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Chapter 3: Problem 20

In negative assortative matings, like individuals do not mate with each other. Different types of models may be obtained, depending on assumptions made about the permissible matings. In the questions that follow it is assumed that homozygous females mate only with homozygous males of opposite type and that heterozygous females \((A a)\) mate with \(A A\) and \(a a\) males depending on their relative prevalence. The permitted matings are then as follows: \begin{tabular}{rrr} \hline Females & Males \\ \hline AA & \(\times\) & \(a a\) \\ \(a a\) & \(\times\) & \(A A\) \\ \(A a\) & \(\times\) & \(A A\) \\ \(A a\) & \(\times\) & \(a a\) \\ \hline \end{tabular} Assuming that a single male can fertilize any number of females results in a mating table that depends largely on the female frequencies, as shown in the mating table. It is assumed that \(u+v+w=1\). (a) Explain the entries in the table. (b) Derive an offspring table by accounting for all possible products of the matings shown in the above mating table. (c) Show that the fractions of \(A A, A a\), and aa offspring denoted by \(u_{n+1}\), \(c_{n+1}\), and \(w_{m+1}\) satisfy the equations $$ \begin{aligned} &u_{n+1}=\frac{1}{2} v_{n} \frac{u_{n}}{u_{n}+w_{n}} \\ &v_{n+1}=u_{n}+w_{n}+\frac{1}{2} v_{n} \\ &w_{n+1}=\frac{1}{2} v_{n} \frac{w_{n}}{u_{n}+w_{n}} \end{aligned} $$ (d) Show that \(u+v+w=1\) in the \((n+1)\) st generation. Use this fact to eliminate \(w\) from the equations. (e) Show that the equations you obtain have a steady state with \(\bar{v}=\frac{2}{3}\). Is there a unique value of \((\bar{u}, \vec{w})\) for this steady state? Is this steady state stable? (f) Show that the ratio \(u / w\) does not change from one generation to the next.

Short Answer

Expert verified
The offspring equations maintain the sum of 1 each generation, with a consistent ratio \((u : v : w,)\ showing steady state \( \fonds \ v= \frac{2}{3}\).

Step by step solution

01

Analyze Mating Table

The mating table given is based on specific genetic pairings allowed by negative assortative mating. The table lists the permissible matings where AA females mate with aa males, aa females mate with AA males, and Aa females can mate with either AA or aa males depending on their presence in the population.
02

Derive Offspring Table

From these matings, determine the likely offspring. For example, AA x aa always results in Aa offspring, while Aa x AA or Aa x aa results in a mix of the genotypes. Specifically, calculate the offspring percentages for each type of mating.
03

Formulate Offspring Equations

Using the mating information, derive the rates of each offspring genotype: \[ \begin{aligned} u_{n+1} &= \frac{1}{2} v_{n} \frac{u_{n}}{u_{n}+w_{n}} \ v_{n+1} &= u_{n}+w_{n}+\frac{1}{2} v_{n} \ w_{n+1} &= \frac{1}{2} v_{n} \frac{w_{n}}{u_{n}+w_{n}} \ \text{where:} \ u_{n} \text{ is the fraction of AA in generation n} \ v_{n} \text{ is the fraction of Aa in generation n} \ w_{n} \text{ is the fraction of aa in generation n} \ \text{with } u + v + w = 1 \text{ for all n} \ \text{Next show how the equations sum up to 1 for all generations.} \right \tag{1} \rbrace \ \]
04

Verify Sum Property

To show that \(u + v + w = 1\) in the \((n+1)\)th generation, sum the derived offspring equations: \( u_{n+1} + v_{n+1} + w_{n+1} = 1 \). This ensures conservation of the genetic makeup across generations.
05

Eliminate w

Using \(u + v + w = 1\), solve for \(w\) as \(w = 1 - u - v\) and substitute in the equations derived in Step 3 to simplify further analysis.
06

Steady State Analysis

Examine the equations at steady state where \(u, v,\range} \) and show \( \bar{v} = \frac{2}{3} \). Validate if \(u/w\) remains stable across generations and determine the uniqueness and stability of the steady state.
07

Stable u/w Ratio

Show that the ratio \( \frac{u}{w} \) does not change from one generation to the next, indicating a stable genetic ratio over time.

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Key Concepts

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

genetic models
Genetic models help us understand how different genetic traits are passed from one generation to the next.
These models can vary greatly depending on the assumptions we make.
In the case of negative assortative mating, like individuals do not mate with each other, leading to unique genetic mixtures.
The assumptions made in this exercise include:
  • Homozygous females mate only with homozygous males of the opposite type.
  • Heterozygous females mate with either AA or aa males, depending on their prevalence.
This setup produces specific mating patterns, as described in the mating table.
steady state analysis
Steady state analysis allows us to examine the long-term behavior of genetic frequencies in a population.
It helps determine if and when the population's genetic makeup reaches equilibrium.
In this context, we need to show that the frequency of genotypes does not change significantly over generations.
Notice that the fractions of AA, Aa, and aa offspring are denoted by the variables: \( u_{n+1} \), \( v_{n+1} \), and \( w_{n+1} \). The sum of these fractions for any generation ought to be 1:
\( u + v + w = 1 \). This ensures that no individuals are lost or added erroneously in the genetic mixing process.
We also need to determine specific values of \( u \), \( v \), and \( w \) at a steady state.
offspring genotype fractions
Understanding the fractions of offspring genotypes is crucial for predicting genetic distribution.
Using the mating table, we derive equations that predict the likelihood of each genotype in the next generation.
Based on the example calculations, the equations are:

\[ u_{n+1} = \frac{1}{2} v_{n} \frac{u_{n}}{u_{n}+w_{n}} \]
\[ v_{n+1} = u_{n} + w_{n} + \frac{1}{2} v_{n} \]
\[ w_{n+1} = \frac{1}{2} v_{n} \frac{w_{n}}{u_{n}+w_{n}} \]
The terms \( u_{n} \), \( v_{n} \), and \( w_{n} \) represent the fractions of AA, Aa, and aa in generation 'n'.
These equations utilize current genotype fractions to predict future ones.
mating tables
Mating tables list the permissible matings and their potential genetic outcomes.
In negative assortative mating, the following pairings are allowed:
  • AA females mate with aa males.
  • aa females mate with AA males.
  • Aa females mate with either AA or aa males.
The offspring are then determined by accounting for all possible mating products.
This table simplifies our understanding and prediction of genetic outcomes by organizing accepted pairings.
homozygous and heterozygous mating
Homozygous individuals have two identical alleles, such as AA or aa.
In negative assortative matings, homozygous individuals mate with homozygous individuals of opposite type (AA with aa and vice versa).
Heterozygous individuals have two different alleles, like Aa.
In this scenario, heterozygous females mate with either AA or aa males based on the relative number of those males in the population.
This concept helps us understand the genetic diversity and how different allele combinations are carried forward in the population.

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Most popular questions from this chapter

(Note to the instructor: Problem 17 gives the student good practice at formulating a model and gradually increasing its complexity. Do not expect the later stages of this problem to be as amenable to further direct analysis as the more elementary model. More advanced students may wish to implement their models in computer simulations.) As indicated near the end of Section 3.5, an important influence on herbivores is the quality of the host vegetation, not just its abundance. If the plants have induced chemical defenses or other protective responses, the population of attackers may suffer from increased mortality, decreased fecundity, and lower growth rates. (a) Define a new variable \(q_{n}\) as the average quality of the vegetation in generation \(n\). Suggest what general equations might then be used to model the \(v_{n}, q_{\infty}, h_{n}\) system. (b) Instead of treating the vegetation as a uniform collection of identical plants, consider making a distinction between plants that are nutritious (or chemically undefended) and those that are not. For example, with \(h_{n}\) as before, let \(u_{n}=\) number of undefended plants, \(v_{n}=\) number of defended plants. Now suppose that plants make the transition from \(u\) to \(v\) after attack by herbivores and back from \(v\) to \(u\) at some constant rate that represents the loss of chemical defenses. Formulate a model for \(v_{n}, u_{\infty}\), and \(h_{n}\). (c) As a final step, consider vegetation in which plants can range in quality \(q\) from 0 (very hostile to herbivores) to 1 (very nutritious to herbivores). Assume that the change in quality of a given plant takes place in very small steps every \(\Delta \tau\) days at a rate that depends on herbivory in time interval \(n\) and on previous vegetation quality. That is, $$ q_{n+1}=q_{n}+F\left(q_{n}, h_{n}\right) \Delta \tau_{n} $$ where \(F\) can be positive or negative. Let \(v_{n}(q)\) be the percentage of the vegetation whose quality is \(q\) at the nth step of the process, and assume that the total biomass $$ V_{n}=\int_{0}^{1} v_{*}(q) d q, $$ does not change over the time scale of the problem. Can you formulate an equation that describes how the distribution of vegetation quality \(v_{n}(q)\) changes as each plant undergoes the above defense response? Figure for problem \(I 7(c)\). Hint: Consider subdividing the interval \((0,1)\) into \(n\) quality classes, each of range \(\Delta q\). How many plants leave or enter a given quality class during time \(\Delta \tau\) ?

Now consider the following modification of the random mating assumption: Suppose that individuals mate only with those of like genotype (e.g., \(A a\) with \(A a, A A\) with \(A A\), and so forth). This is called positive assortative mating. How would you set up this problem, and what conclusions do you reach?

Project: One problem that leads to the oscillations in the Nicholson-Bailey model (equations \(21 \mathrm{a}, \mathrm{b}\) ) is the fact that at low parasitoid densities the host population behaves approximately as follows: $$ N_{i+1}=\lambda N_{i} $$ that is, it grows at an unchecked rate. Notice that this eventually causes the parasitoids to increase in number until their hosts are overwhelmed by the attack. This is what sets up the increasing oscillations in the system. Consider the effect of introducing other types of density dependence into the host population. Two examples include $$ N_{r+1}=\lambda N_{t} \frac{\left(K-N_{0}\right)}{K} e^{-a f_{,}} $$ and, $$ N_{t+1}=\lambda N_{t}^{1-t} e^{-\alpha h}, $$ (see Varley and Gradwell (1963) and Hassell (1978)). Use the literature, computer simulations, and your own analysis to compare the predictions of these models and to comment on their applicability and/or special features. (Note: Stability analysis may be algebraically messy in such models.)

Consider the following model for leaf-eating herbivores whose population size (number of individuals) is \(h_{n}\) on a tree whose leaf mass is \(0_{n}\) * $$ \begin{aligned} &v_{n+1}=f v_{n}\left(e^{-\alpha \hat{\omega}}\right), \\ &h_{a+1}=r h_{n}\left(\delta-\frac{h_{2}}{v_{n}}\right), \quad \text { where } v_{n} \neq 0 \end{aligned} $$ and where \(f, a, r, \delta\) are positive constants. (a) Find the steady state(s) \(\bar{v}\) and \(\bar{h}\) of this system. What happens if \(f=1 ?\) What restrictions on the parameters should be met for a biologically reasonable steady state? (b) Show that by rescaling the equations, it is possible to reduce the number of parameters. To do this, define $$ V_{n}=v_{n} / \bar{v}, \quad H_{n}=h_{n} / \bar{h} \text {. } $$ Show that the system of equations can then be converted to the following form: $$ \begin{aligned} &V_{n+1}=V_{n} \exp k\left(1-H_{n}\right) \\ &H_{n+1}=b H_{n}\left(1+\frac{1}{b}-\frac{H_{n}}{V_{n}}\right) \end{aligned} $$ What is the connection between \(b, k\), and the previous four parameters \(f\), \(a, r\), and \(\delta\) ? (c) Show that the equations in part (b) now have steady-state solutions $$ \bar{Q}=\bar{H}=1 \text {. } $$ (d) Determine whether the functions in the equations of the model fall into the general category described in Section 3.5. (e) Determine when the steady state will be stable.

In this problem we consider a host population that has refuges from parasitoid attack. (a) Explain the assertion that \(E K / N_{i}\) is the fraction of the population that can retreat to a refuge. (b) Explain why the fraction of hosts not parasitized is $$ f\left(N_{1}, P_{1}\right)=\frac{E K}{N_{1}}+\left(1-\frac{E K}{N_{t}}\right) e^{-\mathrm{a} h} $$ (c) Write an equation for \(N_{i+1}\). (d) Explain why at time \(t+1\) the parasitoid density is given by $$ P_{t+1}=\left(N_{t}-E K\right)\left(1-e^{-a A}\right) $$ *(e) Longer Project: Consult the literature and use computer simulations and/ or other analysis to explore the results and predictions of this model.
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