The influence of trade-off shape on evolutionary behaviour in classical ecological scenarios

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Abstract

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.

Introduction

Life-history theory has long recognised the importance of trade-offs in determining evolutionary behaviour (see Stearns, 1992; Roff, 2002 for reviews). It is also increasingly recognised that the shape of trade-offs, in addition to the level of costs, is crucial in determining the evolutionary dynamics (see Levins, 1962, Levins, 1968; de Mazancourt and Dieckmann, 2004; Rueffler et al., 2004; Bowers et al., 2005). By definition, in all trade-off relationships, benefits in one life-history trait come at a cost in terms of another component of fitness. In general then as benefits through one trait increase, the costs due to the change in the other trait may increase at the same rate, leading to an exactly linear trade-off; alternatively the costs may accelerate (increase quicker than the benefits) or decelerate (increase slower than the benefits), so that benefits become increasingly or decreasingly costly. When the benefits of a trait are met with accelerating costs in the correlated trait we define an ‘acceleratingly costly trade-off’. Conversely we define ‘deceleratingly costly trade-offs’ when the costs decelerate. Here our aim is to understand how these different shapes of trade-offs influence evolutionary outcomes in a range of scenarios described by a number of classic ecological models.

The importance of the shape of trade-off relationships was first made clear in the work of Levins, 1962, Levins, 1968. He developed a graphical technique that plots the fitness landscape from the fitness contours for two traits onto which the trade-off relationship between them is superimposed. Applying these techniques to the evolution of reproductive effort it was shown that the optimal strategy for a trade-off with decelerating costs is at the maximum reproductive effort whereas for a trade-off with accelerating costs it is at an intermediate state (see Stearns, 1992). However, optimisation approaches, such as Levin's are not appropriate, when there is frequency-dependent (density-dependent) selection (Maynard Smith, 1982; de Mazancourt and Dieckmann, 2004; Rueffler et al., 2004; Bowers et al., 2005). Different ecological interactions result in particular feedbacks that may clearly lead to different selection pressures on traits (Abrams, 2001) that in turn depend on the nature of the trade-off connections between traits. Here we will examine how trade-off shapes influence evolutionary behaviour in a number of fundamental ecological models using a geometric approach that incorporates frequency-dependent selection.

Under frequency-dependent selection, the evolution of traits is dependent on the ecological feedbacks in the system since for a mutation to be successful it must be able to invade a population whose ecological characteristics are being determined by the resident strain (Metz et al., 1996; Geritz et al., 1998). Successful mutant invasion necessarily changes the resident and therefore also reshapes the characteristics of the population. This approach has been applied to a number of specific ecological scenarios in which trade-off relationships have been explicitly considered (Boots and Haraguchi, 1999; Kisdi, 2001; Day et al., 2002; Bowers et al., 2003; Egas et al., 2004; de Mazancourt and Dieckmann, 2004; White and Bowers, 2005; Rueffler et al., 2006). In particular, the importance of the trade-off shape in characterising evolutionary behaviour has recently been examined in detail with the development of general geometric methods for analysing the evolutionary dynamics (de Mazancourt and Dieckmann, 2004; Rueffler et al., 2004; Bowers et al., 2005). Rueffler et al. (2004) extended the Levins fitness landscape approach to allow for frequency-dependent selection for specific trade-off functions. This was further extended to enable visualisation of the effect of general trade-off functions on evolutionary outcomes (de Mazancourt and Dieckmann, 2004). The method of trade-off and invasion plots (TIPs) developed by Bowers et al. (2005) and first used in Boots and Bowers (2004) is similar to that of de Mazancourt and Dieckmann (2004) in that it is a geometric technique that allows the visualisation of the role of the trade-off shape. From TIPs given a specific ecological model, it is easy to determine which trade-off shapes (or cost structures) produce for example, evolutionary branching. Although globally the curvature of the trade-off can change sign, for example in sigmoidal trade-offs, TIPs focus on the shape of the trade-off locally about the evolutionary singularity; in this local region the curvature of the trade-off stays relatively constant and hence falls into one of three shapes: decelerating, accelerating or straight (linear).

In this study, we use TIPs to explore which ecological characteristics lead to different evolutionary behaviours with trade-offs of different shapes. We use classic models of increasing ecological complexity in order to reveal the ecological features that underlie four fundamental types of TIPs. By classic we mean Lotka–Volterra-type continuous time models, i.e. where the dynamics are linear in terms of the ‘evolving’ parameters (those involved in the trade-off) and of order one or two (linear, bilinear or quadratic) in terms of densities. We choose this framework as it underpins many model structures in theoretical ecology. In particular, we examine classical single species models, Lotka–Volterra models that include species interactions with one class of individuals for both species and host–parasite systems with multi-class interactions. These TIPs in turn define all the evolutionary implications of different trade-off shapes. In this way we outline how the incorporation of additional ecological mechanisms can alter the topology of the invasion boundaries on a TIP and therefore also the possible evolutionary outcomes. For our range of models, we classify, by means of three criteria, the necessary general ecological characteristics and trade-off set-up required to produce different types of evolutionary behaviour. By focussing on the classical models that underpin much ecological theory, our aim is to provide a baseline that may inform the understanding of more complex ecological scenarios that can be modelled by these type of systems.

Section snippets

The approach: trade-off and invasion plots

A detailed description of the use of TIPs to determine evolutionary behaviour has been given elsewhere (Bowers et al., 2005) and their derivation is outlined in Appendix A. An example of a TIP showing its key features can be seen in Fig. 1 (where Table 5 (see Appendix A) has been used to determine whether a region is ES or CS). The mutant–resident invasion boundaries (f1, f2) are plotted in trait space and the trade-off line (f) is superimposed such that the position of the trade-off line in

Single species

We begin with a basic maturation model consisting of two stages, a non-reproducing juvenile stage and a reproducing mature stage. This, for two strains x and y, is defined by the continuous time age-structured dynamicsdX1dt=axX2-qxX2(X1+X2+Y1+Y2)-bxX1-mxX1,dY1dt=ayY2-qyY2(X1+X2+Y1+Y2)-byY1-myY1,dX2dt=mxX1-exX2,dY2dt=myY1-eyY2,where X1 and X2 denote the number of juveniles and matures, respectively, for strain x, and similarly Y1 and Y2 for strain y. Also, a represents the per capita

Discussion

The feedback between ecological and evolutionary processes is crucial to understand how ecological interactions generate natural selection and how evolutionary change further modifies the ecological interactions (MacArthur, 1972; Roughgarden, 1979; Bulmer, 1994). By applying a geometric approach we have developed a theory for how different trade-off shapes affect evolutionary outcomes in a number of classical ecological scenarios. The work clearly demonstrates the importance that the shape of

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