### 经济代写|博弈论代写Game Theory代考|ECON3050

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Two-Habitat Model

An asexual species occupies an environment that consists of two local habitats linked by dispersal. This species has discrete, non-overlapping generations with a generation time of 1 year. Individuals are characterized by a genetically determined trait $x$ where $0 \leq x \leq 1$. An individual with trait $x$ in habitat $i$ leaves $N g_i(x)$ offspring where $N$ is large. The functions $g_1$ and $g_2$ are illustrated in Fig. 4.6a. As can be seen, it is best to have a low trait value in habitat 1 and a high value in habitat 2. Each newborn individual either remains in its birth habitat (with probability $1-d$ ) or disperses to the other habitat (with probability $d$ ). Here $0<d<0.5$. Each habitat has $K$ territories, where $K$ is large. After the dispersal phase each of these territories is occupied by one of the individuals present, chosen at random from all individuals currently in the habitat. Those individuals left without territories die, while the $K$ that possess territories grow to maturity.

Note that for this example the per-capita growth rate of any resident population (i.e. the mean number of offspring obtaining a territory per resident parent) is 1 since the population is at constant size. Frequency dependence acts through competition for territories. For example, suppose the resident trait value is $x=0$ and consider the growth rate of a subpopulation comprising individuals with mutant trait value $x^{\prime}=1$ while the size of this subpopulation is small compared with the resident population. Since residents will leave more offspring in habitat 1 than in habitat 2 after the dispersal phase, there will be greater competition for territories in habitat 1 . However, since mutant individuals do better in habitat 2 mutants will tend to accumulate in habitat 2. They will then suffer less competition than the average competition experienced by residents and so will have a growth rate that is greater than 1 . The mutant strategy will thus invade the resident population.

The derivàtion of the preecise formula for invàsion fitnesss $\lambda\left(x^{\prime}, x\right) \equiv W\left(x^{\prime}, x\right)$ of a mutant $x^{\prime}$ in a resident $x$ population is deferred to Section 10.7. From this payoff function one can obtain the strength of selection. Figure 4.6b illustrates this function for two values of the dispersal probability $d$. As can be seen, for each of these dispersal probabilities the trait $x^*=0.5$ is the unique convergence stable point under adaptive dynamics.

Figure $4.7$ shows the pairwise invasibility plots for each of the dispersal probabilities considered in Fig. 4.6b. We can also infer the convergence stability of $x^*=0.5$ from this figure. To see this, note that when the resident trait $x$ is less than $0.5$, mutants with trait values just greater than $x$ can invade, so that $x$ will increase under adaptive dynamics. Similarly when $x>0.5$ mutants with trait values just less than $x$ can invade, so that the resident trait value will decrease. This reasoning applies to both of the cases $d=0.3$ and $d=0.2$. However, the two cases differ fundamentally. For dispersal probability $d=0.3$ (Fig. 4.7a), if the resident trait value is $x^=0.5$ then any mutant with a different trait value has lower fitness. Thus when the resident trait is $x^=0.5$, mutant fitness has a strict maximum at the resident trait value, so that $x^=0.5$ is an ESS as well as being convergence stable (and is hence a CSS). The evolutionary simulation in Fig. 4.8a illustrates convergence to this trait value. In contrast when $d=0.2$ (Fig. 4.7b), if the resident trait value is $x^=0.5$ then any mutant with a different trait value has higher fitness. In other words, mutant fitness is minimized at the resident trait value, so that evolution leads the population to a fitness minimum.
As the next section describes, once evolution takes a population to a fitness minimum there is then further evolution.

## 经济代写|博弈论代写Game Theory代考|Evolutionary Branching

Consider a population that evolves to a fitness minimum under adaptive dynamics. When the resident population is at or close to this minimum any mutant that arises has greater fitness than residents, so that there will be disruptive selection, leading to trait values having a bimodal distribution with peaks either side of the fitness minimum. This phenomenon is referred to as evolutionary branching.

Figure $4.8 \mathrm{~b}$ illustrates evolutionary branching for the case of two habitats linked by dispersal. In this model the population always evolves towards the trait value $x^=0.5$. This represents a generalist strategy because individuals with this trait value do reasonably well in both habitats. If the dispersal probability is sufficiently high (Fig. 4.8a) and this generalist strategy is the resident strategy, then all other strategies have lower invasion fitness, so that the strategy is an ESS and hence a CSS. In contrast, for low dispersal probability (Fig. 4.8b), once the population approaches $x^=0.5$ there is disruptive selection and the population evolves to be polymorphic. In this polymorphic population half the population members do very well in habitat 1 but badly in habitat 2 . These individuals are also mostly found in habitat 1 (not shown). The other population members are specialists for habitat 2 and are mostly found there. When there are more than two habitats, with different optima on each, further branching into habitat specialists can occur. Not surprisingly, the number of stable polymorphisms can never exceed the number of different habitat types (Geritz et al., 1998). In the model the population size on each habitat is regulated by limiting resources on that habitat (soft selection). Similar results on generalism versus specialism can be found in a number of models in which there is soft selection and limited dispersal (e.g. Brown and Pavlovic, 1992; Meszéna et al., 1997; Debarre and Gandon, 2010).

The standard adaptive dynamics framework assumes clonal inheritance; the phenotype is inherited. This is a major limitation when investigating evolutionary branching in a diploid sexually reproducing population. However, the framework can be extended to such populations. Kisdi and Geritz (1999) considered a two-habitat model with soft selection in which the phenotypic trait was determined by the two alleles at a single diploid locus. They again concluded that evolutionary branching could occur resulting in specialist phenotypes.

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