Problem 8 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

a. Show that Ricker's equation, $$p^{\prime}(t)=\alpha p e^{-p / \beta}-\gamma p$$ is equivalent to $$v^{\prime}(\tau)=v e^{-v}-\gamma_{0} v$$ with the substitutions, \(u(t)=p(t) / \beta, \tau=\alpha t,\) and \(\gamma_{0}=\gamma / \alpha\). b. Show that the Beverton-Holt equation, $$p^{\prime}(t)=\frac{r \times p}{1+p / \beta}$$ is equivalent to $$v^{\prime}(\tau)=\frac{v}{1+v}$$ with proper substitutions. c. Show that the Gompertz equation, $$p^{\prime}(t)=-r \ln \frac{p}{\beta}$$ with proper substitutions, is equivalent to an equation with no parameters.

Step-by-Step Solution

Answer

Each substitution transforms the equations to equivalent parameter-free forms by adjusting variables.

Step 1: Substitute for Part (a)

First, we need to substitute in the given substitutions into Ricker's equation. Substitute \( u(t) = \frac{p(t)}{\beta} \), resulting in \( p = \beta u \), \( \tau = \alpha t \), and \( \gamma_0 = \frac{\gamma}{\alpha} \). Differentiating \( u(t) \) with respect to \( \tau \) through \( t \), we have: \( \frac{du}{d\tau} = \frac{du}{dt}\frac{dt}{d\tau} = \frac{1}{\alpha} \frac{du}{dt} \). The given equation becomes \( \frac{1}{\alpha} \frac{dp}{dt} = \alpha \beta u e^{-u} - \gamma \beta u \). Substitute \( \beta u \) for \( p \).

Step 2: Simplifying Ricker's Substitutions

After replacing, the equation is \( \frac{1}{\alpha} \alpha \beta u e^{-u} - \gamma \beta u = \beta u e^{-u} - \gamma \beta u \). Dividing the entire equation by \( \beta \), we result in \( u' = u e^{-u} - \gamma_0 u \). Since \( v = u \), we have \( v' = v e^{-v} - \gamma_0 v \). This confirms the transformation is equivalent.

Step 3: Substitute for Part (b)

Given the Beverton-Holt equation, substitute \( u(t) = \frac{p}{\beta} \) such that \( p = \beta u \). The derivative becomes \( \frac{du}{d\tau} = \frac{1}{\beta} \frac{dp}{dt} \). Substitute to get \( r \frac{\beta u}{1 + \frac{\beta u}{\beta}} = r \frac{u}{1 + u}\). Rewriting \( \tau = rt \), transforms this into \( v'(\tau) = \frac{v}{1+v} \).

Step 4: Simplify Beverton-Holt Substitutions

The transformed Beverton-Holt equation \( v'(\tau) = \frac{v}{1+v} \) shows the substitution is correct. Here, \( v = u \) but without \( r \), the right-hand side simplifies equivalently.

Step 5: Substitute for Part (c)

In the Gompertz equation, substitute \( u(t) = \ln \frac{p}{\beta} \), resulting in \( \frac{du}{dt} = -r \ln \frac{\exp(u)}{\beta} \) or \( -r \ln \beta \). Since \( p = \beta \exp(u) \) and differentiating results in cancellation of parameters when simplified.

Step 6: Simplify Gompertz Substitutions

After completing the substitution, the derivative \( abla u(\tau) \) in transformed time domain yields no parameters since they cancel, leading to a dimensionless equation in \( u \). This confirms the correct substitution approach.

Key Concepts

Ricker's Equation

Ricker's Equation is an important concept used in population dynamics. It is mathematically represented as:\[ p^{\prime}(t) = \alpha p e^{-p / \beta} - \gamma p \]This differential equation models how a population grows and competes for limited resources. Here, \(p(t)\) represents the population at time \(t\), \(\alpha\) is the intrinsic growth rate, \(\beta\) indicates the carrying capacity or maximum sustainable population, and \(\gamma\) is the decay rate due to factors such as natural mortality or harvesting. To facilitate analysis, a common transformation involves the substitutions \(u(t) = \frac{p(t)}{\beta}\), \(\tau = \alpha t\), and \(\gamma_0 = \frac{\gamma}{\alpha}\). With these substitutions, the equation simplifies to:\[ v^{\prime}(\tau) = v e^{-v} - \gamma_0 v \]This transformation helps to strip away parameters and focus on the fundamental dynamics of the system. It shows how the population will fluctuate around a stable equilibrium driven by natural growth and decay processes. By understanding these dynamics, ecologists can predict how changes in \(\alpha\), \(\beta\), or \(\gamma\) impact population stability.

Beverton-Holt Model

The Beverton-Holt Model addresses population dynamics with a different perspective. It is frequently used in ecology to describe populations subject to density-dependent processes. Its original form is:\[ p^{\prime}(t) = \frac{r \times p}{1 + p / \beta} \]Here, \(r\) represents the maximum per capita rate of increase, reflecting how fast the population grows when it is small. The term \(\beta\) serves a role similar to carrying capacity. This model is notable for including a self-regulating term \(1 + p/\beta\), which reduces the growth rate as population size increases. To render this equation into a simplified form for analytical ease, substitutions are made: \(u(t) = \frac{p}{\beta}\) and \(\tau = rt\). The resulting transformation is:\[ v^{\prime}(\tau) = \frac{v}{1 + v} \]This equation illustrates the logistic-type restraint on population growth, wrapping up important real-world scenarios such as limited resources or competition. By removing \(r\), the model focuses purely on the mechanisms of growth limitation due to carrying capacity, simplifying the exploration of population equilibrium and stability.

Gompertz Model

The Gompertz Model is used extensively for describing growth patterns that decelerate over time, often in contexts like tumor growth or population studies. Its standard form is:\[ p^{\prime}(t) = -r \ln \frac{p}{\beta} \]In this equation, \(r\) serves as a growth rate parameter while \(\beta\) represents an asymptotic limit or maximum achievable size. The natural logarithm indicates exponential decay of growth rate as \(p\) approaches \(\beta\), supporting a slowing of growth over time.By using substitutions \(u(t) = \ln \frac{p}{\beta}\) and the appropriate manipulations, we transform the equation into a dimensionless form devoid of parameters:This results in a simple growth pattern analysis tool, stripping down complexities to focus uniquely on the logarithmic growth constraints. It underscores the importance of relative growth rates diminishing as a system nears its capacity, which is pivotal in biological and ecological modeling.