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1. 群体遗传学中的缓慢选择 考虑具有特定基因的两个等位基因 a 和 b 的群体。每个人 有两个基因拷贝,选项是 aa、ab 或 bb。作为在类中派生的, A、2H 和 B 的方程,具有每个 基因型 aa、ab 和 bb 分别是 其中 b 是出生率(假设所有三种基因型都相同),di 是 每个基因型的死亡率,以及 F(A;H;B) = b d1A 2d2H d3B: (1) (a) 假设死亡率的形式为 di = d + i,其中 << 1。 tities = A+H 和 = B +H 分别是 a 和 b 的基因频率 (种群中所有基因分别为 a 和 b 的比例)。显示 与 A、H 和 B 相比,这些量变化缓慢 相应的慢时间尺度,A;H 和 B 达到准稳态 A = 2 ,H = , 并且 B =
2 (这些被称为哈代-温伯格均衡)。 (b) 假设 1 = 3 = d 和 2 = 0。写一两句话描述什么 这个假设意味着每个基因型的相对性。派生 方程 0 = d (1) 的缓慢变化。从这个相线 ODE for ,确定如果种群中有几个 b 等位基因会发生什么 以a等位基因为主。将此与值联系起来。 2. 下载此作业的 Canvas 代码页上发布的三个 .ode 文件。为了 每个模型/.ode 文件,生成一个显示稳态的分叉图和 作为主要参数的函数的周期解(Iapp 或取决于 模型)。 les 为你设置了几乎所有的东西。你只需要修改 AUTO 中的轴限制如每个 ode 文件的前几行所述。提交一个 图表的草图、屏幕截图或导出。 对于每个模型,用文字描述周期解的分支是如何“诞生的”和 它是如何“死亡”的。对于某些人来说,您可以简单地命名分岔(尽可能详细地 可能的,例如超临界 Hopf)。对于 mechanochemForA6.ode,这是可选的。


1. Slow selection in population genetics
Consider a population with two alleles, a and b, of a particular gene. Every individual
has two copies of the gene, the options being either aa, ab or bb. As derived in class,
the equations for A, 2H, and B, the proportions of the population having each of the
genotypes aa, ab, and bbrespectively are

where b is the birth rate (assumed to be the same for all three genotypes), di is the
death rate for each genotype, and
F(A;H;B) = b d1A 2d2H d3B: (1)

(a) Assume that the death rates take the form di = d + i where  << 1. The quan-
tities = A+H and  = B +H are the gene frequencies for a and b respectively
(the fraction of all genes in the population that are a and b respectively). Show
that these quantities change slowly compared to A;H and B and that on the
corresponding slow time scale, A;H and B achieve quasi-steady states A = 2
,H = , and B = 2
(these are known as the Hardy-Weinberg Equilibrium).

(b) Assume that 1 = 3 = d and 2 = 0. Write a sentence or two describing what
this assumption means in terms of the relative tness of each genotype. Derive
the equation 0
= d(1 ) for the slow change of . From the phase line of this
ODE for , determine what happens if there are a few b alleles in a population
with predominantly a alleles. Relate this to the  values.

2. Download the three .ode les posted on the Canvas code page for this assignment. For
each of the models/.ode les, generate a bifurcation diagram showing steady states and
periodic solutions as a function of the primary parameters (Iapp or depending on the
model). The les set almost everything up for you. You will just have to modify the

Axes limits in AUTO as described in the rst few lines of each ode le. Submit a
sketch, screenshot or export of the diagram.

For each model, describe in words how the branch of periodic solutions is \born” and
how it \dies”. For some, you can simply name the bifurcation (with as much detail as
possible, e.g. supercritical Hopf). For mechanochemForA6.ode this is optional.

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