Coexistence times in the Moran process with environmental heterogeneity

Abstract

Populations evolve in spatially heterogeneous environments. While a certain trait might bring a fitness advantage in some patch of the environment, a different trait might be advantageous in another patch. Here, we study the Moran birth–death process with two types of individuals in a population stretched across two patches of size N , each patch favouring one of the two types. We show that the long-term fate of such populations crucially depends on the migration rate μ between the patches. To classify the possible fates, we use the distinction between polynomial (short) and exponential (long) timescales. We show that when μ is high then one of the two types fixates on the whole population after a number of steps that is only polynomial in N . By contrast, when μ is low then each type holds majority in the patch where it is favoured for a number of steps that is at least exponential in N . Moreover, we precisely identify the threshold migration rate μ ⋆ that separates those two scenarios, thereby exactly delineating the situations that support long-term coexistence of the two types. We also discuss the case of various cycle graphs and we present computer simulations that perfectly match our analytical results.

Fixation times on directed graphs

“…While the focus of this work is to identify regimes and population structures that lead to fast fixation times, population structures with long fixation times may also be desirable, e.g. in conservation ecology to maintain high levels of ecological diversity [ 72 – 76 ]. Our results imply that spatial structures that support coexistence of two competing types on exponential time-scales all have a common feature.…”

“…We note that throughout this work we considered the standard model of Moran process with Birth-death updating. A natural direction for future research is to consider related models, such as those with location-dependent fitness [ 76 – 79 ] or those with death-Birth updating [ 20 , 63 , 80 , 81 ]. It is known that in terms of fixation probabilities the Birth-death and the death-Birth processes behave quite differently for undirected graphs [ 37 , 45 , 82 ].…”

Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating

“…Each edge is assigned a weight that represents the strength of the connection. Such network-based spatial structures can represent island models, metapopulations, lattices, as well as other arbitrarily complex structures [5][6][7][8][9]. Spatial structures that increase the fixation probability of a randomly occurring advantageous mutant beyond the constant 1 − 1/r are called amplifiers of selection [10].…”

The speed of neutral evolution on graphs

“…The trade-off between fixation probability and fixation time is present even for constant selection [16]. In recent years, the rate of evolution has received increasing attention [22][23][24][25]. It is believed that conditional fixation time is important for the rate of evolution [16,23,26].…”