17.1 .1 that \(m\left(p_{0}\right)=p^{\prime}(0) \gg r p_{0} .\) Show, however, that $$\lim _{p_{0} \rightarrow 0} m\left(p_{0}\right)=0$$

Step-by-Step Solution

Answer

The limit is 0 as \(p_{0} \rightarrow 0\).

Step 1: Understanding the Gompertz Model

The Gompertz model is used to describe growth and is given by the differential equation \(p'(t)=-r p(t) \ln (p(t)/\beta)\). Here, \(r\) is the growth rate, \(\beta\) is the carrying capacity, and \(p(t)\) is the size at time \(t\).

Step 2: Initial Condition

We are given the initial condition \(p(0)=p_{0}\), which means that at time \(t=0\), the population size is \(p_{0}\).

Step 3: Expression for m(p_0)

From the given differential equation, it's stated that \(m(p_{0})=-r p_{0} \ln (p_{0}/\beta)\). This represents the derivative at \(t=0\), \(p'(0)\).

Step 4: Explore Behavior When \(p_{0}/\beta \ll 1\)

When \(p_{0}/\beta\) is much smaller than 1, the expression \(m(p_{0}) = -r p_{0} \ln (p_{0}/\beta)\) simplifies. Since \(\ln(x) \rightarrow -\infty\) as \(x\) approaches zero, we suspect that as \(p_{0}\) approaches zero, \(\ln(p_{0}/\beta)\) becomes very negative.

Step 5: Taking the Limit as \(p_0 \rightarrow 0\)

We need to evaluate the limit \(\lim_{p_{0} \rightarrow 0} -r p_{0} \ln (p_{0} / \beta)\). As \(p_{0}\) approaches 0, \(p_{0} \ln(p_{0}/\beta)\) approaches 0 since \(p_{0} \rightarrow 0\) dominates the logarithmic term becoming very large negatively.

Step 6: Conclusion of the Limit

Therefore, the limit \(\lim_{p_{0} \rightarrow 0} -r p_{0} \ln (p_{0} / \beta) = 0\), proving that \(m(p_{0})\) goes to zero as \(p_{0}\) approaches zero.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They express a relationship between a function and its rate of change. In the context of growth models, like the Gompertz model, differential equations help describe how a certain population changes over time.

The Gompertz model represents this change with the equation \(p'(t) = -r p(t) \ln (p(t)/\beta)\). Here, \(p(t)\) is the population size at time \(t\), \(r\) is the growth rate, and \(\beta\) represents the carrying capacity, or the maximum population size that the environment can sustain. The differential equation indicates that the rate of change of the population decreases as the population approaches its carrying capacity.

• The term \(p'(t)\) is the derivative of the population size with respect to time, indicating how the population size changes.
• Negative signs in the equation suggest a decrease in the growth rate as size increases.
• Logarithmic function \(\ln (p(t)/\beta)\) reflects how population growth slows down as it gets larger.

Differential equations like this one are crucial for models concerning dynamic systems that change over time, giving us a powerful tool for predicting future behavior based on current trends.

Growth Models

Growth models are mathematical representations that describe how populations grow over time. They are essential in fields like biology, economics, and ecology.

The Gompertz model is a specific type of growth model that predicts the growth of a population. It is unique because it considers that growth slows exponentially as a population reaches its carrying capacity. This means that populations grow rapidly at first but then start to slow down due to factors like limited resources or increased competition.

• The Gompertz model is often used to model biological systems, like tumor growth, where initial growth is rapid.
• This model helps us understand how a population stabilizes at a particular size over time.
• Parameters like the growth rate \(r\) and the carrying capacity \(\beta\) are key to determining the growth trajectory.

By understanding and applying growth models, researchers can anticipate changes and navigate complex systems more effectively.

Limits in Calculus

Limits are a fundamental concept in calculus, used to describe the behavior of functions as they approach certain points. In the provided problem, we investigate what happens to \(m(p_0) = -r p_0 \ln (p_0/\beta)\) as \(p_0\) approaches 0.

Limits help us make sense of values that are not immediately obvious or are undefined at a certain point. When solving \(\lim_{p_0 \rightarrow 0} -r p_0 \ln (p_0/\beta)\), we find that as \(p_0\) gets very small, the term \(\ln(p_0/\beta)\) tends to \(-\infty\). However, because \(p_0\) itself tends to zero even more rapidly, the entire expression resolves to zero.

• Limits provide a way to analyze the behavior of functions at the edges or boundaries where direct calculation is difficult.
• This illustrates an important idea in calculus: the balance between competing trends in mathematical expressions.
• In the Gompertz model scenario, limits help conclude that the initial growth rate \(m(p_0)\) diminishes to nothing as the starting population approaches zero, which aligns with intuitive expectations in real-world contexts.

Through understanding limits, we gain valuable insights into the continuous behavior of functions near critical points.