Host–pathogen coevolution in the presence of predators: fluctuating selection and ecological feedbacks

Abstract

Host–pathogen coevolution is central to shaping natural communities and is the focus of much experimental and theoretical study. For tractability, the vast majority of studies assume the host and pathogen interact in isolation, yet in reality, they will form one part of complex communities, with predation likely to be a particularly key interaction. Here, I present, to my knowledge, the first theoretical study to assess the impact of predation on the coevolution of costly host resistance and pathogen transmission. I show that fluctuating selection is most likely when predators selectively prey upon infected hosts, but that saturating predation, owing to large handling times, dramatically restricts the potential for fluctuations. I also show how host evolution may drive either enemy to extinction, and demonstrate that while predation selects for low host resistance and high pathogen infectivity, ecological feedbacks mean this results in lower infection rates when predators are present. I emphasize the importance of accounting for varying population sizes, and place the models in the context of recent experimental studies.

1. Introduction

Antagonistic coevolution between hosts and their pathogens is central to shaping the structure and function of biological communities [1,2]. A rich field of experiment and theory has been developed to understand the drivers of host–pathogen coevolution and its impact on ecological dynamics [3–6]. However, for tractability, the vast majority of studies assume that the host and pathogen exist in isolation. In reality, host–pathogen interactions will be embedded within complex communities with an array of biological interactions. These community interactions will have significant impacts on the host–pathogen interaction, which will in turn feed back to the community dynamics [6]. Predation will be particularly significant owing to the direct effects on host population size, as well as indirect links between infection and predation. Classic empirical work has shown that hosts with higher pathogen burdens are more likely to be predated [7,8], potentially altering selection pressure on both antagonists, and thus impacting the community structure itself.

Theoretical studies on the coevolution of host resistance and pathogen infectivity have found a range of possible qualitative outcomes, including long-term stable investment (continuously stable strategies), branching to polymorphism and coevolutionary cycles (fluctuating selection dynamic (FSD)), depending on the ecological and evolutionary context [9–20]. A particular focus has been on FSD, given its importance to the maintenance of diversity [21], evolution of sex [22] and local adaptation [23]. It is well known that highly specific, ‘matching-allele’, infection mechanisms give rise to FSD owing to negative frequency-dependent selection [17,18], while gene-for-gene mechanisms (variation between specialists and generalists) can lead to FSD if there are costs [19,20]. Recent work including explicit ecological dynamics found that cycles of host and pathogen investment could occur even without specificity [14]. However, we have little understanding of how robust theoretical predictions are to including community interactions.

There is increasing awareness in experimental literature of the importance of community interactions to host–pathogen coevolution [1,2,6], and there have been some direct experimental tests [24–26]. Friman & Buckling [24] found that the arms race dynamic between a bacteria (Pseudomonas flourescens) and its phage (Φ2) appeared to break down when a predatory protist (Tetrahymena thermophila) was present, while Örmälä-Odegrip et al. [26] found that selection owing to predatory protists led to lower susceptibility to phage infection in both Serratia marcescens and P. flourescens. Alongside this experimental work, there is increasing theoretical focus on how the evolution of hosts and pathogens [27–32] are separately impacted by an immune predator (a predator that cannot be infected by the parasite). These studies have shown that pathogens invest in higher virulence and transmission when a predator is present [27], while hosts maximize defence to parasitism at intermediate predation rates [31]. In contrast with standard models, predation allows for evolutionary branching to coexistence in pathogens (if virulence and predation are linked; [28] versus [33]) and the pathogen can be eradicated through host evolution ([30] versus [34]). These studies provide a broad examination of the separate evolutionary properties of hosts and pathogens in the presence of a predator. However, given the importance of the coevolutionary setting to the potential for FSD [14,16–20], the differing predictions of the impact of predation on parasites [27] to hosts [31] and the importance of changing population sizes to host–parasite coevolution [5], it is vital that we investigate the full coevolutionary dynamics in the presence of a predator. Here, I present a model of the coevolution of host resistance (through reduced susceptibility) and pathogen transmission with non-specific infection, and respective costs to host birth rate and virulence.

2. Methods

I use a standard model of the population dynamics of susceptible (S) and infected hosts (I), adding an immune predator (P), as given by the following ordinary differential equations:

2.1

2.2

2.3

Susceptible hosts reproduce at birth rate b which is reduced due to crowding by a factor q (H = S + I). All hosts die at natural death rate d. Transmission is a density-dependent term with coefficient β. As well as the natural death rate, infected hosts suffer an additional mortality, which I define as virulence, at rate α, and can recover back to susceptibility at rate γ. Both susceptible and infected hosts are at risk of predation with coefficient c, with a functional response given by ρ(S, I) = 1/(1 + ch(S + ϕI)) (see the electronic supplementary material and figure S1). If h = 0 (i.e. there is no ‘handling time’), the functional response is linearly dependent on the effective host density, S + ϕI (type I). If h > 0, then the response is saturating at higher effective host densities (type II). In what follows, I assume the type I response unless otherwise stated. I also allow the predator to selectively predate infected hosts by the inclusion of the parameter ϕ > 1. Predators convert energy from eating hosts in to births through parameter e, and die at rate μ. Note that I do not assume any link between virulence and predation, as in [28].

When there is linear (type I) predation, the full host–pathogen–predator equilibrium (where it exists) is always stable. However, for a type II response, population cycles can occur. In the type I case, the resident equilibrium for and can be found as

2.4

and

2.5

Therefore, the susceptible density will always increase as the predator is introduced, while the infected density will always decrease (the total host density, , also decreases with increasing ). Note that this relationship is independent of whether ϕ is greater than or less than unity. This is because, as in classic host–parasite models, the susceptible density is regulated by the parasite [35]. Therefore, the increase in predation ultimately benefits susceptible hosts by reducing the density of infecteds. Models with different underlying assumptions, such as an explicit carrying capacity in the host [36], may yield different feedbacks.

I assume that the host can evolve its susceptibility to infection, and the pathogen its infectivity. As such, I need to determine how the two jointly control transmission. Here, I use a multiplicative function, β(σ, τ) = στ + k, where σ is the host's susceptibility and τ the pathogen's transmission. Such a ‘universal’ infection function has been commonly used in theoretical studies [11,12,15,17], and is representative of systems where infection is not specific to certain combinations of host and parasite strains [37–39]. I assume that investment in lower susceptibility and higher transmission incur respective costs for the host (lowered birth rate) and pathogen (increased virulence). Examples of the trade-offs are plotted in the electronic supplementary material, figure S2; also see the electronic supplementary material, and figure legends for the form of the trade-off functions. I model coevolution using the evolutionary invasion analysis framework of adaptive dynamics [40–42], assuming that small, rare mutants (σ_m, τ_m) arise and attempt to invade a resident equilibrium. The success of the mutant is given by its invasion fitness, which is defined as its growth rate while rare. As described in the electronic supplementary material, assuming a type I functional response, this is given for the host by

2.6

where

and for the pathogen,

2.7

where all population densities are evaluated at the resident equilibrium (denoted by hats).

Assuming small mutations, the coevolutionary dynamics of the traits σ and τ over evolutionary time can then be approximated by a pair of ordinary differential equations [42] (see the electronic supplementary material):

2.8

and

2.9

The possible long-term outcomes are: (i) a continuously stable strategy (CSS) in both antagonists where the host and pathogen both invest in a stable level of investment, (ii) coevolutionary cycles (FSD), (iii) evolutionary branching in one or both species, and (iv) maximization/minimization to the imposed (physiological) limits of the trait by one or both species. In the latter two cases, one species may exhibit this outcome, while the other could exhibit any of behaviours (i), (iii) or (iv) [14]. Further details of the methods are given in the electronic supplementary material.

3. Results

(a). Qualitative outcomes

In figure 1, I show the qualitative outcome from simulations as the host and pathogen trade-off curvatures (p_h and p_p) are varied, for (i) linear (type I), and (ii)–(iv) saturating (type II) predation (h = 0.4, 0.45, 0.5). Note that accelerating (increasingly costly) trade-offs occur for p_h > 0 but p_p < 0 (marked ‘(acc.)’ in figure 1; see also the electronic supplementary material, figure S2). A range of qualitatively different outcomes are possible (see sample outputs in the electronic supplementary material, figure S3). In all cases, while the pathogen's trade-off is accelerating, if the host's trade-off is also accelerating, there is a coevolutionary CSS, while if the host's trade-off decelerates, the host branches (and the parasite remains at its CSS). The potential for cycles (FSD) and pathogen branching depend on the handling time. For type I predation (figure 1a), if both trade-offs decelerate (marked ‘(dec.)’; p_h < 0, p_p > 0), then FSD is common. Initially introducing a handling time (figure 1a,b) shifts the region of FSD to higher parasite trade-off curvatures but any host trade-off shape, suggesting the parasite trade-off must be reasonably decelerating for selection to be destabilized. This also introduces greater regions of pathogen branching, either on its own or together with the host. However, figure 1b–d shows that cycles rapidly disappear once the handling time reaches a threshold value (here between h = 0.4 and h = 0.5). Comparing these figures, the cycles are lost in two ways. First, the dynamics can be stabilized towards an evolutionary branching point, generally resulting in both species branching. Alternatively, the predator can go extinct during the cycle (after this, the host maximizes susceptibility and the pathogen minimizes infectivity). The irregular nature of these transitions (their ‘scattered’ nature) is owing to small stochastic variations between simulations—small amplitude cycles being close enough to a singular point to branch, or low predator densities during a cycle being approximated to zero. Why does saturating predation cause coevolution to stabilize towards a branching point? When predation is linear, mortality is higher (electronic supplementary material, figure S1). With selective predation of infecteds, this will strengthen selection for host resistance, pushing host investment, temporarily, to higher levels and continuing the cycles. When predation saturates and mortality is lower, this effect is reduced and the dynamics are stabilized.

Figure 1.

Qualitative output from numerical simulations of the coevolutionary dynamics for differing handling times, (a) h = 0, (b) h = 0.4, (c) h = 0.45, (d) h = 0.5, as the shape of the host and parasite trade-offs vary. Accelerating (acc.) and decelerating (dec.) trade-offs are highlighted on the plots. The simulations were run (see the electronic supplementary material) and the output analysed and classified. CSS, continuously stable strategy; BR, branching; MX, maximization of trait; MN, minimization of trait; FSD, fluctuating selection/cycles. See colourbar for classifications. Parameter values: q = 0.5, d = 0.2, γ = 0.2, ϕ = 3, k = 0.5, μ = 0.5, c = 0.15. The trade-offs, linking transmissibility and virulence in the pathogen, and susceptibility and birth rate in the host, are given by , where p_p and p_h are varied along the x- and y-axes, respectively.

Figure 2 shows how FSD depends on the predation rate, c, and selective predation, ϕ. Here, we see that FSD is most common when there is high selective predation but low general predation. This means that infected hosts suffer much higher mortality than susceptible hosts, fitting with the above argument that this increases selection for host resistance, thus destabilizing selection. This region is bounded on both sides by regions where one or both species branches. We also see that when both selective and general predation are low, the predator dies out and when both are high, the pathogen dies out.

Figure 2.

Qualitative output from numerical simulations as the predation rate, c, and selective predation, ϕ, are varied. Parameters are as in figure 1 with p_h = −0.5, p_p = 0.5. See colourbar in figure 1 for classifications.

(b). Extinction of the predator or pathogen

Invasion/exclusion thresholds exist for the pathogen and predator ([30]; see the electronic supplementary material). This allows for one of the species to be driven to extinction. A particularly interesting example of pathogen extinction can be seen in the phase portrait of figure 3, highlighting regions where the pathogen (red) or predator (blue) cannot persist (a case of predator extinction is in the electronic supplementary material, figure S4). The solid line shows a trajectory that tends to intermediate host and high pathogen investment when all three species coexist (blue dot). However, changing only the initial condition, the dashed line crosses the threshold for pathogen persistence, at which point the pathogen goes extinct. Note that this extinction occurs because of the host increasing its susceptibility to infection, a rather unintuitive result. This occurs because increasing susceptibility leads to a greater predator density, pushing the infected host population to ever lower densities. Again, note that increased predator density always leads to increased susceptible and decreased infected densities, regardless of selective predation.

Figure 3.

Phase portrait of coevolution showing regions where the pathogen (red) or predator (blue) cannot persist. Parameter values are as in figure 1a, except ϕ = 2.25, k = 0.35. The trade-offs are α(τ) = 1.56 − 1 (1 − τ) / (1 − 0.23τ), b(σ) = 1.87 + 0.21σ/ (0.59 + 0.41σ). (Online version in colour.)

(c). Continuously stable strategies

Figure 4 explores how predation impacts host and pathogen investment at a CSS. Figure 4a shows the host (solid) and pathogen (dashed) strategies as predation rate, c, is varied, with the overall transmission coefficient, β, in figure 4b and the resulting per capita rates of infection, , and predation, , in figure 4c. For low predation, the predator cannot persist and there is a fixed level of investment. Once the predator can persist, the pathogen increases its investment, while the host displays a ‘U’-shaped curve (figure 4a), leading to an overall increase in the transmission coefficient (figure 4b). However, figure 4c shows that the negative feedback from predation to the infected density means that the per capita rate of infection, , is significantly reduced. Thus, high rates of predation lead to high host susceptibility and high pathogen infectivity, yet relatively low rates of infection in the population. Similar patterns are found for varying other parameters (electronic supplementary material, figure S5).

Figure 4.

How the co-CSS varies with predation, c. (a) Host, σ (solid) and pathogen, τ (dashed) strategies, (b) transmission coefficient, β, and (c) per capita rate of infection, (solid) and predation, (dashed). Parameter values are as in figure 1a with p_p = −0.25, p_h = 0.25.

(d). Evolutionary branching

Purely host–parasite models with ecological dynamics and universal transmission have found that branching can occur such that two hosts and one pathogen, or two of each antagonist, coexist [12,15]. Further work found that adding a predator means the pathogen can branch against a monomorphic host when there is a link between virulence and predation [28]. Here, I find the stronger result that the pathogen can branch (against a monomorphic host) even without this link when predation saturates (figures 1 and 2). This indicates the emergence of a negative feedback to pathogen selection once predation is saturating. Further branching is not possible and the maximum level of diversity remains two hosts–two pathogens. After the pathogen has branched, the system stabilizes. In particular, the predator cannot be driven to extinction without one of the pathogen strains first being excluded (because standard host–parasite models cannot support two pathogen strains [12]). Examining simulation results, after host branching, it seems there is never extinction of either the predator or pathogen.

4. Discussion

There is increasing focus on understanding how community interactions impact host–pathogen coevolution [1,2,6]. I have examined the coevolution of host resistance (reduced susceptibility) and pathogen transmission, with respective costs to birth rate and virulence, in the presence of a predator. FSD is a particularly important coevolutionary behaviour because it is the only sustained dynamic outcome in a constant environment, and is the focus of much theoretical study [14,16–20]. I have found that while FSD is common when the predator's functional response is linear, if predation saturates at high host densities FSD becomes an increasingly rare outcome, with evolutionary branching of the pathogen occurring instead. FSD is also promoted when there is strong selective predation of infected hosts. The driver of both results is that mortality of infected hosts is higher when predation is selective and does not saturate, destabilizing selection near an evolutionary attractor. Thus, host–pathogen FSD may be expected in communities with highly selective predators with low handling times. In an experimental study of a microbial system, the addition of a predatory protist appeared to breakdown an arms race dynamic, but there was no conclusive evidence that the dynamics shifted to FSD [24]. It would be interesting to conduct explicit experimental tests of how host–pathogen systems that exhibit FSD behave when a predator is added.

In standard models, hosts cannot cause pathogen extinction through the evolution of costly resistance [34], but can when a predator is present [30]. Here, I have shown a particularly unintuitive example of pathogen extinction caused by the host lowering its resistance. This drives an ecological feedback whereby the predator density increases and pushes pathogen numbers to extinction. It is notable that there is no evolutionary rescue of the pathogen. This is in fact intuitive because as the pathogen numbers decrease, the relative speed of mutation also decreases. Host-driven pathogen extinction, in the absence of predation, has been found in experimental studies when further pressures, for example, population bottlenecks [43] or reduced resource availability [44], are placed on the pathogen. This appears consistent with the result that extinction may occur when a predator is introduced. Intriguingly, in their experimental study of bacteria–phage coevolution in the presence of a predatory protist, Friman & Buckling [24] report a case of phage being driven to extinction, and it would be fascinating to see whether such a result is repeated elsewhere.

I have shown that while the introduction of a predator may lead to lower host resistance and higher pathogen infectivity at a coevolutionary CSS compared to when no predator is present, the negative feedback from predators to the infected density means that there are in fact lower per capita rates of infection than when the predator is absent. This has important consequences for how infection rates are measured in empirical studies, suggesting opposing patterns of infection may be predicted depending on whether population sizes are controlled or not. Previous theory has shown, when only one antagonist evolves, that the pathogen should increase transmission when a predator is added [27], but the host should maximize defence at intermediate predation rates [31]. These results remain broadly true here, but give a misleading impression of the full coevolutionary outcome when feedbacks to population sizes are not included. Interestingly, experimental results from two bacteria–phage–protist systems found hosts exhibited lower susceptibility to phage infection when a predatory protist was present [26]. This host response is consistent with the results here and earlier [31], assuming predation rates are not too high or coevolution and ecological feedbacks are not fully present. More generally, the prediction here that overall infection rates may be lower when a predator is present is consistent with two key experimental studies [24,26]. Interestingly, Friman & Buckling [24] also reported that the introduction of the protist lowered overall host numbers, as would be expected here. It would be interesting to see whether direct experimental tests in the presence and absence of predators, including measures of population sizes, confirm the findings here.

Almost all natural and managed populations are part of communities, and this work is likely to have important implications to understanding a range of empirical systems, not least in microbial communities [2,4,45,46]. However, understanding antagonistic coevolution in the context of complex communities is still an emerging field, and many open questions remain. For example, here I assumed no specificity in infection. Previous theory has shown that such specificity has implications for both static and transient diversity [14,15], and this may be more realistic for modelling certain systems. Further, I have assumed that the additional interaction is with an immune predator, but other interactions, such as mutualisms or competitors, may lead to different feedbacks. A broader assessment of the impacts of community interactions on antagonistic coevolution should be a long-term goal of both experiment and theory [6].

Data accessibility

C++ code for the simulations is available as the electronic supplementary material.

Competing interests

I declare I have no competing interests.

Funding

I received no funding for this study.