The co-evolution of signalling and defence in prey populations consisting of two species: An account of stability and inter-specular interaction

对防御和防御信号（也称为警告）的共同进化进行简洁的数学解释一直是生态学中长期存在的问题。Broom等人（2006）的单物种博弈论模型的完成及其对两物种情况的扩展将对一系列学科产生影响：从数学生态学和实验生物学（例如参见Cortesi和Cheney 2010， Lee等人2010和Lee等人2011，在参考书目部分）到博弈论和复杂系统理论。拟建项目的具体目标是1。2.广泛查阅相关文献（包括在参考书目部分）；2 .完成单物种案例（通过揭示ESSs的完整列表并根据其稳定性对其进行表征）；3 .建立由两个物种组成的猎物种群处理的一般模型；确定两种情况下的ESSs（并根据其稳定性来描述这些情况），其中在外观上存在收敛（也称为两点解）；和5。为贝叶斯拟态建立一个模型（特别强调这方面的具体模型）。背景：为了提高自身的适应性，被捕食物种对攻击者进行了结构性和诱导性的防御。对于在短时间内发生的攻击，构成防御承认避免捕食的主导机制，可以分为初级和次级（Broom et al. 2006）。前者的目的是减少被捕食物种被攻击的几率，而后者的目的是双重的：减少骑马攻击导致死亡的概率(K)和增加捕食者对相似外貌的猎物的厌恶信息(I)。对二级防御的投入可能不容易被捕食者评估（就像化学防御一样），并且通常伴随着警告显示，表明猎物物种对这些防御的投入（Broom et al. 2006）。在接下来的讨论中，我们将考虑那些需要信号的次级防御，以便被捕食者评估（Broom et al. 2010, Broom et al. 2013, Ruxton et al. 2009）。我们提议的该项目的目的是完成room等人（2006）对单物种情况的警告作用的研究，并为两物种情况开发一般模型。后者将为研究生物学中感兴趣的贝叶斯模仿（一种模仿形式，无害猎物与有害猎物相似，以提高其适应性）的特定模型开辟可能性（Broom和Rychtar 2013， Cortesi和Cheney 2010， Lee等人2010和Lee等人2011）。次级防御和这些防御信号的共同进化已经得到了一定的理论关注，但不是很重要（与对这些防御的作用机制的研究相比- Broom et al. 2006）。Leimar等人（1986）首先提出了这种共同进化的博弈论模型，并继续在警告论中占据主导地位（Broom等人，2006）。尽管它很新颖，但该模型无法解释由于存在不断增加的抑制梯度而导致捕食者增加对猎物的攻击概率的可能性。此外，许多结果依赖于某些限制性函数形式，这使得很难评估模型的普遍性（Broom et al. 2006）。Leimar等人（1986）考虑了一组（最初）幼稚的捕食者（Broom等人，2006），与之相反，Broom等人（2006）的（也是博弈论的）模型将一组捕食者建模为具有学习、饥饿等平衡状态的群体，而猎物物种（平均局部相关性为a:）由居民和变种人组成，采取两种不同的策略之一。对该模型至关重要的是进化稳定策略（ESSs）

A succinct mathematical account of the co-evolution of defence and signalling of that defence (also known as aposematism) has been a long standing problem of ecology. The completion of the single-species game-theoretic model of Broom et al (2006) and its extension to the two-species case will have impact in a range of disciplines: from mathematical ecology and experimental biology (see for example Cortesi and Cheney 2010, Lee et al. 2010 and Lee et al. 2011, in the bibliography section) to game theory and complex systems theory. The specific aims of the proposed project are to1. Carry out an extensive review of the relevant literature (included in the Bibliography section);2. Complete the one-species case (by uncovering the complete list of ESSs and characterising these in terms of their stability);3. Develop the general model for the treatment of prey populations consisting of two species;4. Determine the ESSs for two-species case (and characterise these in terms of their stability) for the case in which there is convergence in appearance (also known as two-point solution); and 5. Produce a model for Batesian mimicry (with strong emphasis in specific models of this).Background Prey species invest in constitutive and induced defences against attackers in order to increase their fitness. For attacks that occur over short time-scales constitutive defences admit the dominant mech-anism to avoid predation and may be classified as primary or secondary (Broom et al. 2006). The aim of the former is to reduce the rate at which prey species are attacked whereas the aim of the latter is two-fold: to reduce the probability that a mounted attack results in death (K) and to increase the aversive information (I) that predators have on similar looking prey. Investment in secondary defences may not be easily evaluated by predators ( as is the case with chemical defences) and is often coupled with a warning display which signals a prey species' investment in these defences (Broom et al. 2006). For the remainder of this discussion we will consider those secondary defences that require signalling in order for them to be evaluated by predators (Broom et al. 2010, Broom et al. 2013, Ruxton et al. 2009). The aim of the project that we are proposing is to complete the work on aposematism byroom et al. (2006) for the one-species case and to develop the general model for the two-species case. The latter will open up the possibility for studying particular models of Batesian mimicry (a form of mimicry in which harmless prey resemble harmful prey in order to increase their fitness) that are of interest in biology (Broom and Rychtar 2013, Cortesi and Cheney 2010, Lee et al. 2010 and Lee et al. 2011). Literature Review The co-evolution of secondary defences and signals of those defences have received certain, but not significant theoretical attention ( compared with studies on the mechanisms of action of these defences - Broom et al. 2006). A game-theoretic model of this co-evolution was first devised by Leimar et al.(1986) and continues to constitute dominant work in aposematism (Broom et al. 2006). Despite its novelty, this model is unable to account for the possibility of predators increasing their attack proba-bility towards prey species due to the presence of an ever-increasing inhibitory gradient. Additionally, many of the results rely on certain restrictive functional forms, which make it difficult to assess the generality of the model (Broom et al. 2006).In contrast with Leimar et al. (1986), which considers a set of (initially) naive predators (Broom et al. 2006), the (also game-theoretic) model of Broom et al. (2006), considers a set of predators modelled as a group in equilibrium for states of learning, hunger etc. and prey species (with average local relatedness a:) consisting of residents and mutants playing one of two distinct strategies. Of key importance to this model are evolutionarily stable strategies (ESSs) which are