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We investigate how the loss of previously evolved diversity in host resistance to disease is dependent on the complexity of the underlying evolutionary trade-off. Working within the adaptive dynamics framework, using graphical tools (pairwise invasion plots, PIPs; trait evolution plots, TEPs) and algebraic analysis we consider polynomial trade-offs of increasing degree. Our focus is on the evolutionary trajectory of the dimorphic population after it has been attracted to an evolutionary branching point. We show that for sufficiently complex trade-offs (here, polynomials of degree three or higher) the resulting invasion boundaries can form closed 'oval' areas of invadability and strategy coexistence. If no attracting singular strategies exist within this region, then the population is destined to evolve outside of the region of coexistence, resulting in the loss of one strain. In particular, the loss of diversity in this model always occurs in such a way that the remaining strain is not attracted back to the branching point but to an extreme of the trade-off, meaning the diversity is lost forever. We also show similar results for a non-polynomial but complex trade-off, and for a different eco-evolutionary model. Our work further highlights the importance of trade-offs to evolutionary behaviour., (Copyright © 2015 Elsevier Inc. All rights reserved.)

Variation for resistance to infectious disease is ubiquitous and critical to host and parasite evolution and to disease impact, spread and control. However, the processes that generate and maintain this diversity are not understood. We examine how ecological feedbacks generate diversity in host defence focussing on when polymorphism can evolve without co-evolution of the parasite. Our key result is that when there is heritable variation in hosts in both their transmissibility and susceptibility along with costs to resistance, there is the possibility of the evolution of polymorphism. We argue that a wide range of behavioural or physiological mechanisms may lead to relationships between transmissibility and susceptibility that generate diversity. We illustrate this by showing that a tendency for higher contacts between related individuals leads to polymorphism. Only dimorphisms can evolve when infection is determined only by an individuals' susceptibility or when transmissibility and susceptibility are simply positively or negatively correlated., (© 2012 Blackwell Publishing Ltd/CNRS.)

In order to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary in an ab initio calculation to find the fitness and its derivatives at a singular point. It has been suggested that the possible evolutionary behaviour can be predicted directly from the resident population dynamics, without the need for calculation, by applying three criteria-one based on the form of the density dependent rates and two on the role played by the evolving parameters. The existing arguments for these criteria are rather limited: they apply to systems in which individuals enter an initial class and can then move through any number of other population classes sequentially. (Extensions are included but only apply for systems of two and three classes.) Additionally, many of the arguments depend on the use of a phenomenologically motivated fitness (shown equivalent to the standard form but in a rather long and indirect manner). The present paper removes all these flaws-the criteria are established directly from the standard definition of fitness and individuals can enter any class and move through the classes non-sequentially without restriction on their number. The criteria thus established underlie a geometric description of the singular behaviour in adaptive dynamics which allows direct inferences to be made from population dynamics to the possible singular behaviour depending on which of the criteria apply and on the nature of the trade-off between evolving parameters. The method has the great advantage of leaving the trade-off explicit but unspecified.

Many studies of the evolution of life-history traits assume that the underlying population dynamical attractor is stable point equilibrium. However, evolutionary outcomes can change significantly in different circumstances. We present an analysis based on adaptive dynamics of a discrete-time demographic model involving a trade-off whose shape is also an important determinant of evolutionary behaviour. We derive an explicit expression for the fitness in the cyclic region and consequently present an adaptive dynamic analysis which is algebraic. We do this fully in the region of 2-cycles and (using a symbolic package) almost fully for 4-cycles. Simulations illustrate and verify our results. With equilibrium population dynamics, trade-offs with accelerating costs produce a continuously stable strategy (CSS) whereas trade-offs with decelerating costs produce a non-ES repellor. The transition to 2-cycles produces a discontinuous change: the appearance of an intermediate region in which branching points occur. The size of this region decreases as we move through the region of 2-cycles. There is a further discontinuous fall in the size of the branching region during the transition to 4-cycles. We extend our results numerically and with simulations to higher-period cycles and chaos. Simulations show that chaotic population dynamics can evolve from equilibrium and vice-versa.

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria - one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.

Trade-off shapes are crucial to evolutionary outcomes. However, due to different ecological feedbacks their implications may depend not only on the trade-off being considered but also the ecological scenario. Here, we apply a novel geometric technique, trade-off and invasion plots (TIPs), to examine in detail how the shape of trade-off relationships affect evolutionary outcomes under a range of classic ecological scenarios including Lotka-Volterra type and host-parasite interactions. We choose models of increasing complexity in order to gain an insight into the features of ecological systems that determine the evolutionary outcomes. In particular we focus on when evolutionary attractors, repellors and branching points occur and how this depends on whether the costs are accelerating (benefits become 'increasingly' costly), decelerating (benefits become 'decreasingly' costly) or constant. In all cases strongly accelerating costs lead to attractors while strongly decelerating ones lead to repellors, but with weaker relationships, this no longer holds. For some systems weakly accelerating costs may lead to repellors and decelerating costs may lead to attractors. In many scenarios it is weakly decelerating costs that lead to branching points, but weakly accelerating and linear costs may also lead to disruptive selection in particular ecological scenarios. Using our models we suggest a classification of ecological interactions, based on three distinct criteria, that can produce one of four fundamental TIPs which allow for different evolutionary behaviour. This provides a baseline theory which may inform the prediction of evolutionary outcomes in similar yet unexplored ecological scenarios. In addition we discuss the implications of our results to a number of specific life-history trade-offs in the classic ecological scenarios represented by our models.

In this study we use the theory of adaptive dynamics firstly to explore the differences in evolutionary behaviour of a generalist predator (or more specifically an omnivorous or intraguild predator) in a predator-prey model, with a Holling Type II functional response, when two distinct forms for the carrying capacity are used. The first of these involves the carrying capacity as an emergent property, whilst in the second it appears explicitly in the dynamics. The resultant effect this has on the intraspecific competition in each case is compared. Taking an identical trade-off in each case, we find that only with an emergent carrying capacity is evolutionary branching possible. Our study then concentrates solely on the case where the carrying capacity appears explicitly. Using the same model as above, but choosing alternate trade-offs, we find branching can occur with an explicit carrying capacity. Our investigation finishes by taking a more general functional response in an attempt to derive a condition for when branching can or cannot occur. For a predator-prey model, branching cannot occur if the functional response can be separated into two components, one a function of the population densities, X and Z, and the other a function of the evolving parameter z (traded off against the intrinsic growth rate), i.e. if F(z,X,Z) = F(1)(z)F(2)(X,Z). This search for evolutionary branching is motivated by its possible role in speciation.

The purpose of this paper is to take an entirely geometrical path to determine the evolutionary properties of ecological systems subject to trade-offs. In particular we classify evolutionary singularities in a geometrical fashion. To achieve this, we study trade-off and invasion plots (TIPs) which show graphically the outcome of evolution from the relationship between three curves. The first invasion boundary (curve) has one strain as resident and the other strain as putative invader and the second has the roles of the strains reversed. The parameter values for one strain are used as the origin with those of the second strain varying. The third curve represents the trade-off. All three curves pass through the origin or tip of the TIP. We show that at this point the invasion boundaries are tangential. At a singular TIP, in which the origin is an evolutionary singularity, the invasion boundaries and trade-off curve are all tangential. The curvature of the trade-off curve determines the region in which it enters the singular TIP. Each of these regions has particular evolutionary properties (EUS, CS, SPR and MI). Thus we determine by direct geometric argument conditions for each of these properties in terms of the relative curvatures of the trade-off curve and invasion boundaries. We show that these conditions are equivalent to the standard partial derivative conditions of adaptive dynamics. The significance of our results is that we can determine whether the singular strategy is an attractor, branching point, repellor, etc. simply by observing in which region the trade-off curve enters the singular TIP. In particular we find that, if and only if the TIP has a region of mutual invadability, is it possible for the singular strategy to be a branching point. We illustrate the theory with an example and point the way forward.

We examine the evolutionary dynamics of resistance to parasites through acquired immunity. Resistance can be achieved through the innate mechanisms of avoidance of infection and reduced pathogenicity once infected, through recovery from infection and through remaining immune to infection: acquired immunity. We assume that each of these mechanisms is costly to the host and find that the evolutionary dynamics of innate immunity in hosts that also have acquired immunity are quantitatively the same as in hosts that possess only innate immunity. However, compared with resistance through avoidance or recovery, there is less likely to be polymorphism in the length of acquired immunity within populations. Long-lived organisms that can recover at intermediate rates faced with fast-transmitting pathogens that cause intermediate pathogenicity (mortality of infected individuals) are most likely to evolve long-lived acquired immunity. Our work emphasizes that because whether or not acquired immunity is beneficial depends on the characteristics of the disease, organisms may be selected to only develop acquired immunity to some of the diseases that they encounter.

Our understanding of the evolution of diseases has been greatly aided by the use of baseline criteria. Here we examine the theoretical and biological relationships of the well known baseline criteria for the evolution of disease (R0) and the recently introduced corresponding criterion for the evolution of resistance in hosts (D0). We show that there is a formal theoretical equivalence between the two criteria and discuss the characteristics of seperability that determine whether the criteria define the course of evolution. These theoretical determinants correspond biologically to whether strains compete for resources or not. We discuss the biological application of the criteria and argue that D0 may be less widely applicable than R0, but does determine the evolution of resistance in populations with fixed carrying capacities.

We analyse the adaptive dynamics of a generalised type of Lotka-Volterra model subject to an explicit trade-off between two parameters. A simple expression for the fitness of a mutant strategy in an environment determined by the established, resident strategy is obtained leading to general results for the position of the evolutionary singular strategy and the associated second-order partial derivatives of the mutant fitness with respect to the mutant and resident strategies. Combinations of these results can be used to determine the evolutionary behaviour of the system. The theory is motivated by an example of prey evolution in a predator-prey system in which results show that only (non-EUS) evolutionary repellor dynamics, where evolution is directed away from a singular strategy, or dynamics where the singular strategy is an evolutionary attractor, are possible. Moreover, the general theory can be used to show that these results are the only possibility for all Lotka-Volterra systems in which aside from the trade-offs all parameters are independent and in which the interaction terms are of quadratic order or less. The applicability of the theory is highlighted by examining the evolution of an intermediate predator in a tri-trophic model.