The interplay of selection and dormancy in a Moran model can lead to coexistence of types

TL;DR

It turns out that the dormancy trait can not only invade and subsequently fixate under suitable parameter assumptions despite its selective disadvantage, but that there is also a novel regime of stable coexistence of both types due to a frequency-dependent balancing effect that did not arise in the previous setup with Lotka--Volterra type symmetric competition.

Abstract

In this paper we propose a Moran model that describes the population dynamics of two types: While the first type has a selective advantage during reproduction, the second type can avoid replacement during reproduction with some positive probability by switching temporarily into a dormant state. We investigate the interplay of both evolutionary strategies by studying the invasion dynamics of the dormant type into the resident (selectively advantageous) population in the large population limit of the system. It turns out that the dormancy trait can not only invade and subsequently fixate under suitable parameter assumptions despite its selective disadvantage (a phenomenon that has already been observed in a related context in Blath and Tóbiás (2020)), but that there is also a novel regime of stable coexistence of both types due to a frequency-dependent balancing effect that did not arise in the previous setup with Lotka--Volterra type symmetric competition. The emergence of a coexistence regime here rests in part on specific properties of the Moran modelling framework, in particular its fixed overall population size that enforces instant re-colonization after death events, as well as on the (positive) mortality and resuscitation rates of the dormant state. We provide heuristic explanations for the observed types of behaviour and the corresponding proofs, which involve comparisons to suitable branching processes, approximations by dynamical systems, and an analysis of asymptotic behaviour of the latter.

Figures (5)

• Figure 1: We plot, for $p=0.4$ and $s=0.1$, the evolution of $\bar{x}_{2a}$, $\tilde{x}_{2a}$ and $\tilde{x}_1$ according to $\kappa$, for two values of $\sigma$, the first one less and the second one greater than $\frac{2s^2}{p(1-s)}$. The light-red region corresponds to founder control (in the first case) and coexistence (on the second) coinciding with the existence of $(\tilde{x}_{2a},\tilde{x}_{2d})$. Moreover, the pictures verify the properties of $(\tilde{x}_{1},\tilde{x}_{2a})$ stated in Lemma \ref{['lemma-xtildeexistence']}. We also remark the apparition of an asymptote (red vertical line) that will be made explicit in the proof of Lemma \ref{['lemma-xtildeexistence']} in Section \ref{['sec-stability']}.
• Figure 2: In the case $p>\frac{2s}{1+s}$ and $\sigma < \frac{2s^2}{p(1-s)}$, we have $\tilde{\kappa} >\kappa'$. In the founder control regime where $\kappa \in ( \kappa', \tilde{\kappa})$, both equilibria $(0,\bar{x}_{2a})$ and $(1,0)$ are locally asymptotically stable, and the coordinate-wise positive equilibrium $(\tilde{x}_1, \tilde{x}_{2a})$ also exists (but we expect it to be unstable).
• Figure 3: In the case $p>\frac{2s}{1+s}$ and $\sigma > \frac{2s^2}{p(1-s)}$, we have $\tilde{\kappa} < \kappa'$. In the coexistence regime where $\kappa \in (\tilde{\kappa}, \kappa')$, both equilibria $(0,\bar{x}_{2a})$ and $(1,0)$ are unstable, and the coordinate-wise positive equilibrium $(\tilde{x}_1, \tilde{x}_{2a})$ emerges (and we expect it to be asymptotically stable, see Conjecture \ref{['conj-Moran']}).
• Figure 4: For fixed values of $p=0.1$, and $s=0.05$ (so that $p > \frac{2s}{1+s}$), we plot $\tilde{\kappa}$ (blue) and $\kappa'$ as functions of $\sigma$. Light green regime: fixation of type $2$; blue regime: founder control; gray regime: fixation of type $1$; yellow regime: stable coexistence. The vertical line $\{ \sigma = \frac{2s^2}{p(1-s)}\}$ separates the cases (b) (left) and (c) (right). The gray regime corresponds to scenario (i), the light green one to scenario (ii), the yellow one to scenario (iii), and the blue one to scenario (iv).
• Figure 5: We plot, for $p$ and $s$ satisfying $p>\frac{2s}{1+s}$, the maximum of the size of the interval where we have coexistence or founder control according to $\sigma$. Here, we generated a large number of random pairs $(s,p)$ such that $p>\frac{2s}{1+s}$ and then chose the maximizer of $|\tilde{\kappa}-\kappa'|$ from a finite set of values of $\sigma \in (0,1)$.

Theorems & Definitions (28)

• Remark 2.1
• Lemma 2.2: Stability of monomorphic equilibria
• Lemma 2.3: Conditions for the existence of a polymorphic equilibrium
• Remark 2.4
• Remark 2.5: Reverse invasion of type 1 and fixation of type 2
• Definition 2.6: Invasion scenarios
• Remark 2.7
• Theorem 2.8: Invasion, fixation, coexistence and founder control
• Remark 2.9: General comments on the assertions of Theorem \ref{['theorem-mainMoran']}
• Remark 2.10: Stability of the polymorphic equilibrium