2.1 Stable equilibria

The simplest non-trivial evolutionary outcome is stable monomorphism, where a single type prevails within a population. For example, the host may evolve to a particular level of resistance and the parasite to a particular level of infectivity (Figure 1a). In a quantitative genetic framework (see ‘Genetic Structure’), where traits are continuous, stable monomorphism is usually referred to as an evolutionarily singular strategy (ESS) (Maynard Smith, 1972), or a continuously stable strategy (CSS) if it is also convergence stable (i.e. it can be approached by small mutations) (Eshel, 1983). Although a single type may be optimal, it is possible for variation to be maintained due to a mutation-selection balance. Alternatively, one or both populations may exhibit stable polymorphism, where two or more types coexist at equilibrium within a population. Stable polymorphism occurs when different alleles at a given locus coexist in the population (Figure 1b) (Ashby et al., 2019; Sasaki, 2000; Tellier & Brown, 2007b), or when disruptive selection leads to a branching process, creating two sub-populations with different trait values (Best et al., 2009).

2.2 Directional selection

Directional selection occurs when there is a continual increase or decrease in traits such as resistance or infectivity (Figure 1). Such dynamics may produce a coevolutionary arms race, leading to trait escalation in both species (Dawkins & Krebs, 1979). For example, bacteria and phage often exhibit escalatory arms races under laboratory conditions (Buckling & Rainey, 2002) and plant–pathogen coevolution has led to an escalatory series of defence (e.g. R genes, pathogen-associated molecular pattern (PAMP) recognition, effector-triggered immunity) and counter-defence (e.g. avirulence genes, effector-triggered susceptibility, toxin production) mechanisms (Jones & Dangl, 2006). Directional selection may also take the form of de-escalation (Sasaki & Godfray, 1999).

Unlike stable monomorphism and polymorphism, directional selection is a dynamic outcome of coevolution. In principle, directional selection may continue indefinitely, for example when resistance or infectivity increases without bound (Lopez Pascua et al., 2014) or if parasites ‘chase’ hosts through phenotype space without cycling back to previous phenotypes. Directional selection may therefore provide an explanation for the existence of extreme attack and defence traits in parasites and their hosts (Iseki et al., 2011). However, directional selection cannot necessarily be maintained indefinitely due to escalating fitness costs and physiological constraints. As such, periods of directional selection may eventually give way to stable monomorphism, polymorphism or fluctuating selection (Ashby & Boots, 2017; Tellier et al., 2014; Figure 1). For example, Hall et al. (2011) found that although coevolving bacteria and phage initially exhibit directional selection, this eventually gives way to fluctuating selection.

2.3 Fluctuating selection

Fluctuating selection (also referred to as coevolutionary cycling or Red Queen Dynamics) occurs when either the direction of selection changes, resulting in host and parasite traits varying non-monotonically through time, or when there is no stable distribution of genotypes, resulting in oscillations in allele frequencies (Figure 1c). These dynamics tend to be driven by negative frequency-dependent selection (i.e. rare advantage) (Dybdahl & Lively, 1998; Hamilton, 1980), fitness costs (Ashby & Boots, 2017; Sasaki, 2000), environmental fluctuations (Mostowy & Engelstädter, 2011) or stochasticity (Stephan & Tellier, 2020). Empirically, fluctuating selection has been observed using time-shift experiments (Gaba & Ebert, 2009), where hosts from one time point are exposed to parasites from another, leading to oscillatory patterns of resistance and infectivity (Decaestecker et al., 2007; Hall et al., 2011; Jokela et al., 2009; Stahl et al., 1999). However, fluctuating selection can be difficult to detect as it may appear as directional selection over short time scales (Figure 1c), where periods of escalation alternate with periods of de-escalation (Gaba & Ebert, 2009; Gandon et al., 2008). Fluctuating selection may also occur at different levels simultaneously. For example, fluctuations may occur in the range of hosts which parasites can infect (specialism-generalism) while the specific hosts that they can infect vary as well (Ashby & Boots, 2017). Fluctuating selection may also occur in the short term in the form of damped cycles, which diminish in amplitude and eventually tend towards stable polymorphism (Ashby & Gupta, 2014; MacPherson & Otto, 2018).

Fluctuating selection is closely associated with the Red Queen Hypothesis for sex (Bell, 1982; Lively, 2010), which argues that sexual reproduction is maintained despite being less efficient than asexual reproduction due to coevolution with parasites. This is because sex generates diverse offspring, which allows sexually reproducing hosts to adapt more rapidly than asexual hosts when targeted by coevolving parasites. Understanding when host–parasite coevolution leads to fluctuating selection and investigating how the nature of the oscillatory dynamics (e.g. amplitude, frequency) selects for sex have therefore been major drivers of theoretical research on host–parasite coevolution (reviewed in Ashby & King, 2015; Lively, 2010).

4.1 Genetic structure

Coevolutionary models must make assumptions about the underlying genetics of host and parasite traits and how they are inherited. Will the model track individual genotypes and allele frequencies or will coevolution be modelled phenotypically? Are hosts and parasites haploid, diploid or polyploid? Is there epistasis or are effects between loci additive? The underlying genetics of a system may be modelled in one of two ways. A population genetics approach (used in 145 models in the literature survey) assumes that the traits under selection are determined by a small number of genes with relatively large, potentially epistatic, effects. Population genetic models therefore assume that there are a finite number of host and parasite genotypes which may be haploid, diploid or polyploid. However, even a small number of genotypes can render a model analytically intractable and so population genetic models tend either to neglect population dynamics for simplicity, or to be solved numerically or through simulations. In contrast, a quantitative genetics approach (used in 38 models in the literature survey) assumes that the traits under selection are determined by many loci with relatively small, additive effects. Quantitative traits are generally modelled as continuous phenotypes, with analysis focusing on how characteristics of the trait distributions evolve (e.g. mean and variance). By modelling continuous traits and ignoring epistasis, quantitative genetic models can provide analytical insights which complement population genetic models and can often include population dynamics while maintaining tractability (see Appendix S1). Whether a population genetic or a quantitative genetic approach is more appropriate depends entirely on the focus of the model. A population genetic approach is clearly more appropriate when traits are determined by a few major genes, especially when epistasis is involved. Major gene effects have been observed, for instance, in bacteria-phage systems (Scanlan et al., 2011), viral infections of fruit flies (Cogni et al., 2016) and many other infections of plants and animals (Wilfert & Schmid-Hempel, 2008). A quantitative genetic approach may be more appropriate if many loci are involved, as has been observed in many cases of plant resistance (Corwin & Kliebenstein, 2017), if the genetic details are unknown or if the focus is on evolution at the phenotypic level.

Direct comparisons between population genetic and quantitative genetic models are rare (Frank, 1993b). There is some evidence that the different approaches may produce qualitatively different results but this may be due to other factors, such as whether models include population dynamics. It is also possible for models to include both population genetics and quantitative genetics, for example by considering a multi-step infection process (Nuismer & Dybdahl, 2016) or by modelling one species using a continuous trait and the other a discrete trait (Akçay, 2017; Yamamichi & Ellner, 2016).

Epistasis can play an important role in coevolutionary dynamics, as mutations may have little effect on fitness in isolation but together may have a very large effect. For instance, negative epistasis has been shown to result in increased fluctuating selection in some models (Fenton & Brockhurst, 2007) and affects the ability of parasites to adapt to their hosts (Ashby et al., 2014a). Population genetic models must also choose whether hosts and parasites are haploid or diploid, which can have both qualitative and quantitative effects on coevolution. For instance, diploidy may reduce the incidence of cycling (Nuismer, 2006), widen the conditions under which local adaptation is observed (Gandon & Nuismer, 2009) or favour assortative mating (Greenspoon & M’Gonigle, 2014). The precise nature of these effects depends upon the model in question (Mostowy & Engelstädter, 2012). Alternatively, ploidy itself may be allowed to evolve, with theoretical models predicting that parasites generally evolve to be haploid whereas their hosts evolve to be diploid (Nuismer & Otto, 2004). Similar conditions have also been shown to favour the evolution of parasitism (M’Gonigle & Otto, 2011). Discussion of the effects of epistasis and ploidy on host–parasite coevolution can be found in Nuismer (2017) (Chapters 5 and 6).

4.2 Infection genetics

The genetic basis of infection and its effects on coevolutionary dynamics are a major focus of the theoretical host–parasite literature. The infection genetics of a model describe the interactions between all combinations of hosts and parasites, detailing who can infect whom and to what extent. These generally fall into a few main classes (Table 1; Figure 2). In a population genetic model, there are a finite number of host and parasite genotypes and so the infection genetics can be represented in a table of genotype × genotype interactions, where each host and parasite genotype is defined by a certain set of alleles at one or more genetic loci (Figure 2a). In a quantitative genetic model, the infection genetics are represented by a continuous function of host and parasite traits (Figure 2b). In either case, the infection genetics may vary in the level of specialism or generalism (specificity) with which parasites infect different hosts (and with which hosts can resist different parasites). In most cases, the infection genetics take one of three forms depending on the specificity between host and parasite types: (1) high specificity (Figure 2i); (2) variation in specificity (Figure 2ii) or (3) no specificity (Figure 2iii). A small number of studies have also explored multi-step infection processes, combining one or more frameworks, but these are relatively rare (Agrawal & Lively, 2003; Fenton et al., 2012; Nuismer & Dybdahl, 2016).

When there is high specificity, either infectivity or resistance is maximized when there is a ‘match’ between the host and parasite traits (Figure 2i). For example, the immune systems of vertebrates are able to detect foreign bodies by means of a self-nonself recognition system (Medzhitov & Janeway, 2000). Similar ‘matching’ has been observed in bacterial infections of crustaceans (Luijckx et al., 2013), where the parasite must match the host for infection to occur. This is clearly a case of a highly specialized parasite, as it can only infect hosts which it matches (genetically or phenotypically). In a population genetic model, infection occurs if a host and parasite match at some or all loci (Frank, 1991a, 1994b; Gandon et al., 1996; Seger, 1988), which is known as ‘matching alleles’ (MA) (Figure 2ai). Variations exist where the probability of infection depends on the number of matching loci (Nuismer et al., 2017) or where infection occurs if a sufficiently large number of loci match (Shin & MacCarthy, 2016) or if there is a long enough continuous sequence of matching loci (Carrillo-Bustamante et al., 2015). Conversely, a host may need to ‘match’ the parasite in order to recognize it and mount an immune response, as in the case of the vertebrate major histocompatibility complex (MHC) (Frank, 2002). Such cases are described by ‘inverse matching alleles’ models (Nuismer, 2006). This is also an example of high specificity. Similar infection genetics can be modelled quantitatively, where each host and parasite has a continuous trait value and the probability of infection depends on the similarity between trait values (with infection more likely when they are similar). This is typically referred to as a ‘matching’ or ‘bidirectional’ function (Figure 2bi) and often takes the form of a bell (Gaussian) curve which peaks when host and parasite trait values match (Boots et al., 2014; Nuismer et al., 2005).

When there is variation in specificity, some parasites are able to infect a broader range of hosts than other parasites (conversely, some hosts are able to resist a broader range of parasites than other hosts). As such, there may be variation in the level of generalism among host and parasite types. Such systems have been found to occur in plants (Brown & Tellier, 2011; Flor, 1956), bacteria (Flores et al., 2011; Scanlan et al., 2011), fruit flies (Wilfert & Jiggins, 2010) and crustaceans (Little et al., 2006). In a population genetic framework, a host generally has either a resistant or susceptible allele at each locus, whereas a parasite may or may not possess a corresponding allele at each locus to counter host resistance (note that these are often called ‘virulence’ alleles in the literature, but we refer to these instead as ‘infectivity’ alleles to avoid confusion with disease severity). These are typically referred to as ‘gene-for-gene’ (GFG) models (Figure 2aii). In contrast to MA models, GFG models usually assume that the probability of infection is determined by the number of loci at which both the host is resistant and the parasite does not possess a corresponding infectivity allele (Ashby & Boots, 2017; Ashby et al., 2014b; Frank, 1992). In a quantitative genetic framework, variation in specificity is modelled by assuming that the probability of infection is determined by the extent to which an infectivity trait in the parasite exceeds a resistance trait in the host (Best et al., 2010; Nuismer et al., 2007), and is often referred to as a ‘range’ or ‘unidirectional’ model (Figure 2bii).

There has been much debate regarding the relevance of the various infection genetic frameworks to real systems. For example, Parker argued that the GFG system was prevalent in plants (Parker, 1996) whereas Frank believed that their infection genetic systems followed the MA framework (Frank, 1996a,b). However, there is evidence that both systems exist across taxa (Dybdahl et al., 2014). Some have argued that the MA and GFG models lie at either end of a spectrum, with the most realistic models somewhere in between (Agrawal & Lively, 2002), but MA models can also be considered as a subset of GFG models (Ashby & Boots, 2017) or GFG models as a subset of inverse matching allele models (Dybdahl et al., 2014). Similarly, comparisons between matching (bidirectional) and range (unidirectional) functions have been studied extensively (Best et al., 2017; Macpherson et al., 2018; Ridenhour & Nuismer, 2007; Yoder & Nuismer, 2010). In general, models that feature fixed, high specificity (i.e. ‘matching’) tend to produce fluctuating selection dynamics more readily. Oscillations are often rapid and may either persist indefinitely (typically as neutrally stable or stochastically driven cycles; Best et al., 2017; Frank, 1991b) or decay towards a stable polymorphic population (damped cycles; Ashby & Gupta, 2014; MacPherson & Otto, 2018). These models typically assume that there are no trade-offs, so that types only differ in their susceptibility or infectivity profiles. Oscillations are therefore driven by negative frequency-dependent selection (rare advantage) and occur either between types that have identical levels of specialism or as the result of a ‘chase’ through phenotype space (Best et al., 2017). In contrast, GFG and range (unidirectional) models generally include trade-offs (see ‘Pleiotropy and trade-offs’) and types vary in their degree of generalism. These models can therefore produce a far broader range of outcomes, with stable monomorphic or polymorphic populations in either population (Ashby et al., 2019; Cortez et al., 2017; Fenton et al., 2009; Tellier & Brown, 2007a), rapid oscillations occurring within levels of specialism (driven by negative frequency-dependent selection) or slower oscillations occurring between levels of specialism (driven by trade-offs) (Ashby & Boots, 2017). Moreover, the oscillations tend to be either stable limit cycles or damped cycles and are not structurally unstable as is common in matching models (Best et al., 2017; Kawecki, 1998; Kwiatkowski et al., 2012). Differences between the outcomes of models with different infection genetic systems, while not surprising, emphasize the need for caution when drawing general conclusions about host–parasite coevolution.

Although most studies explore models with high specificity or variation in specificity, the infection genetics may also be nonspecific. This means that for any two parasite types, A and B, if parasite A is more infective on one host type than parasite B, then it is more infective on all host types (hence, it is said to be ‘universally’ more infective, sensu Boots et al., 2014). In a population genetic framework, the universal and GFG models are identical when only one locus is considered but are distinct for multiple loci (Figure 2aiii). For a GFG model, the probability of infection is determined by the number of corresponding loci at which the host is resistant and the parasite is not infective, whereas in a nonspecific model, it is the overall proportion of resistance and infectivity alleles that determines the probability of infection. In a quantitative genetic framework, the difference between universal and range models is more subtle. Range models are typically close to step functions, with parasites having very high infectivity on a subset of hosts and very low infectivity on all others (Figure 2bii), whereas universal models generally incorporate a broader spectrum of levels of infectivity, with different parasites able to infect different hosts to varying degrees (Figure 2biii).

4.3 Pleiotropy and trade-offs

Coevolutionary models often incorporate pleiotropy through trade-offs between life-history traits. This reflects empirical observations where, for example, increased host resistance may be associated with decreased growth or reproduction (Bartlett et al., 2018; Wright et al., 2016) or a higher within-host replication rate may lead to greater parasite transmissibility but may also be associated with higher disease-induced mortality (and hence, a shorter infectious period) (de Roode et al., 2008; Thrall & Burdon, 2003). Such trade-offs are usually not necessary in models with high specificity because selection may be driven by negative frequency dependence. In contrast, if there were no trade-offs associated with resistance or infectivity in models with variable or no specificity, then the most resistant host and the most infective parasite would always have the highest fitness. Trade-offs can be incorporated into models by letting other life-history parameters vary as functions of a focal trait (Figure 3). The precise trade-off will depend on the system being modelled and may be hypothetical or estimated from empirical results (Jessup & Bohannan, 2008). Where empirical data are not available, the effects of varying the shape and magnitude of any trade-offs should be considered in the model analysis, as these can both quantitatively and qualitatively affect the results.

It is well known that the shape and strength of trade-offs can have significant effects on evolutionary dynamics (Bowers et al., 2005; Kisdi, 2006). For example, changing the strength of fitness costs can cause the dynamics to switch between stable equilibria and fluctuating selection in models of host–parasite coevolution (Best et al., 2010; Fenton & Brockhurst, 2007; Sasaki, 2000). Trade-off shapes (Figure 3) can also cause qualitative shifts in coevolutionary dynamics, with stable monomorphism more likely when there are diminishing fitness returns (e.g. costs accelerate; Kisdi, 2006) and evolutionary branching (leading to stable polymorphism) more likely when trade-offs are close to linear or are slightly decelerating (Bowers et al., 2005).

4.4 Population dynamics

Population dynamics (also referred to as ecological or epidemiological dynamics) describe changes in the number (or density) of individuals in a population over time. Models of host–parasite coevolution can be broadly divided into two categories based on whether they include (84 models in the literature survey) or exclude (101 models in the literature survey) population dynamics (Figure 4b). Many models neglect population dynamics by assuming either that they do not influence fitness, or that the host and parasite population sizes are constant or infinite (Fenton et al., 2009; Mostowy & Engelstädter, 2012; Sasaki, 2000). In the absence of population dynamics, selection is only frequency-dependent, as fitness depends on the relative proportions of individuals with different genotypes/phenotypes and not on their absolute abundance (Ashby et al., 2019; MacPherson et al., 2021a). Population dynamics are often neglected for simplicity, especially to allow for more detailed genetic effects (e.g. epistasis, ploidy) without the model becoming analytically intractable. In contrast, eco-evolutionary models assume that population dynamics are integral to (co)evolution, as the evolution of a trait will generally affect, and be affected by, the population dynamics of hosts and parasites. Selection is therefore both frequency- and density-dependent, with feedback loops existing between population dynamics and evolutionary dynamics (Ashby et al., 2019; Figure 4a). However, eco-evolutionary feedback loops can limit analytical tractability and so it is often necessary to make simplifying assumptions about the genetics or mutations (Dieckmann & Law, 1996; Metz et al., 1995), or use numerical or simulation-based approaches.

Population dynamics were first incorporated into coevolutionary models by Pimentel (1961), who also suggested that their effects on coevolution might be significant in an empirical context (Pimentel et al., 1978). Several early studies of host–parasite coevolution also emphasized the importance of eco-evolutionary feedbacks (Frank, 1991a; May & Anderson, 1983). In some cases, population dynamics have little or no impact on coevolution (Nuismer, 2017). However, it is well established that population dynamics often have significant qualitative and quantitative effects on host–parasite coevolution models due to eco-evolutionary feedbacks (Frank, 1991a). A number of recent studies have directly explored the effects of introducing eco-evolutionary feedbacks into (or removing them from) models of host–parasite coevolution, leading to fundamental changes in model outcomes (Ashby et al., 2019; Gokhale et al., 2013; MacPherson & Otto, 2018). To see why eco-evolutionary feedbacks can be so important, consider selection for an allele that confers resistance to a common parasite. As the allele increases in frequency, fewer parasites will be able to infect hosts and so there is likely to be a reduction in parasite prevalence, and hence weaker selection for resistance. At the same time, the strength of selection for counter-adaptations in the parasite will tend to increase, although there may be a reduction in the mutation supply (due to the smaller parasite population). In a model that lacks population dynamics, there is no change in the parasite population size and so there is no impact on the strength of selection in the host, nor is there any impact on mutation supply.

Our survey of the literature suggests that models which include population dynamics are more likely to lead to stable polymorphism (see Appendix S1). Indeed, in some models, population dynamics have been shown to be necessary for stable polymorphism (Ashby et al., 2019). Theoretical studies have also shown that population dynamics tend to dampen oscillations in allele frequencies or make fluctuating selection less likely (Ashby et al., 2019; MacPherson et al., 2021b; MacPherson & Otto, 2018). Population dynamics can also cause a quantitative shift in evolutionary outcomes, although the precise effects depend on the model.

4.5 Time

An important, but rarely discussed, aspect of many models is whether time proceeds continuously (76 models in the literature survey) or in discrete steps (108 models in the literature survey). When modelling a specific biological system, there may be strong motivation to take one approach over the other, but in a general model the choice of how to represent time may be made arbitrarily or for convenience. The question of how to model time is often considered in the context of generational overlap: some models assume that generations of hosts (and sometimes also parasites) are separate, whereas others allow them to overlap. If generations do not overlap, then the entire population must be replaced at each time step.

The use of discrete or continuous time can have significant effects on model dynamics (May, 1973a). Discrete time models have a greater tendency to ‘overshoot’ an equilibrium because time proceeds in fixed jumps, whereas continuous time models may approach an equilibrium smoothly. Mathematically, the stability of an equilibrium is determined by the eigenvalues of a linear approximation to the system (for an introduction to eigenvalues, see Otto & Day, 2007). In a discrete time model, an equilibrium is only stable if all eigenvalues have absolute value less than 1. All eigenvalues must therefore lie within a circle of radius 1, centred at the origin, in the complex plane (Figure 5a). In a continuous time model, stability only requires the real part of all eigenvalues to be negative. All eigenvalues must therefore lie to the left of the imaginary axis (Figure 5a). Hence, whether a model is implemented in discrete time or continuous time can greatly affect the stability of an equilibrium. For example, Kouyos et al. (2007) showed that adapting a discrete time model of host–parasite coevolution to continuous time causes cycles in allele frequencies to become damped. Given that cycling is of special interest in models of host–parasite coevolution (in part due to links with the Red Queen Hypothesis for sex (Lively, 2010), although cycling is not necessary for the maintenance of sex (Ashby, 2020b)), one must be careful to ensure that oscillations are not simply artefacts that arise due to discrete time. Conversely, if a biological system is better approximated by a discrete time model, then a continuous time model may underestimate the potential for cycles.

4.6 Stochasticity

In nature, there is an element of chance to all processes. Whether or not an individual reproduces or is infected at a particular moment in time will depend on a variety of factors, many of which will be subject to randomness (e.g. finding a mate, encountering a parasite). Deterministic models (97 models in the literature survey; Figure 6a), such as those in 1-4, assume that stochasticity is relatively unimportant, thus greatly simplifying the analysis. This is often a reasonable approximation in large populations, as the effects of demographic stochasticity decrease with the square root of population size (May, 1973b). Stochasticity (88 models in the literature survey; Figure 6b) may be incorporated using stochastic differential equations, which include noise terms, or through simulations (for instance by allowing infection to occur with a certain probability). However, stochastic models are computationally more intensive than their deterministic counterparts, with computational time increasing rapidly with population size. Moreover, many replicates may be required to reveal representative dynamics. Hybrid models which combine deterministic and stochastic methods may be computationally more efficient when modelling processes that occur at contrasting spatial or temporal scales. For example, one may choose to model short-term population dynamics deterministically, but model rare mutations stochastically (Ashby et al., 2019; Best et al., 2010).

Incorporating stochasticity can have significant effects on coevolutionary dynamics. Crucially, stochasticity can cause rare types to go extinct, potentially causing systems with negative frequency-dependent selection to go to fixation rather than cycle indefinitely (Gokhale et al., 2013). However, stochasticity can also induce cycling by repeatedly pushing a system away from a stable equilibrium (Figure 6; Kouyos et al., 2007; Lythgoe, 2000; M’Gonigle et al., 2009). Stochasticity may also prevent stable polymorphism by causing sub-populations to go extinct (Schenk et al., 2018; Xue & Goldenfeld, 2017).

4.7 Spatial structure

Models of host–parasite coevolution often assume that the populations are well-mixed (as in 158 models in the literature survey) so that all individuals have an equal probability of encountering or interacting with all others. By using the law of mass action and replacing many individual interactions with an average over the population (known as a mean-field approximation), well-mixed models can reduce a complex many-body problem to a relatively simple one-body problem. In other words, one can approximate the transmission dynamics of a large number of randomly mixing infectious and susceptible hosts by a single term, βSI, where β is the transmission rate and S and I are the densities of susceptible and infectious hosts. Although this approach often works well, individuals may instead interact strongly with a small subset of the population due to social or sexual contact networks, or due to proximity arising from spatial structure. Models may incorporate spatial structure (27 models in the literature survey) in a number of different ways, including metapopulations (Figure 7a), where migration occurs between distinct, well-mixed patches (Frank, 1991; Gandon et al., 1996; Gomulkiewicz et al., 2000; Nuismer et al., 1999; Lively, 1999), individual-based models on lattices or networks (Figure 7b), where the contact structure of the population is represented by a collection of edges (contacts) between nodes (individuals; Ashby et al., 2014b; Lion & Gandon, 2015) and models with continuous space (Figure 7c) where the probability of infection decreases with distance (known as a dispersal kernel).

Spatial structure has been shown to have several important effects on host–parasite coevolution (Lion & Gandon, 2015). For example, spatial structure tends to promote fluctuating selection (Gómez et al., 2015), greater host resistance (Ashby et al., 2014b) and lower infectivity (Best et al., 2011). Spatial structure has also been shown to favour the evolution of more nested infection genetic systems (Valverde et al., 2017). Models predict that greater environmental heterogeneity promotes generalism in hosts and parasites (Hesse et al., 2015) and increases both global and local polymorphism (Frank, 1991b; Tellier & Brown, 2011). Unlike well-mixed models, spatially structured models allow for local adaptation, especially in metapopulations (Nuismer, 2006; Thrall et al., 2016). For example, parasites are predicted to be most highly locally adapted when they have high migration rates relative to their hosts (Gandon et al., 1996; Morgan et al., 2005). Local maladaptation can also occur (see Chapter 10 of Nuismer (2017)) and appears to be more common in population genetic models than in quantitative genetic models (Ridenhour & Nuismer, 2007).