Fixation in competing populations: Diffusion and strategies for survival

Open Access

Fixation in competing populations: Diffusion and strategies for survival

Tapas Singha, Prasad Perlekar, and Mustansir Barma

Phys. Rev. Research 2, 023412 – Published 29 June 2020

Abstract

How should dispersal strategies be chosen to increase the likelihood of survival of a species? We obtain the answer for the spatially extended versions of three well-known models of two competing species with unequal diffusivities. Though identical at the mean-field level, the three models exhibit drastically different behavior leading to different optimal strategies for survival, with or without a selective advantage for one species. With conserved total particle number, dispersal has no effect on survival probability. With a fluctuating number, faster dispersal is advantageous if intraspecies competition is present, while moving slower is the optimal strategy for the disadvantaged species if there is no intraspecies competition: it is imperative to include fluctuations to properly formulate survival strategies.

Received 14 November 2019
Accepted 9 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023412

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Ecological population dynamics Evolutionary & population dynamics Organismal, population, evolutionary & ecological systems Population genetics

Physics of Living Systems

Authors & Affiliations

Tapas Singha ^*, Prasad Perlekar, and Mustansir Barma

TIFR Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Gopanpally, Hyderabad 500046, India

^*Corresponding author: tapas134@gmail.com

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 2, Iss. 2 — June - August 2020

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Space-time plots for the three models, namely, voter model with diffusion (VMD), fluctuating voter model with diffusion (FVMD), and competitive Lotka-Volterra model with diffusion (CLVMD), with equal (top panels) and unequal (bottom panels) diffusivities and with zero selective advantage. Red (species A) and green (species B) indicate two competing species, with green diffusing slower (bottom panels), with the intensity of color being an indicator of local density. Yellow indicates empty space. At left (a), the top and bottom panels are shown when $A$ wins, and similarly, at right (b), top and bottom panels are shown when $B$ wins. The initial state has well-mixed populations with equal densities $ρ_{A} = ρ_{B} = 8$ of the two species, where $ρ_{A} = N / (2 L)$ . The diffusion constant for the slower species is taken to be zero for the VMD and FVMD, resulting in pillarlike structures localized at sites. For CLVMD, the slower species has diffusion constant 0.96. It is evident that the dynamics of fluctuations is very different in the three cases.
Reuse & Permissions
Figure 2
In the neutral case $s = 0$ , the fixation probability $F_{B}$ does not vary with respect to relative diffusivity $Δ D$ for VMD and FVMD, while it shows a strong variation with $Δ D$ for the CLVMD.
Reuse & Permissions
Figure 3
Cartoon of some moves and reactions to illustrate the contrasting effects of $Δ D > 0$ in the three models when $s > 0$ . Starting from the same initial configuration (top row) we depict configurations which occur just before (second row) and just after (third row) a possible $A - B$ interaction event. $A$ particles (open red circles) are stronger and faster; $B$ particles (solid green circles) are weaker and slower. VMD: Each reaction event can be mapped onto a step in the Moran process irrespective of diffusion constants. FVMD: In this model, an individual can die on its own. When the lifetime of an $A$ particle is less than $τ_{Diff}$ , it is likely to die before interaction (depicted by the faint red circle in the second row), which is advantageous for the weaker $B$ species, as the number of interactions would drop. Thus the optimal strategy for the weaker species is to make $τ_{Diff}$ large, which it can do by moving as slowly as possible. CLVMD: The lifetime of a cluster of slow $B$ 's formed by chance on a site is larger than that of a corresponding $A$ particle cluster, leading to more intraspecies extinction, in turn making it easier to fixate $A$ .
Reuse & Permissions
Figure 4
Strategy plot: The change of $Δ F_{B} (s, Δ D)$ of the slower $B$ species for given values of selective advantage $s$ for the three models; recall that $s > 0 (s < 0)$ implies selective advantage for $A (B)$ . VMD: $Δ F_{B}$ vanishes for all $s$ , as depicted by the thick red line in (a). Diffusing faster or slower does not change the fixation probability. FVMD: When $B$ is weaker, $Δ F_{B}$ is positive, implying that moving slowly is a better strategy than moving fast. $Δ F_{B}$ reverses sign when $s < 0$ , implying reversal of the strategy. CLVMD: the faster species gets a benefit for all values of $s$ as shown in (b).
Reuse & Permissions
Figure 5
Plot of $Δ F_{B} (s, Δ D)$ with respect to $s$ for different $Δ t$ and with $Δ D = 1$ fixed.
Reuse & Permissions
Figure 6
(a) Long-time dynamics showing extinction for our simulation with $L = 8$ and $ρ = 2$ . (b) Average number of particles $〈 N (t) 〉$ vs time $t$ for equal (black solid line) and unequal (red solid line) diffusivities. The angular brackets denote averaging over 10 000 independent realizations. Average fixation ( $〈 t_{A} 〉$ , $〈 t_{B} 〉$ ) and extinction ( $〈 t_{Ext} 〉$ ) times are marked with arrows.
Reuse & Permissions

Physical Review Research