Coevolution Maintains Diversity in the Stochastic "Kill the Winner" Model

TL;DR

This paper introduces a stochastic coevolutionary model for the 'Kill the Winner' hypothesis, showing that demographic stochasticity disrupts stable coexistence but coevolution promotes persistent diversity through ongoing species flux.

Contribution

It develops a novel stochastic coevolutionary model demonstrating how predator-prey dynamics sustain diversity despite stochastic extinction risks.

Findings

Demographic stochasticity destroys stable coexistence in the deterministic model.

Predator-prey coevolution induces persistent species flux and high diversity.

Stochastic effects lead to ongoing extinction and recolonization cycles.

Abstract

The "Kill the Winner" hypothesis is an attempt to address the problem of diversity in biology. It argues that host-specific predators control the population of each prey, preventing a winner from emerging and thus maintaining the coexistence of all species in the system. We develop a stochastic model for the "Kill the Winner" paradigm and show that the stable coexistence state of the deterministic "Kill the Winner" model is destroyed by demographic stochasticity, through a cascade of extinction events. We formulate an individual-level stochastic model in which predator-prey coevolution promotes high diversity of the ecosystem by generating a persistent population flux of species.

Figures

Equations

$$X_i$$

$$X_i+X_j$$

$$Y_i+X_i$$

$$Y_i$$

$$\dot{B_i}$$

$$\dot{V_i}$$

$$B_i^{\ast }$$

$$V_i^{\ast }$$

$$X_i$$

$$Y_i$$

$$S=-\sum _{i=1}^m{f}_i\ln f_i.$$

Full text

The high diversity of coexisting species in most ecosystems has been a major puzzle for more than 50 years. In a classic paper, Hutchingson articulated the so-called Paradox of the Plankton for the case of marine ecosystems Hutchinson (1961): why do many species of plankton that feed on the same nutrients coexist, instead of one species outcompeting all the others? This latter expectation might seem to be intuitive, and has also been formulated precisely as the so-called competitive exclusion principle Hardin (1960). The Paradox of the Plankton is not limited to marine ecosystems, but has been generalized to terrestrial systems and expressed as the biodiversity paradox Clark et al. (2007); Wilson (1990).

The various tentative resolutions of the paradox can be divided into two classes Wilson (1990); Roy and Chattopadhyay (2007). In the first, the ecosystem is argued to have failed to reach a fixed point equilibrium state in which the competitive exclusion principle applies, due to temporal or/and spatial factors. For example, the time needed for the system to reach equilibrium might be much longer than the time over which the system undergoes significant changes in its boundary conditions, such as weather Levins (1979). Spatial heterogeneity can increase the global diversity of the system by maintaining local patches that each obey the competitive exclusion principle but globally support the coexistence of multiple species Richerson et al. (1970); Levin (1974). In the second class of resolutions, interactions such as predation, in conjunction with competitive exclusion, promote the co-existence of species through time-dependent or stochastic steady states Scheffer et al. (2003); Roughgarden and Feldman (1975); Thingstad and Lignell (1997); Vetsigian et al. (2011). One widely celebrated example of this behavior, which is seen in both natural ecosystems as well as some laboratory systems such as chemostats Fernández et al. (1999); Louca and Doebeli (2017), is the continual succession of different community members known as “Kill the Winner” (KtW) dynamics Thingstad and Lignell (1997); Thingstad (2000); Winter et al. (2010).

In the KtW hypothesis Thingstad and Lignell (1997); Thingstad (2000); Winter et al. (2010) there are two groups of resource consumers, for example bacteria and plankton. The plankton community generally has a lower efficiency of resource usage than bacteria. They remain in the system, only because a protozoan consumes the bacteria non-selectively and thus limits the bacterial population, leaving room for plankton to thrive. Inside the bacterial community, different strains have distinct growth rates. They coexist, with no dominating winners, due to host-specific viruses controlling the corresponding strains. This results in two layers of coexistence through KtW dynamics (bacteria-plankton coexistence and bacterial strain coexistence), rested like Russian dolls Winter et al. (2010).

The original KtW model Thingstad and Lignell (1997); Thingstad (2000) was formulated as deterministic Lotka-Volterra type equations for the species biomass concentrations. The high diversity of the system is exhibited in the steady state where multiple species coexist with positive biomass; these calculations assume that the system is spatially homogeneous and that the number of individuals is large enough that it is valid to use a continuous density to describe the population. However, this is not appropriate when the population is finite, because large fluctuations are able to drive the system towards extinction, an outcome that cannot be captured by a continuous density that is allowed to become arbitrarily small. Requiring the population size to be integer-valued leads inexorably to shot noise, referred to in the ecological context as demographic stochasticity.

The purpose of this Letter is to explore the effect of demographic stochasticity on the “Kill the Winner” paradigm and demonstrate that the stochasticity causes the coexistence steady state in the deterministic KtW model to break down through a cascade of extinctions, leading to a loss of diversity. This extinction can be avoided by allowing the predators and prey to coevolve. We propose a stochastic model of this coevolution and show that it generically maintains the diversity of the ecosystem, even in the absence of spatial extension. Our results strongly suggest that diversity reflects the dynamical interplay between ecological and evolutionary processes, and is driven by how far the system is from an equilibrium ecological state (as could be quantified by deviations from detailed balance). The surprisingly deep role of demographic stochasticity uncovered here is consistent with earlier demonstrations of strong fluctuations and qualitatively new phenomena in ecosystems where local populations are small. For example, it is now well-understood how individual-level minimal models can account for a wide variety of ecological phenomena, including large-amplitude persistent population cycles McKane and Newman (2005), anomalous phase shifts due to the emergence of mutant sub-populations Yoshida et al. (2003); Shih and Goldenfeld (2014), spatial patterns Täuber (2012); Butler and Goldenfeld (2009); Biancalani et al. (2010, 2017) and even reversals of the direction of selection Constable et al. (2016) without requiring overly detailed modeling of inter-species interactions.

Model:- The key component of the KtW hypothesis is that, for each resource competitor, there is a corresponding predator that can prevent it from becoming a dominant winner. The Russian doll-like hierarchy is not essentially important for the basic idea. Thus we will focus on only a single layer of KtW interaction, the host-specific viral infection, and ignore the multilevel structure.

We write down the individual reactions for a simplified system of $m$ pairs of prey and predators, which we will take to be bacteria and viruses (phages), as follows:

[TABLE]

Here $i,j=1,2,\dots,m$ are strain indices. Bacterial individuals $X_{i}$, have strain-specific growth rate $b_{i}$. They compete with each other for an implicit resource with strength $e_{ij}$. Viruses of the $i$th strain $Y_{i}$, infect the corresponding host $X_{i}$ with rate $p_{i}$ and burst size $\beta_{i}$, and decay to nothing $\varnothing$ with rate $d_{i}$. These reactions form a minimal model, which we refer to as the generalized KtW model, and ignore many biological details that are present in ecosystems.

The corresponding mean-field rate equations are shown below.

[TABLE]

The dot operator stands for the time derivative. $B_{i}$ and $V_{i}$ represent the density of the $i$th bacterial and viral strains respectively. For simplicity, we set $e_{ij}\equiv e$.

The nonzero steady state of Eq. (2) is given by the following equations.

[TABLE]

Linear stability analysis shows that the steady state Eq. (3) is exponentially stable, with all eigenvalues of the linear stability matrix having negative real parts, as long as the quantity $x_{i}\equiv\beta_{i}p_{i}^{2}B_{i}^{*}V_{i}^{*}=d_{i}(b_{i}-e\sum_{i=1}^{m}\frac{d_{i}}{\beta_{i}p_{i}})$ is distinct for each $i$.

In Fig. 1, we show in the first row the time series of prey and predator densities obtained from a numerical evolution of Eq. (2) for $m=10$ pairs of bacteria and phages. Species densities are initially perturbed away from the steady state by a small random amount. As shown in the figure insets, species densities decay back to the steady state at long times, confirming the result of the linear stability analysis. The oscillatory behavior on the short time scale is due to the imaginary parts of the eigenvalues of the linear stability matrix.

To reveal the effect of demographic noise, we also conduct the stochastic simulation of the corresponding individual level reactions (1) with the same parameter set, using the Gillespie algorithm Gillespie (1976). The resultant species density time series are shown in the second row of Fig. 1. In contrast to the deterministic behavior of oscillatory decay, species go extinct in a short time. Bacterial strains become extinct due to random fluctuation; this consequentially triggers the extinction of the corresponding viral strains, due to a lack of food. The number of species monotonically decreases in the process, and the system diversity undergoes a cascade.

The reason that the stable deterministic steady state of the generalized KtW model cannot be maintained in the presence of demographic stochasticity lies in the fact that species populations in the stochastic model are all finite, and the probability of the population reaching zero due to random drift is always nonzero.

Ecosystems have evolved many potential mechanisms to get around the path to extinction, as introduced at the beginning of the article. Here, we discuss one possibility: prey and predator coevolve with each other so that fit mutants are constantly being introduced into the system, thus preventing the elimination of the species. Specifically, prey improve their phenotypic traits (e.g. strengthening the shell) to escape from predators, and predators also adjust their corresponding traits (e.g. sharpening the claws) to catch prey. This coevolutionary arms race has been well-documented in many systems Bergelson et al. (2001); Chisholm et al. (2006); Person (1959); Thompson and Burdon (1992); Buckling and Rainey (2002a, b); Råberg et al. (2014); Luijckx et al. (2013); Adema and Loker (2015). Previous theoretical studies focused on the dynamics of the traits of prey and predator groups Dieckmann et al. (1995); Dieckmann and Law (1996); Agrawal and Lively (2002); Weitz et al. (2005), and the structure of the predation network Beckett and Williams (2013), under different coevolving modes. Here, we study how coevolution affects the diversity of the host-specific predation system.

Coevolution model:- We modify the stochastic generalized KtW model (1) by adding in the following two sets of reactions to describe mutations of the prey $X_{i}$ from strain $i$ to $i\pm 1$, and similarly those of the predator $Y_{i}$.

[TABLE]

We assume that the mutation rates are strain independent and one individual can mutate into its two neighbor strains with the same rate, $\mu_{1}/2$ for bacteria or $\mu_{2}/2$ for viruses. We set the boundary condition to be open, so that mutations out of the index set $\{1,2,\dots,m\}$ are ignored. We will refer to Eq. (1) and (4) as the coevolving KtW (CKtW) model.

For sufficiently high mutation rates, the absorbing extinction state in the generalized KtW model can be avoided, in the sense that a strain can reemerge as mutants are generated from its neighbor relatives after its population drops to zero. Therefore, mutation can stimulate a flux of population through different strains and promote coexistence.

We define the diversity of the system in the CKtW model using the Shannon entropy

[TABLE]

Here, $f_{i}$ is the fraction of the $i$th bacterial (viral) strain in the entire bacterial (viral) community. The above expression reaches the maximum, when all strains coexist at their deterministic steady state values Eq. (3), and the minimum [math], when only one strain exists. We score $S=-1$, if either the bacterial or viral community goes extinct.

We present population density time series in Fig. 2, and the dependence of prey diversity on the mutation rates in Fig. 3. We set $\mu_{1}=\mu_{2}\equiv\mu$ for simplification. For small enough mutation rate (time series not shown), the entire community can go extinct before mutants can emerge, and the system still collapses as in the generalized KtW model. This corresponds to region I in Fig. 3(a). For intermediate mutation rates, most strains stay near extinction, driven by demographic noise, while some mutants can grow to a dominant status if they happen to confront only a few predators when first emerging. Subsequently, the predator population expands, feeding on the dominating winners, thus reducing the winner population, and allowing the next dominator to grow. In this way, we see that winner populations spike alternatively in the time series, as in the first row of Fig. 2. Near the onset of coexistence, the diversity has a large deviation and is very sensitive to the mutation rate, as shown in region II in Fig. 3(a). For large mutation rate, the coevolution-driven population flow is fast enough to compensate for the demographic fluctuations. All strains remain near the steady state, and no one can win over others, as shown by the population time series in the second row of Fig. 2. The diversity approaches slowly the maximum, with small deviations, as demonstrated in region III in Fig. 3(a). For extremely large mutation rate (time series not shown), we can not view the mutation as a perturbation to the ecological population dynamics. Species populations will deviate from the mean-field steady state Eq. (3) due to the large effect of mutations. According to the above discussion, there are three phases of dynamics, as illustrated in Fig. 3(b), the extinction phase at low mutation rate, the winner-alternating phase at intermediate mutation rate, and the coexisting phase at high mutation rate.

Open system model:- Up to now, the models are closed: we have pre-assigned the number of predator-prey pairs in the system, and furthermore set boundaries to the mutation of species. A more realistic approach is to let the system be open, and evolve by itself to establish however many species there can be.

Suppose that a fit species mutates, by changing its phenotypic traits to escape from its predator. As mutants take on new traits, the population spreads in trait space. This expansion usually is associated with a trade-off in the fitness Dieckmann et al. (1995): the further the trait is from the origin, the lower the growth rate becomes. We model the trade-off effect by assigning species discretely at certain trait values, so that the species index basically represents the trait. Specifically, we assume the trait space is 1-dimensional and set up $M$ species in it; we assign the highest birth rate to the species with index $M/2$, which is at the center of the trait space and then is the origin of the trait expansion; we decrease the birth rate as the species index goes from $M/2$ to $1$ and and from $M/2$ to $M$, manifesting the trade-off of mutation. Species $1$ and $M$ have the lowest birth rates that are almost [math], and further mutation of the two will result in mutants with negative birth rates, which can not grow and are thus excluded from the model. This species space $\{1,2,\dots,M\}$ contains all possible species that can potentially exist in the system. However, under conditions of resource limitation, only a few with relatively high growth rates, out of $M$, can eventually be established in the system. In our model, this limit due to the carrying capacity is set by the competition strength $e$. The number of species that manage to thrive corresponds to $m$ in the previous models.

We conduct simulations to test the dependence of diversity on the mutation rate. See the Supplemental Material for the resultant population time series and diversity dependence on the mutation rate. Even though the number of established pairs varies with time and the population leaks out of the region deterministically allowed by the carrying capacity, the system still exhibits three phases depending on the mutation rate, similar to the CKtW model with fixed number of species.

Discussion:- In the intermediate and fast mutation regions of the CKtW model, the ecological and evolutionary dynamics are coupled to each other and occur on the same time scale Yoshida et al. (2003). This type of coupling can most easily be observed in microbial systems, in which organisms have a high mutation frequency Bohannan and Lenski (1997); Shih and Goldenfeld (2014); Yoshida et al. (2007). Recent work has shown clearly the existence of genomic islands, where genomes of different strains vary in loci thought to be associated with phage resistance Rodriguez-Valera et al. (2009). Both host-specific predation and mutation are important in generating the observed diversity of the bacterial genome. The minimal CKtW model can in principle describe the diversity in the above system. For example, by controlling the mutation rate through an inducible promoter, using molecular techniques pioneered in Ref. Kim et al. (2016), we envisage a fast bacterium-phage coevolution experiment to test the phase diagram predicted by our model.

In addition to inevitable simplification of biological details, both the generalized KtW and the coevolving KtW models assume that the system is well mixed, ignoring any spatial dispersion. Consequently, they can not capture the reservoir effect Shmida and Ellner (1984) present in an ecosystem, which means that for any local community, organisms in its surrounding environment can move into it, keeping it supplied and refreshed. Specifically, even if a species goes extinct in a local community, it can be reseeded there by the surrounding reservoir. Well-mixed models should be thought of as describing not the entire system, but a much smaller correlation volume, in which local demographic stochasticity can be significant Butler and Goldenfeld (2011); Täuber (2012); Biancalani et al. (2017).