Problem 11 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Parameter reduced population models are shown below. In the Ricker equation, find a condition on \(\gamma_{0}\) that will insure that there is a value of \(v\) for which the population is growing the fastest. In the Beverton-Holt equation show that there is no value of \(v\) for which the population is growing the fastest. In the Gompertz equation, find the value of \(v\) for which the population is growing the fastest. $$\text { a. } \quad v^{\prime}=v e^{-v}-\gamma_{0} v \text { Ricker }$$$$\begin{array}{l} \text { b. } \quad v^{\prime}=\frac{v}{1+v} \\\\\text { c. } v^{\prime}=-v \ln (v)\end{array}$$ Beverton-Holt Gompertz The three previous problems have important implications for wildlife management, at least conceptually. Suppose you are managing a wildlife population, salmon, for example, as a renewable resource, and wish to annually harvest as many salmon as possible. If you harvest too severely, the next years spawn will be low, and four years later the harvest will be limited. Your optimum strategy is to maintain the population at the level where the growth is the greatest.

Step-by-Step Solution

Answer

Condition for Ricker: \( e^{-v}(1-v) > \gamma_{0} \). No fastest growth for Beverton-Holt. Fastest growth at \( v = \frac{1}{e} \) for Gompertz.

Step 1: Analyze the Ricker Equation

The Ricker model for population growth is given by the differential equation: \[ v' = ve^{-v} - \gamma_{0}v \]To find the condition on \( \gamma_{0} \), observe that for maximum growth, the derivative of \( v' \) with respect to \( v \) should be zero. Differentiate with respect to \( v \): \[ \frac{d}{dv}(ve^{-v} - \gamma_{0}v) = e^{-v} - ve^{-v} - \gamma_{0} \]Set the derivative to zero to find critical points: \[ e^{-v}(1 - v) = \gamma_{0} \]For maximum growth, \( e^{-v}(1-v) > \gamma_{0} \) must hold, which is dependent on the parameters ensuring a non-negative expression. Therefore, there should exist a \( v \) for which growth is maximum.

Step 2: Analyze the Beverton-Holt Equation

The Beverton-Holt equation is given by: \[ v' = \frac{v}{1+v} \]Differentiate \( v' \) with respect to \( v \) to find any point of fastest growth: \[ \frac{d}{dv}\left(\frac{v}{1+v}\right) = \frac{1 + v - v}{(1 + v)^2} = \frac{1}{(1+v)^2} \]The derivative \( \frac{1}{(1+v)^2} \) is positive for all \( v \geq 0 \), implying it is always increasing. There is no specific value of \( v \) ensuring fastest growth. Hence, no specific fastest growing point exists in this model.

Step 3: Analyze the Gompertz Equation

The Gompertz equation is given by: \[ v' = -v \ln(v) \]Differentiate to find where growth is fastest: \[ \frac{d}{dv}(-v \ln v) = -\ln v - 1 \]Setting this derivative equal to zero gives: \[ -\ln v - 1 = 0 \rightarrow \ln v = -1 \]Solving \( \ln v = -1 \) yields \( v = e^{-1} = \frac{1}{e} \).Thus, the fastest growth rate occurs at \( v = \frac{1}{e} \).

Key Concepts

Ricker equation

The Ricker equation is a mathematical model used to describe population dynamics. Specifically, it helps in understanding how populations grow and fluctuate over time. The differential equation for the Ricker model is \( v' = ve^{-v} - \gamma_0 v \). In this equation, \( v' \) represents the change in population size, \( v \) is the population size at a given time, and \( \gamma_0 \) is a parameter that can affect the rate of growth or decline.

To find where the population is growing the fastest in this model, we need to find the value of \( v \) that maximizes the growth rate \( v' \). This is done by setting the derivative of \( v' \) with respect to \( v \) to zero and solving. In our case, this means finding when \( e^{-v}(1 - v) = \gamma_0 \).

• If \( \gamma_0 \) is too large, the population may not grow quickly as the overriding factor could be population decline due to overcrowding or resource limitation.
• For \( e^{-v}(1-v) \) to be greater than \( \gamma_0 \), certain conditions must be met so the expression remains positive.

Hence, this equation is useful in identifying critical points where interventions can help in optimizing population growth.

Beverton-Holt equation

The Beverton-Holt equation models population growth in environments with limited resources. It is represented by \( v' = \frac{v}{1+v} \). This equation is often used in fishery models to understand how fish populations respond to fishing pressures.

The structure of the Beverton-Holt equation introduces a form of carrying capacity, which limits the population size. When analyzing this equation, we differentiate with respect to \( v \) to find where this function grows the fastest, expecting a standout point of rapid growth.

Interestingly, differentiating \( \frac{v}{1+v} \) shows us \( \frac{1}{(1+v)^2} \), which is always positive, indicating a monotonic increase as \( v \) increases. This positivity means there is no specific maximum growth value because the rate continuously increases.

• No single \( v \) results in the fastest growth. Instead, it limits the size naturally as it gets larger.
• This model is beneficial to understand equilibria in maximum sustainable yield conditions.

The Beverton-Holt equation offers insights into managing resources by highlighting the absence of a singular optimal growth point.

Gompertz equation

The Gompertz equation is another model used to describe population growth, often applied to populations like tumors, microbial growth, or market dynamics, where the growth rate decreases exponentially over time. It is expressed as \( v' = -v \ln(v) \). This model is interesting because it emphasizes growth inhibition as the population reaches maturity or saturation levels.

In this context, finding the fastest growth involves differentiating \( -v \ln(v) \) to pinpoint where this rate peaks. Solving \( -\ln v - 1 = 0 \), we find that the fastest growth occurs at \( v = \frac{1}{e} \).

• Here, \( \ln(v) = -1 \), which highlights the nature of the Gompertz model as focused on growth deceleration with increasing size.
• This peak explains how systems behave under constraints that naturally regulate their expansion.

The Gompertz model is instrumental for understanding how internal and external pressures shape the growth pattern of various populations.