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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|演化博弈论代写game theory in biology代考|Model: Group Defence and Dispersal to Empty Areas

We develop a dispersal model in which $N$ areas, each of which may contain a group of individuals, are linked by dispersal. Each area has $K$ food territories. Each individual must occupy its own territory in order to survive. Thus the group occupying an area has maximum size $K$. At the beginning of each year each territory owner produces one offspring. It then dies with probability $m$, otherwise it retains its territory until next year. The offspring either remains in its natal area (with probability $1-d$ ) or immediately disperses to a randomly selected area (with probability $d$ ). Those offspring that remain compete (scramble competition) for the territories that have become vacant due to the death of previous owners. Those individuals that disperse to a non-empty area die, since local group members oppose them. If a dispersing individual is lucky enough to disperse to an empty area it competes for a territory (again scramble competition) with others that also dispersed to this area. Those individuals that gain a territory are founder members of a new group. Individuals that do not gain territories immediately die without leaving offspring. Each year a threat to the whole group in an area occurs with probability $\beta$. If a threat appears, each group member decides how much effort to expend in group defence (its prosocial trait). Let $u_i$ be the defence effort of the individual on territory $i$ (set to zero if there is no individual on the territory). The probability the whole group is wiped out is a function $H\left(u_1, u_2, \ldots, u_K\right)$ that decreases with the defence efforts of the individuals. If the group is successful in defending against the threat, individual $i$ dies with probability $h\left(u_i\right)$ that increases with its group defence effort.

In this model there is a trade-off-to defend the group or save oneself. We might expect that when the dispersal probability is high, members of a group are not highly related and expend little effort on group defence. Consequently, there will be many areas where the group has been wiped out, and hence many opportunities for dispersing offspring to find empty areas, so that dispersal remains high. Conversely, when the dispersal probability is low, individuals will be related and much more prosocial, putting more effort into group defence. This results in few empty patches and selects for low dispersal (Fig. 6.9). Figure $6.10$ shows a case in which high initial levels of group defence lead to even higher levels evolving, accompanied by rare vacant areas and low dispersal rates. In contrast, low initial levels of group defence lead to lower levels evolving accompanied by more empty areas and higher dispersal. Overall this suggests that when dispersal and defence effort are allowed to co-evolve there may be two different stable endpoints. Whether this is so depends very much on the functions $H$ and $h$; often there is just one endpoint.

## 经济代写|演化博弈论代写game theory in biology代考|Co-evolution of Species

As we mentioned, in evolutionary biology co-evolution usually refers to reciprocal evolutionary changes in the traits of two or more species, as a consequence of their interactions. This is a large and complex area of investigation, in which game theory has only played a minor role. The species interactions can involve harm, as for parasites and hosts or predators and prey, or they can be mutualistic such that they benefit each of the species involved.

Here we illustrate the latter possibility using a model of the evolution of Müllerian mimicry. In this mimicry two or more species that are aposematic (i.e. unprofitable as prey, with striking appearances that signal their unprofitability) evolve to become similar in their appearance (Fig. 6.11), which benefits the prey because of reduced costs of predator education. The species interact through predators, who can learn more quickly to avoid them as prey by generalizing between their appearances. The first hypothesis about this form of mimicry evolution was put forward by Müller (1878) and it played a role in early discussions about evolution by natural selection. Müller suggested that learning by predators to avoid unpalatable prey worked such that a predator would attack a given number of prey after which it would stop attacking. It then follows that by having the same appearance, two species can divide the cost of this predator education between them, and that the gain in per capita prey survival would be greater for a species with lower population density.

In Box $6.5$ we present a reinforcement learning model that shows some similarity to Müller’s original assumptions about learning. The learning mechanism works by updating an estimate of the value of performing a certain action (like attack) in a given situation (like a particular prey appearance). The preference for attack is assumed to be proportional to the estimated value $(\mathrm{eq}(6.33)$ ), which is simpler but possibly less biologically realistic than actor-critic learning (Box 5.1). The learning model in Box $6.5$ also includes generalization between different prey appearances, using a Gaussian generalization function (eq (6.35)). Let us now consider the evolution of prey appearances, while the predator learning rule is unchanging. Generalization influences mimicry evolution by causing predators to treat prey with similar appearances in similar ways. This is illustrated in Fig. 6.12a, showing the probability of survival for mutants with different appearances $\xi$, which could be a measure of prey colouration, when there are two resident populations with average appearance $\xi=2$ and $\xi=8$, respectively.

From the perspective of game theory, the best response in this situation is a mutant with appearance near the average of the population with highest population density, which is $\xi=8$ in this example. The light grey arrows in the figure illustrate evolutionary changes for mutants from the less numerous population that could invade. This might suggest that mimicry can only evolve through major mutational changes. There is, however, an alternative evolutionary scenario. Note that the survival peaks around the population means in Fig. 6.12a are slightly asymmetric, with a skew in the direction of the other population. The skew is a consequence of generalization by predators. It will only be noticeable if the width of the generalization function $(\sigma$ in eq (6.35)) is not too small compared with the separation of the population appearances. Such gradual evolution of mimicry is illustrated in Fig. 6.12b, using individual-based simulation.

# 博弈论代考

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## 有限元方法代写

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## MATLAB代写

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