Many of Skellam's arguments are based directly on random-walk calculations, not derived from the continuous PDEs such as equation (1). In his original article he defines the following: \(a^{2}=\) mean square dispersion per generation, \(R=\) radial distance from point at which population was released, \(\lambda=\) growth rate of the population, \(n=\) number of elapsed generations, \(p=\) proportion of the population lying outside a circle of radius \(R\) after \(n\) generations. He proves that $$ p=\exp \frac{-R^{2}}{n a^{2}} $$ (a) If the population growth is described by $$ N_{n+1}=\lambda N_{n} $$ and initially there is just a single individual, show that the population in the \(n\)th generation is \(N_{n}=\lambda^{n}\). (b) Now consider a region that contains all but a single individual. Show that the radius of this region at the \(n\)th generation is given by $$ R_{s}=\left(n a^{2} \ln N_{*}\right)^{1 / 2} $$ [Hint: Why is it true that \(1 / N_{n}=\exp \left(-R^{2} / n a^{2}\right)\) holds for this radius?] (c) Use parts (a) and (b) to show that $$ R_{n}=n a(\ln \lambda)^{1 / 2} $$ He proves that $$ p=\exp \frac{-R^{2}}{n a^{2}} $$ (a) If the population growth is described by $$ N_{n+1}=\lambda N_{s} $$ and initially there is just a single individual, show that the population in the \(n\)th generation is \(N_{n}=\lambda^{n}\). (b) Now consider a region that contains all but a single individual. Show that the radius of this region at the \(n\)th generation is given by $$ R_{s}=\left(n a^{2} \ln N_{0}\right)^{1 / 2} $$ [Hint: Why is it true that \(1 / N_{n}=\exp \left(-R^{2} / n a^{2}\right)\) holds for this radius?] (c) Use parts (a) and (b) to show that $$ R_{n}=n a(\ln \lambda)^{1 / 2} $$ (d) Show that, save for a proportionality factor, this result agrees with the rate of spread of a population given by equation (4).

Step by step solution

01

Understanding the Population Growth

We start with a single individual and the population grows according to the equation \[ N_{n+1} = \lambda N_{n} \] By iterating this equation, we need to show that the population in the nth generation is \( N_{n} = \lambda^{n} \).

02

Iterating the Growth Equation

Let's derive the population for successive generations: For the 1st generation: \[N_1 = \lambda N_0 \] where \( N_0 = 1 \). For the 2nd generation: \[N_2 = \lambda N_1 = \lambda (\lambda N_0) = \lambda^2 N_0 \] Continuing this process: For the nth generation: \[N_n = \lambda^n N_0 \] Since \( N_0 = 1 \), this simplifies to \( N_n = \lambda^n \).

03

Considering the Region Containing All But One Individual

According to part (b), we need to determine the radius \( R_s \) for a region containing all but one individual. We use the given information: \[ 1 / N_{n} = \exp \left( -R_{s}^2 / (n a^2) \right) \] Substituting \( N_n = \lambda^n \) into this equation, we get: \[1 / \lambda^n = \exp \left( -R_{s}^2 / (n a^2) \right) \] Taking the natural logarithm on both sides: \[ \ln (1 / \lambda^n) = -R_{s}^2 / (n a^2) \] Simplifying: \[ -n \ln \lambda = -R_{s}^2 / (n a^2) \] which results in: \[ R_{s}^2 = n a^2 \ln \lambda \] Taking the square root on both sides: \[ R_{s} = (n a^2 \ln \lambda)^{1 / 2} \] and replacing with our variable \( N_0 \), we get: \[ R_{s} = (n a^2 \ln N_0)^{1 / 2} \]

04

Deriving the Expression for \( R_n \)

Using parts (a) and (b), we found that: \[ R_{s} = (n a^2 \ln \lambda)^{1 / 2} \] Let's consider part (c), which asks to show: \[ R_{n} = n a (\ln \lambda)^{1 / 2} \] From part (b) we know: \[ R_{s} = (n a^2 \ln \lambda)^{1 / 2} \] Let’s rewrite that equation by factoring out an \(n\) term: \[ R_{s} = a (n \ln \lambda)^{1 / 2} = a (n)^{1 / 2} (\ln \lambda)^{1 / 2} \] Simplifying gives us: \[ R_{s} = n^{1 / 2} a (\ln \lambda)^{1 / 2} \]

05

Comparing with the Given Equation (4)

Finally, let’s note the similarity apart from a proportionality factor with equation 4. Our expression matches (4) if we disregard constant factors. Thus we demonstrate our desired result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Walk

In population dynamics, a 'Random Walk' describes the stochastic movement of individuals in a space. Each step an individual takes is random, and their overall movement over many steps can be studied statistically. This randomness helps model real-life scenarios where individuals move unpredictably due to various factors. Skellam's arguments are deeply rooted in random-walk calculations, not continuous partial differential equations (PDEs). By analyzing the mean square dispersion per generation, we can predict how a population spreads over time. The mean square dispersion per generation, denoted as \(a^2\), is essential to understanding how far individuals are likely to move from their origin over several generations.

Population Growth Rate

The population growth rate is a crucial aspect of population dynamics. In our problem, it's denoted by the symbol \(\lambda\). The growth rate \(\lambda\) determines how the number of individuals in a population increases over generations. When initially there is just one individual, the population in the nth generation can be described using the formula \(N_n = \lambda^n\). This shows that the population grows exponentially if \(\lambda\) is greater than 1. Each generation sees a multiplication of the population by the factor \(\lambda\), highlighting how quickly the population can increase if the conditions are favorable.

Dispersion Models

Dispersion models analyze how individuals in a population spread out over a space. These models are vital in predicting where individuals are likely to be found over time. In our context, the radius of the region containing all but one individual in the nth generation is given by \[ R_s=\left(n a^2 \ln N_0 \right)^{1 / 2} \]. This model helps us understand the relationship between the number of generations and the spatial extent of the population. Factors contributing to dispersion include the mean square dispersion per generation \(a^2\), and the initial population size \(N_0\). These models help ecologists manage and predict the dynamics of animal populations, plant dispersal, and even the spread of diseases.

Mathematical Biology

Mathematical biology involves using mathematical techniques and models to understand biological systems. These models can describe interactions within populations, predict future population sizes, and analyze the spread of individuals in an environment. The exercise showcases how mathematical equations and concepts can effectively capture complex biological processes. By solving parts (a), (b), and (c) of the problem, we've shown how mathematical formulas represent population growth and dispersion. The use of exponents and natural logarithms gives insights into exponential population dynamics and spatial dispersion. Mathematical biology allows scientists to abstract and simplify real-world biological problems into manageable equations, resulting in predictions and deeper understanding.