Question

The following questions consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells.When does population increase the fastest for the Gompertz equation \(P(t)^{\prime}=\alpha \ln \left(\frac{K}{P(t)}\right) P(t) ?\)

Short answer

The population increases the fastest at \(P(t) = \frac{K}{e}\).

Step-by-step solution

Understanding the Gompertz Equation

The Gompertz growth model is given by the equation \(P'(t) = \alpha \ln\left(\frac{K}{P(t)}\right) P(t)\), where \(P(t)\) is the population at time \(t\), \(K\) is the carrying capacity, and \(\alpha\) is the growth rate constant.

Finding Critical Points

To find when population increases the fastest, we need to maximize \(P'(t)\). We first take the derivative of \(P'(t)\) with respect to \(t\) to find critical points. The derivative of \(P'(t) = \alpha \ln\left(\frac{K}{P(t)}\right) P(t)\) needs to be set to zero or undefined to find such points.

Applying the Chain Rule

Apply the chain rule to differentiate \(P'(t)\). We note that since \(P(t)\) affects both the logarithmic term and the outside multiplication, differentiation should be performed using the product rule and the chain rule. This full derivative gives an equation that involves the derivative of \(P(t)\) itself, indicating where the acceleration of growth changes.

Setting up the Derivative

Communicate clearly the expressions according to the rules of differentiation: \(\frac{d}{dt}[\alpha \ln\left(\frac{K}{P(t)}\right) P(t)] = \alpha P(t)\frac{d}{dt}\ln\left(\frac{K}{P(t)}\right) + \alpha \ln\left(\frac{K}{P(t)}\right) P'(t)\). Use the derivative \(\frac{d}{dt}\ln(x) = \frac{1}{x}\frac{dx}{dt}\).

Simplifying the Expression

Insert the expression for the derivative of \(\ln\left(\frac{K}{P(t)}\right)\): \(-\frac{1}{P(t)}\frac{dP(t)}{dt}\). Solve through this to get: \(\alpha \left(-1 + \ln\left(\frac{K}{P(t)}\right)\right) P'(t) = 0\).

Solving for Maximum Growth

Set the simplified expression to zero to find the critical points: \(\ln\left(\frac{K}{P(t)}\right) = 1\). This implies \(\frac{K}{P(t)} = e\), or specifically that \(P(t) = \frac{K}{e}\).

Conclusion: Fastest Growth Rate

The population increases the fastest at \(P(t) = \frac{K}{e}\), where \(e\) is the base of the natural logarithm.

Key concepts

Logistic Growth

Logistic growth is a model often used to describe how populations grow in nature. In the simplest terms, it assumes that the growth rate of a population decreases as the population size approaches a maximum limit, known as the carrying capacity.
- The equation for logistic growth is: \[ P'(t) = r P(t) \left( 1 - \frac{P(t)}{K} \right) \]
- Here, - \( P(t) \) is the population at time \( t \), - \( r \) is the intrinsic growth rate, and - \( K \) is the carrying capacity.
Logistic growth begins with an exponential increase when populations are small enough. Over time, as resources become scarce, the growth rate slows down, and populations stabilize around the carrying capacity.

Cancer Growth Modeling

Cancer growth modeling describes how cancer cells multiply within a host organism. The Gompertz equation is especially favored for this, as it captures the slowing growth rate of tumors over time.
- One of the reasons that Gompertz is preferred is because it accounts for densities of cells that slow the rate of growth in a realistic manner.
- The formula used is: \[ P'(t) = \alpha \ln\left(\frac{K}{P(t)}\right) P(t) \]
This form reflects that as tumor cells grow and resources become limited, the rate of growth decreases, which is more realistic than constant rate models. Understanding where tumors grow the fastest can provide insights into intervention strategies.

Differential Equations

Differential equations play a crucial role in mathematical modeling of various phenomena, including population growth and cancer. They involve equations that include derivatives of a function.
- A basic form is: \[ \frac{dy}{dt} = f(y, t) \]
- Here, \( y \) is the function of interest, and - \( t \) is time or another independent variable.
- For the Gompertz equation, differentiation is used to find how quickly a tumor population changes concerning time.
Differential equations can be analyzed to find critical points where the behavior of a system changes, such as the point where cancer growth is fastest. This involves setting the derivative to zero to solve for maximum or minimum conditions.