Sources and sinks: a stochastic model of evolution in heterogeneous environments

Abstract

We study evolution driven by spatial heterogeneity in a stochastic model of source-sink ecologies. A sink is a habitat where mortality exceeds reproduction so that a local population persists only due to immigration from a source. Immigrants can, however, adapt to conditions in the sink by mutation. To characterize the adaptation rate, we derive expressions for the first arrival time of adapted mutants. The joint effects of migration, mutation, birth and death result in two distinct parameter regimes. These results may pertain to the rapid evolution of drug-resistant pathogens and insects.

Biological evolution and ecology are intimately linked, because the reproductive success or “fitness” of an organism depends crucially on its ecosystem. The biotic and abiotic factors defining an ecosystem often have complex time- and space-dependent dynamics [1]. Yet, most models of evolution describe homogeneous, fixed-size populations subjected to a constant selection pressure [2]. Even though such minimal descriptions have lead to invaluable insights into some of the major evolutionary forces [2], their scope is clearly limited. An important challenge in evolutionary biology is therefore to understand the interaction between ecology and evolution.

In this Letter, we study how spatial heterogeneities can drive evolution in ecologies displaying so-called source–sink dynamics (SSD) [3-6]. A species’ habitat is called a sink if the local mortality exceeds the local reproduction so that a population can persist only due to continuous immigration from a source habitat, where reproduction exceeds mortality. Sinks often occur at the border of a species’ range: if an environmental variable such as temperature or humidity varies in space and limits a species’ range, the conditions at the border are often poor. SSD results in a sustained presence of poorly adapted immigrants in the sink; this suggests that it could assist adaptation to the sink conditions. Immigrants that, due to mutations, acquire the ability to reproduce efficiently in the sink have an opportunity to establish a population there. Such a mutant can be successful even if it does not have a competitive advantage in its original habitat (the source); it is sufficient if the mutation allows it to colonize the sink. Importantly, this adaptive process driven by the opportunity to establish a new niche is qualitatively different from the conventional notion of a population climbing a fitness gradient [7, 8].

Evolution in source-sink systems is not merely of theoretical interest. For instance, SSD may accelerate the evolution of drug resistance in bacteria [9]. In humans or livestock treated with antibiotics, drug levels can vary between different organs, creating sources and sinks within a host [10]. Also, SSD can emerge when bacteria migrate between treated and untreated individuals [9]. Likewise, the migration of fruit flies between plantations using different (amounts of) insecticides could assist the emergence of insecticide resistance [11, 12]. The evolution of virulence has been associated with SSD as well [13-15].

Here we analyze a minimal stochastic model of adaptation driven by SSD. In particular, we study how the rate of adaptation depends on parameters such as migration and mutation rates. Earlier studies examined source–sink systems from various perspectives. Although the models proposed differ from each other in various respects, they can be divided into three classes. First, several models are based on deterministic population dynamics [16-21]. Such models are convenient but ignore the intrinsic stochasticity of the demographic processes, which, as we will see below, can be important. Second, some studies employ the formalism of quantitative genetics [22, 23]. This type of models describes the response of a quantitative trait to selection in a population characterized by a distribution of phenotypes. This line of attack is appropriate only for multi-locus traits and is again deterministic. Third, individual-based simulations have been used [24]. Unfortunately, such studies are necessarily limited to a narrow set of parameters. In contrast, our model is fully stochastic and yields analytical results that are valid for a wide range of parameter values.

We consider a haploid population in an environment consisting of two “patches” (Fig. 1(a)). Individuals migrate between the patches at a rate ν and die at a rate δ. Each organism has one of two possible genotypes, called “wild-type” (w) and “mutant” (m). Mutations turn a wild-type into a mutant at a rate μ_f; the reverse occurs at a rate μ_b. The reproduction rate of genotype g ∈ {W, M} in patch i obeys the logistic form γ_ig(N_i) ≡ max {r_ig(1 − N_i/K), 0}, where N_i is the population size in patch i. The logistic form keeps N_i finite and introduces competition between organisms sharing a patch. K is called the carrying capacity and r_ig is the maximal reproduction rate of type g in patch i. The wild-type can reproduce in patch 1 but not in patch 2 (r_2w = 0); this introduces SSD. By contrast, mutants can reproduce in both patches. We initially choose r_1w = r_1M ≡ r, so that both types are equally “fit” in patch 1. Later we will relax this to consider the effect of a possible fitness cost conferred by the mutation.

FIG. 1.

Source-sink model. We consider 2 patches and 2 genotypes. The wild-type can reproduce only in patch 1 while the mutant can grow in both patches. (a) Organisms mutate and migrate (indicated by arrows) and die at rate δ. (b) A wild-type population in patch 1 will in time give rise to a mutant in patch 2. Arrows indicate the two competing pathways.

We make a few biologically motivated assumptions. First, we limit our analysis to large populations (K >> 1). Second, we assume that mutation rates are low (μ_b, μ_f << ν, δ). Third, we assume that ν < δ, i.e., organisms are unlikely to migrate multiple times within their lifetime.

The adaptation process consists of two parts. First a mutant has to appear in the sink; next this mutant has to establish a population. The latter process has been treated by (stochastic) models of colony growth [18, 19, 25]. We therefore focus on the question how long it takes before the first mutant arrives in patch 2 starting from a wild-type population in patch 1.

Fig. 1(b) shows the two pathways that can generate a mutant in patch 2. In the upper path (denoted by ↱) first a mutation occurs in patch 1 and later a mutant migrates to patch 2. In the lower path, ⬏, a wild-type first migrates to patch 2 and then mutates. A priori it is unclear which path is more likely. Below, we derive the first arrival time (FAT) distributions for both paths.

We start with path ↱. Let n_ig be the number of organisms with genotype g in patch i. Since K >> 1, N₁ ≡ n_1w + n_1M is approximately constant; it fluctuates around the value N ≡ [1-(δ + ν)/r]K for which the reproduction rate equals the rate at which organisms disappear from the first patch (γ_1m(N) = δ + ν) [26]. Below, we assume that N₁ = N; this allows us to study path ↱ in isolation. We write a Master Equation for the probability P_↱(n, t) that at time t no mutant has yet migrated to patch 2 and n_1M has value n. It reads

$${\partial }_t{P}_{\Rsh }(n,t)=w^{-}(n+1)P_{\Rsh }(n+1,t)+w^{+}(n-1)P_{\Rsh }(n-1,t)-\left[w^{-}\right(n)+w^{+}(n)+u(n\left)\right]P_{\Rsh }(n,t),$$ (1)

with w⁻ ≡ (δ + μ_b)n, w⁺(n) ≡ γ_1M(N)n + μ_f(N − n), u(n) = νn and initial condition P_↱(n, 0) = δ_n,0 (δ_n,m is the Kronecker delta function).

We rewrite Eq. 1 in terms of the generating function $G_{\Rsh }(z,t)\equiv {\Sigma }_n{z}^n{P}_{\Rsh }(n,t)$ and solve the resulting partial differential equation. The probability that at time t no mutant has migrated yet, S_↱(t) = G_↱(1, t), follows as

$$S_{\Rsh }\left(t\right)={\left[\frac{c_{\Rsh }{\mathrm{e}}^{b_{\Rsh }t}}{c_{\Rsh }\cosh \left(c_{\Rsh }t\right)+b_{\Rsh }\sinh \left(c_{\Rsh }t\right)}\right]}^{a_{\Rsh }},$$ (2)

with a_↱ ≡ μ_fN/(δ + ν − μ_f), b_↱ ≡ (μ_b + μ_f)/2, and $c_{\Rsh }\equiv {[b_{\Rsh }^2+\nu (\delta +\nu -{\mu }_{\mathrm{f}}\left)\right]}^{1∕2}$. Now the desired FAT probability density F_↱ (t) ≡ −dS_↱(t)/dt can be expressed as

$$F_{\Rsh }\left(t\right)=\nu \langle n_{1\mathrm{M}}\left(t\right)\rangle S_{\Rsh }\left(t\right),$$ (3)

where 〈n_1M(t)〉 is the mean value of n_1M at time t given that no mutant has migrated yet; it obeys

$$\langle n_{1\mathrm{M}}\left(t\right)\rangle =\frac{{\mu }_{\mathrm{f}}N\tanh \left(c_{\Rsh }t\right)}{c_{\Rsh }+b_{\Rsh }\tanh \left(c_{\Rsh }t\right)}.$$ (4)

These results show that path ↱ is governed by two time scales. First, 〈n_1M(t)〉 builds up in a time scale τ_↱ = 1/2c_↱. Second, at large times F_↱(t) decays with time scale ${\tau }_{\Rsh }^{\prime }={\left[a_{\Rsh }\right(c_{\Rsh }-b_{\Rsh }\left)\right]}^{-1}$ due to the time dependence of S_↱(t); this time scale reflects the total migration rate after 〈n_1M〉 has equilibrated. We define ${\kappa }_{\Rsh }\equiv {\tau }_{\Rsh }∕{\tau }_{\Rsh }^{\prime }$.

We note that b_↱ << c_↱; thus κ_↱ ≈ μ_fN/[2(δ + ν)] and the mean first arrival time (MFAT) T_↱ is given by

$$c_{\Rsh }{T}_{\Rsh }\approx (\sqrt{\pi }∕2)\Gamma \left({\kappa }_{\Rsh }\right)∕\Gamma ({\kappa }_{\Rsh }+1∕2),$$ (5)

where Γ(x) is the Gamma function. Interestingly, only two lumped parameters remain: c_↱ and κ_↱. If κ_↱ << 1 (i.e., μ_fN << δ + ν) the process is limited by the generation of mutants whereas if κ_↱ >> 1 (i.e., μ_fN >> δ + ν) it is migration limited. By expanding Eq. 5 and using Stirling’s approximation we obtain in these limits:

$$T_{\Rsh }\approx \{\begin{array}{cc}\frac{1}{2c_{\Rsh }{\kappa }_{\Rsh }}\approx \frac{\sqrt{\delta ∕\nu +1}}{{\mu }_{\mathrm{f}}N}\hfill & (\text{if}{\mu }_{\mathrm{f}}N<<\delta +\nu ),\hfill \\ \frac{\sqrt{\pi +{\kappa }_{\Rsh }}}{2c_{\Rsh }}\approx \frac{\sqrt{\pi ∕2\nu }}{{\mu }_{\mathrm{f}N}}\hfill & (\text{if}{\mu }_{\mathrm{f}}N>>\delta +\nu ).\hfill \end{array}\phantom{\}}$$ (6)

The plot in Fig. 2(a) clearly reveals these two regimes.

FIG. 2.

Mean first arrival time (MFAT) vs. parameters, for (a) path ↱ and (b) path ⬏. In each case the MFAT depends on only two lumped parameters: c_i and κ_i. In both plots clearly two regimes can be discerned as a function of κ_i.

We now turn to the lower path, ⬏. As we expect the number of mutants in patch 1 to be small we assume that n_1w ≈ N. A Master Equation can then be written for the probability P_⬏(n, t) that at time t no mutation has yet occurred in patch 2 and n_2w = n. It is identical to Eq. 1 but with w⁻(n) ≡ (δ + ν)n, w⁺(n) ≡ νN, and u(n) ≡ μ_fn. The probability S_↱(t) that no mutation occurs before time t and the FAT distribution F_⬏(t) can now be obtained by a similar derivation as for the ↱ case.

Again the process has two time scales: τ_⬏ = 1/c_⬏, with c_⬏ ≡ δ + ν + μ_f, and ${\tau }_{⬏}^{\prime }=c_{⬏}∕\left({\mu }_{\mathrm{f}}N\nu \right)$. We define ${\kappa }_{⬏}\equiv {\tau }_{⬏}∕{\tau }_{⬏}^{\prime }$; the MFAT of path ⬏, called T_⬏, then reads

$$c_{⬏}{T}_{⬏}={(\mathrm{e}∕{\kappa }_{⬏})}^{{\kappa }_{⬏}}\gamma ({\kappa }_{⬏},{\kappa }_{⬏}),$$ (7)

where γ(x, y) is the lower incomplete Gamma function. Fig. 2(b) is a plot of Eq. 7; it is again characterized by two regimes. Indeed, in the limits κ_⬏ << 1 and κ_⬏ >> 1,

$$T_{⬏}\approx \{\begin{array}{cc}\frac{1}{c_{⬏}{\kappa }_{⬏}}\approx \frac{\delta ∕\nu +1}{{\mu }_{\mathrm{f}}N}\hfill & ({\mu }_{\mathrm{f}}N<<{(\delta +\nu )}^2∕\nu ),\hfill \\ \frac{\sqrt{\pi ∕2{\kappa }_{⬏}}}{c_{⬏}}\approx \sqrt{\frac{\pi ∕2\nu }{{\mu }_{\mathrm{f}}N}}\hfill & ({\mu }_{\mathrm{f}}N>>{(\delta +\nu )}^2∕\nu ).\hfill \end{array}\phantom{\}}$$ (8)

We are now in the position to calculate the full FAT distribution F(t) taking into account both paths ↱ and ⬏. Since both paths are nearly independent the probability that neither path has completed at time t is S(t) ≡ S_↱(t)S_⬏(t), from which F(t) and the combined MFAT T follow. However, comparing Eq. 6 to Eq. 8 we recognize that T_↱ < T_⬏ unless κ_⬏ >> 1, in which case T_↱ ≈ T_⬏. Therefore we conclude that path is dominant and we should expect F(t) ≈ F_↱(t) and T ≈ T_↱.

We tested these results using kinetic Monte Carlo simulations. Fig. 3(a) shows T as a function of K and μ_f, and Fig. 3(b) plots T versus ν for various δ. The curves are theoretical predictions calculated by numerical integration of S(t). Fig. 3(c) shows all data of Figs 3(a) and (b) in a single scaling plot. Since T ≈ T_↱, Eq. 5 predicts that all points should collapse on one curve if c_↱T is plotted against κ_↱. This is indeed the case.

FIG. 3.

Mean first arrival time T as a function of parameters. Unless specified, parameters are: r = 1, δ = 10⁻¹, ν = 10⁻³, μ_b = 10⁻⁴, μ_b = 10⁻⁷, K = 10⁵. (a) T as a function of the carrying capacity K, for various mutation rates μ_f. Data points are averages over 10⁴ simulations; lines are corresponding theoretical predictions. (b) T versus migration rate ν, for various death rates δ. (c) As predicted by the theory all data points from Figs (a) and (b) collapse on a single curve after rescaling. The plot shows two regimes, for κ_↱ << 1 and κ_↱ >> 1, corresponding to mutation- or migration-limited dynamics.

So far we assumed that the wild-type cannot reproduce at all in patch 2. One may expect that path ⬏ could become more relevant if it does, albeit at a reduced rate r_2w < r. We now derive results for this extended model. In the derivations for path ↱ the rate r_2w plays no role, but the results for path ⬏ do change. The dynamics of n_2w can be approximated by the rate equation

$$\frac{\mathrm{d}{n}_{2\mathrm{w}}\left(t\right)}{\mathrm{d}t}=\nu N+r_{2\mathrm{w}}(1-n_{2\mathrm{w}}∕K)n_{2\mathrm{w}}-(\delta +\nu )n_{2\mathrm{w}}.$$

With initial condition n_2w(0) = 0 it is solved by

$$n_{2\mathrm{w}}\left(t\right)=\frac{\nu N\tanh \left(c_{⬏}t\right)}{c_{⬏}+b_{⬏}\tanh \left(c_{⬏}t\right)},$$ (9)

with b_⬏ ≡ (δ + ν − r_2w)/2 and $c_{⬏}\equiv {(b_{⬏}^2+\nu r_{2\mathrm{w}}N∕K)}^{1∕2}$. The expression for S_⬏(t) has the same form as Eq. 2, but with a_⬏ ≡ μ_fK/r_2w, and we obtain

$$F_{⬏}\left(t\right)={\mu }_{\mathrm{f}}{n}_{2\mathrm{w}}\left(t\right)S_{⬏}\left(t\right).$$ (10)

Again we find two time scales: τ_⬏ = 1/2c_⬏ and ${\tau }_{⬏}^{\prime }={\left[a_{⬏}\right(c_{⬏}-b_{⬏}\left)\right]}^{-1}$. If ${\tau }_{⬏}∕{\tau }_{⬏}^{\prime }<<1$ (i.e., if μ_fK << r_2w or c_⬏ ≈ b_↱) the first time scale can be ignored and T_⬏ ≈ τ_↱.

In Fig. 4 we test this result with simulations. Evidently, as long as r_2w << δ + ν the MFAT is insensitive to changes in r_2w. Only when r_2w approaches δ + ν, T_⬏ decreases rapidly. Indeed, if r_2w = δ + ν we obtain T_↱ ≈ T_⬏, which means that paths ↱ and ⬏ each contribute about equally to the MFAT T. We therefore conclude that in source–sink systems, where r_2w < δ + ν (i.e., to the left of the vertical line in Fig. 4) a nonzero r_2w can result in at most a two-fold increase in the rate of adaptation.

FIG. 4.

T vs. reproduction rate of wild-type in the sink, r_2w. As long as r_2w ≈ δ + ν, T is insensitive to r_2w because path ⬏ is dominant. When r_2w ≈ δ + ν, T_↱ ≈ T_⬏ and the process speeds up about two-fold. If r_2w ≈ δ + ν patch 2 is no longer a sink. Also shown is the mean time before n_2m = 500. Clearly the growth of the mutant colony slows down dramatically as r_2w approaches r. Parameters used: r = 1, δ = 10⁻¹, ν = 10⁻³, K = 10⁵, μ_f = 10⁻⁷, μ_b = 10⁻⁴.

For r_2w > δ + ν, although the FAT of a mutant in patch 2 is much reduced (see the red line to the right of the vertical line in Fig. 4), it is more difficult for the mutant to conquer patch 2 because it has to compete with the wild-type. To demonstrate this, Fig. 4 also shows the mean waiting time before n_2M = 500, obtained from simulations. Clearly, as r_2w → r the competition dramatically slows down the growth of the mutant population.

Finally, we extend the model to include a fitness cost s of adaptation, i.e., we consider r_1M = (1 − s)r_1w. This means that, in Eq. 1, γ_1M(N) = (δ + ν)(1-s). S_↱ and F_↱ are still given by Eqs 2-4, but with modified constants a_↱ ≡ μ_fN/[(δ + ν)(1−s)−μ_f], b_↱ ≡ [μ_b + μ_f + s(δ + ν)]/2, and $c_{\Rsh }\equiv {[b_{\Rsh }^2+\nu (\delta +\nu \left)\right(1-s)-\nu {\mu }_{\mathrm{f}}]}^{1∕2}$. It follows that, in the parameter regime considered (μ_b, μ_f << δ,ν, and δ ≳ ν), s becomes important only when $s\gtrsim \sqrt{\nu ∕\delta }$. For example, even at a low migration rate ν/δ = 10⁻² (that is, 99% of the organisms never migrate in their lifetime) the fitness cost is noticable only if s ≳ 0.1. In population genetics this is considered a very large fitness pressure [2]. To conclude, the adaptation is not slowed down by a fitness cost conferred by the mutation unless the cost is very large. This apparently surprising result may be understood by noting that the mutant is initially very rare in the source so that a weak selection pressure is overwhelmed the demographic noise [27]. This result underscores the importance of stochasticity in the process.

To summarize, we formulated a stochastic model for adaptation in source-sink ecologies. In contrast with most traditional models of evolution, in which adaptation results from competition within a single well-stirred population, here adaptation is driven by spatial heterogeneity. To characterize the speed of adaptation, we derived analytical results for the arrival time of the first adapted mutant in the sink. The behavior of the system is found to have two qualitatively distinct regimes in which the system is either mutation-limited (μ_fN << δ + ν) or migration-limited (μ_fN >> δ + ν). In the latter regime the mean FAT does not scale as (μ_fN)⁻¹, as one might naively expect, but as (μ_fN)^−1/2. Because real beneficial mutation rates and population sizes vary considerably (mutation rates in the range 10⁻⁵–10⁻¹⁰ per generation and population sizes 10⁴–10¹⁰ are reasonable), both regimes can be relevant [12]. Furthermore the results demonstrate that the first adapted mutant found in the sink usually originates as a neutral mutation in the source which by chance migrates to the sink (the so-called “Dykhuizen–Hartl effect” [28, 29]). Mutations arising in the sink contribute only if the system is migration-limited (κ_⬏ >> 1), or if r_2w ≈ δ + ν (when patch 2 can hardly be called a sink). Strikingly, these results hold even if the mutation is mildly deleterious in the source habitat.

Many variations on the current model can be envisioned. For instance, we assumed that the reproduction rate in patch 2 depended on the genotype while the death rate was assumed constant. Yet, in reality both rates could vary in space. Also, the model can be extended to include more than two patches, which would presumably allow for a stepwise adaptation to an environmental gradient. In future work such variations and extensions could be explored within the formalism presented here.