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

对于防御和防御信号的共同进化(也称为无融合现象)，一个简洁的数学解释一直是一个长期存在的生态学问题。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(例如见Cortesi和Cheney 2010，Lee等人)。2010年和Lee等人提出的。2011年，在参考书目部分)涉及博弈论和复杂系统理论。拟议项目的具体目标是1。对相关文献进行广泛的审查(包括在参考书目部分)；2.完成单一物种的情况(通过揭示ESSS的完整清单，并根据其稳定性描述这些情况)；3.开发处理由两个物种组成的被捕食种群的一般模型；4.确定两个物种情况下的ESSS(并根据其稳定性来表征这些情况)，在出现收敛的情况下(也称为两点解)；和5.建立一个巴特式模仿的模型(重点放在这个特定的模型中)。后台的猎物物种投资于对攻击者的结构性和诱导性防御，以增加它们的适应性。对于发生在短时间尺度上的攻击，构成防御承认了主要的机制，以避免捕食，并可能被归类为主要或次要(Broom等人。2006)。前者的目的是减少猎物物种受到攻击的速度，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猎物的厌恶信息(I)。对次级防御的投资可能不容易被捕食者评估(化学防御就是这种情况)，而且往往伴随着警告显示，表明被捕食物种对这些防御的投资(Broom等人)。2006)。在接下来的讨论中，我们将考虑那些需要信令才能被捕食者评估的次要防御(Broom等人。2010年，布鲁姆等人。2013年，Ruxton等人。2009年)。我们提议的项目的目的是完成由Room等人完成的关于无种族隔离的工作。(2006)对单一物种的情况，并为两种情况发展一般模式。后者将为研究生物学感兴趣的特定巴特斯模仿模型(一种无害的猎物与有害的猎物相似以增加适应性)提供了可能性(Broom和Rychtar 2013，Cortesi和Cheney 2010，Lee等人)。2010年和Lee等人提出的。2011年)。文献综述次生防御的共同进化和这些防御的信号已经得到了一定的理论关注，但没有显著的理论关注(与对这些防御的作用机制的研究相比-Broom等人)。2006)。这种共同进化的博弈论模型首先是由Leimar等人(1986)设计的，并继续构成无种族隔离的主要工作(Broom等人。2006)。尽管这个模型很新颖，但它不能解释捕食者由于不断增加的抑制梯度的存在而增加其对被捕食物种攻击的可能性。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一般性(Broom等人。2006)。与Leimar等人形成对比。(1986)，它认为一组(最初)幼稚的捕食者(Broom等人)。2006)，(也是博弈论的)模型。(2006)，考虑了一组捕食者被建模为处于学习、饥饿等状态的平衡状态的一组捕食者，以及由居民和变种组成的猎物物种(具有平均本地关联度a：)，它们扮演着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)，它们是

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