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Chapter 3: Problem 19

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?

Short Answer

Expert verified
Positive assortative mating results in stable frequencies where genotypes mate with like genotypes, leading to distinct genotype groups only mating within themselves.

Step by step solution

01

- Define the Genotypes

Identify and define the genotypes involved in the population. Assume a simple scenario with two alleles, A and a, resulting in three genotypes: AA, Aa, and aa.
02

- Recognize the Mating Patterns

In positive assortative mating, individuals mate with others of the same genotype. Thus, AA mates with AA, Aa mates with Aa, and aa mates with aa.
03

- Determine the Offspring Contributions

For each mating pattern, determine the probable offspring. AA x AA will always produce AA offspring. Aa x Aa will produce AA, Aa, and aa offspring in a 1:2:1 ratio. aa x aa will always produce aa offspring.
04

- Calculate the Genotype Frequencies in the Next Generation

Let the initial frequencies of AA, Aa, and aa genotypes be p, q, and r respectively. Calculate the contributions of each mating type to the genotypes in the next generation.
05

- Solve for Equilibrium Frequencies

Set up equations for the frequencies of each genotype after mating and solving for when the genotype frequencies become stable over generations.

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Key Concepts

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

Genotype Frequencies
To understand genotype frequencies, let's start by defining the term. Genotype frequencies refer to how often a particular genetic combination appears in a population. In a simplified scenario with two alleles, A and a, there are three possible genotypes: AA, Aa, and aa.
If we let p, q, and r be the frequencies of AA, Aa, and aa in the current generation, respectively, we can study these frequencies to predict future genetic structures.
For example, if the genotype AA appears 40% of the time, Aa 50%, and aa 10%, then our frequencies would be p = 0.4, q = 0.5, and r = 0.1. It's essential to calculate these frequencies correctly as they will help us understand the population's genetic makeup over time.
Equilibrium Frequencies
Equilibrium frequencies occur when the genotype frequencies in a population remain constant across generations.
When studying genetic models like positive assortative mating, we aim to find the point where the population's genetic structure doesn't change over time.
Mathematically, this means setting up equations that describe how the frequencies of AA, Aa, and aa genotypes evolve after mating. We then solve these equations to find the equilibrium point.
For example, in positive assortative mating, if p, q, and r are the initial frequencies of AA, Aa, and aa, we need to find the new frequencies after one round of mating. When these new frequencies match the original frequencies, we have reached equilibrium.
Mating Patterns
Mating patterns describe how individuals in a population choose their partners.
Positive assortative mating is a specific type where individuals mate only with others of the same genotype.
Here are the main points of this pattern:
  • AA mates with AA
  • Aa mates with Aa
  • aa mates with aa
This non-random mating pattern contrasts with random mating, where individuals pair up without regard to genotype.
Understanding these patterns helps us predict how genetic structures will shift, especially when non-random mating rules, like positive assortative mating, apply.
Offspring Contributions
Offspring contributions explain how different mating combinations contribute to the next generation.
In positive assortative mating, the offspring outcomes are predictable due to the defined mating rules.
Let's break it down:
  • AA x AA always produces AA offspring.
  • Aa x Aa produces AA, Aa, and aa offspring in a 1:2:1 ratio, respectively.
  • aa x aa always produces aa offspring.
Knowing these contributions helps us calculate the genotype frequencies in the next generation. By understanding how pairs of each genotype mix and contribute to the gene pool, we can map out future genetic distributions more accurately.

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Most popular questions from this chapter

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.)

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.

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.

The model for density-dependent growth due to Varley, Gradwell, and Hassell (1973) given by equation (3), has the undesirable feature that, for low population levels, the fraction of survivors \(f=N_{s} / N_{t}=\frac{1}{\alpha}\left(N_{t}\right)^{-b}\) is greater than 1 . (a) Explain why this is faulty. (b) Find the critical population level, \(N_{c}\), below which this happens. \(\left(N_{e}\right.\) should be expressed in terms of \(\alpha\) and \(b\).) (c) Suggest how the model might be modified to alleviate this problem.

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.
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