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

一个简洁的数学解释的共同进化的防御和信号的防御（也被称为警戒）一直是一个长期存在的问题生态学。Broom et al（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在一系列学科中产生影响：从数学生态学和实验生物学（例如参见参考书目部分中的Cortesi和切尼2010，Lee et al. 2010和Lee et al. 2011）到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.完成一个物种的情况下（通过揭示ESS的完整列表，并根据其稳定性来描述这些）;3。建立了两种群食饵处理的一般模型;4.确定两物种情况下的ESS（并根据其稳定性对这些ESS进行验证），以用于出现收敛的情况（也称为两点解）;以及5.产生一个模型为Bajuanmimicry（与强烈强调在特定的模型，这）。背景猎物物种投资于组成和诱导防御攻击者，以增加他们的健身。对于发生在短时间尺度上的攻击，组成性防御承认主要机制是避免捕食，可以分为初级或次级（Broom等人，2006）。前者的目的是降低被捕食物种被攻击的速度，而后者的目的是双重的：降低一次安装攻击导致死亡的概率（K）和增加捕食者对相似外观的猎物的厌恶信息（I）。捕食者可能不容易对二级防御的投资进行评估（化学防御就是这种情况），而且往往伴随着一种警告显示，表明被捕食物种对这些防御的投资（Broom等人，2006年）。在接下来的讨论中，我们将考虑那些需要信号的二级防御，以便捕食者对其进行评估（Broom et al. 2010，Broom et al. 2013，Ruxton et al. 2009）。我们所提出的项目的目的是完成byroom et al.（2006）关于单物种情况下的警戒主义的工作，并开发两物种情况下的一般模型。后者将为研究生物学中感兴趣的特定的拟食动物模型（拟食动物的一种形式，其中无害的猎物类似于有害的猎物，以增加其适应性）开辟可能性（Broom和Rychtar 2013，Cortesi和切尼2010，Lee等人2010和Lee等人2011）。文献综述二级防御和这些防御信号的共同进化已经得到了一定的，但不是显著的理论关注（与这些防御的作用机制的研究相比- Broom et al. 2006）。这种共同进化的博弈论模型首先由Leimar等人设计。（1986）并继续构成警戒主义的主导工作（Broom et al. 2006）。尽管该模型具有新奇，但由于抑制梯度的存在，该模型无法解释捕食者增加其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的函数形式，这使得难以评估模型的通用性（Broom等人，2006年）。与Leimar等人（1986年）相比，Leimar等人认为一组（最初）天真的掠食者（Broom等人，2006年），Broom等人（2006年）的（也是博弈论）模型，考虑了一组捕食者，将其建模为学习状态的平衡组，饥饿等和猎物物种（平均本地相关性a：）由居民和突变体组成，发挥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（ESS），即

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