Evolutionary and population dynamics: A coupled approach

Jonas Cremer, Anna Melbinger, and Erwin Frey

Phys. Rev. E 84, 051921 – Published 28 November 2011

Abstract

We study the interplay of population growth and evolutionary dynamics using a stochastic model based on birth and death events. In contrast to the common assumption of an independent population size, evolution can be strongly affected by population dynamics in general. Especially for fast reproducing microbes which are subject to selection, both types of dynamics are often closely intertwined. We illustrate this by considering different growth scenarios. Depending on whether microbes die or stop to reproduce (dormancy), qualitatively different behaviors emerge. For cooperating bacteria, a permanent increase of costly cooperation can occur. Even if not permanent, cooperation can still increase transiently due to demographic fluctuations. We validate our analysis via stochastic simulations and analytic calculations. In particular, we derive a condition for an increase in the level of cooperation.

Received 10 August 2011

DOI:https://doi.org/10.1103/PhysRevE.84.051921

Authors & Affiliations

Jonas Cremer, Anna Melbinger, and Erwin Frey

Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 84, Iss. 5 — November 2011

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The per capita birth and death rates for two different traits, $A$ [light gray (red)] and $B$ [dark gray (blue)]. Each rate depends on a global, trait-independent and a relative, trait-dependent part. While the global and relative fitness terms $g$ and $f_{A / B}$ affect the birth rates, the global and relative weakness terms $d$ and $w_{A / B}$ determine the death rates.Reuse & Permissions
Figure 2
Cooperation in growing populations. Temporal development of ensemble averages. (a) The population size. Starting with $N_{0} = 4$ , the system grows exponentially until the carrying capacity is reached. It then falls again due to selection and a decreasing carrying capacity; see text. The full stochastic solution, gray (red) line, is described well by the deterministic approximation, black line. (b) The fraction of cooperators. It initially increases due to asymmetric amplification of fluctuations, and then falls again due to selection; see text. The level of cooperation $x$ falls below its initial value $x_{0}$ at the cooperation time $t_{C}$ . The transient increase is stronger for larger fluctuations and thus is stronger with a smaller initial population size $N_{0}$ ; see gray (colored) lines. The deterministic approximations do not account for this behavior (cf. black line). Parameters are $s = 0.1$ and $p = 10$ .Reuse & Permissions
Figure 3
The transient increase of cooperation and its dependence on parameters. Encoded in gray (colored) scale, the cooperation time $t_{C}$ is plotted for three different pairs of parameters: ${N_{0}, s}$ , ${N_{0}, p}$ , and ${x_{0}, s}$ in (a), (b), and (c), respectively. The boundary between the regimes of transient increase and immediate decrease are in good agreement given by Eq. (24), plotted as black lines. In the inset of (a), the cooperation time is shown for varying selection strength $s$ : $t_{C}$ sharply drops at the boundary. Not varied parameters are given by $p = 10, x_{0} = 0.5$ in (a); $s = 0.05$ and $x_{0} = 0.5$ in (b); $p = 10, N_{0} = 6$ in (c).Reuse & Permissions
Figure 4
The dilemma of cooperation in the dormancy scenario. (a) The growth dynamics. Initially, the small population grows exponentially until growth is stopped [cf. light gray (red) line]. This behavior is well described by the deterministic equation (27a); see black line. In contrast, for the balanced growth scenario, the dynamics continue and, due to selection, the population size falls again; see dark gray (blue) line. (b) The fraction of cooperators. Equal to the balanced growth scenario, dark gray (blue) line, there is an initial increase of cooperation due to asymmetric amplification within the dormancy scenario. Again, this is not described by the deterministic approximation Eq. (27b). However, in contrast to the balanced growth scenario, the higher level of selection is later fixed due to the stop in growth dynamics. Parameters are given by $s = 0.05$ , $p = 10$ , and $N_{0} = 4$ .Reuse & Permissions
Figure 5
The transient increase of cooperation for the dormancy scenario. The cooperation time $t_{C}$ depending on the initial population size $N_{0}$ and the strength of selection $s$ . The condition for a transient increase of cooperation to occur is still given by Eq. (24) (black line). In addition, due to the stop in growth dynamics, there is an additional regime, where the increase becomes permanent [dark gray (dark blue) area]. The permanent increase is also shown in the inset, where the cooperation time is shown for varying strength of selection. If, for a given initial population size, selection is sufficiently slow compared to fixation of the growth dynamics, the increase of cooperation becomes permanent. Parameters are given by $p = 10, x_{0} = 0.5$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics