Delays in Fitness Adjustment Can Lead to Coexistence of Hierarchically Interacting Species

Abstract

Organisms that exploit different environments may experience a stochastic delay in adjusting their fitness when they switch habitats. We study two such organisms whose fitness is determined by the species composition of the local environment, as they interact through a public good. We show that a delay in fitness adjustment can lead to coexistence of the two species in a metapopulation, although the faster growing species always wins in well-mixed competition experiments. Coexistence is favored over wide parameter ranges, and is independent of spatial clustering. It arises when species are heterogeneous in their fitness and can keep each other balanced.How biodiversity -for example, the surprising coexistence of more than 10 000 species in a single gram of soil [1-3] -is stabilised is one of the most fundamental questions in ecology [4]. Known stabilising factors derived from resource competition models include metabolic trade-offs [5] and reciprocal oscillations in population sizes [6]. Cyclic competition models and their derivatives are also frequently employed to model biodiversity [7][8][9]. However, recent experiments on a variety of different soil species in well-mixed pair-wise competition experiments found that these species could not be represented by cyclic competition models; instead, a few species outcompeted the others [10]. Thus, microbial diversity in soil is thought to be supported by the highly porous and fragmented structure of this habitat [11], and the fluctuating environmental conditions that individual bacteria experience there [10].Here, we ask how the influence of spatial structure, characterized by intrinsic variation between local environments, and the delayed adjustment to changes of these environments, can affect the long-term behaviour of species in a simple model system. Indeed, it is wellknown that such delays in physiological responses (here, adjustment of fitness or growth rate) occur in microbes, following externally imposed changes in the environment [12][13][14][15], such as nutrient composition, or antibiotic stress [16][17][18][19][20][21][22]. Here, we focus on changes that occur because species move between different habitats. We consider a system with two species, in which the dominant strain (fast grower) depends on the slower growing strain for its fitness. Thus, the fitness of both species depends on the intrinsic population structure of the local habitat. We assume that the fitness of an individual is not instantaneously reset when the environment changes, but is initially retained from its previous environment, as we discuss below.We show that coexistence can arise in such a minimal two-species model, as a result of delays in fitness adjustment after a change of local habitat, and nonlinearity of fitness functions. In particular, we conclude that the combination of underlying spatial structure -not spa-

Cooperator driven oscillation in a time-delayed feedback-evolving game

“…Accordingly, such time-delay feature is different from the relative timescale of strategy and resource dynamics introduced in previous works [17,24], which characterizes the relative strength of strategic versus resource change. On the other hand, we stress that delays in fitness adjustment have been already considered by Bauer and Frey who observed a coexistence of two competing species in a metapopulation [30]. But studying the direct consequence of time delay on a renewable environment which is subject to the battle of competing consuming strategies remained unexplored.…”

Snowdrift game induces pattern formation in systems of self-propelled particles

“…A key factor for the dynamics is the spatial clustering of individuals, which has already been argued by Hamilton to support cooperation in populations [27], because cooperating individuals are more likely to interact with each other and therefore are less likely to be exploited by defectors. To what degree this determines the composition of a population has been studied extensively on different kinds of lattices as well as for different kinds of interaction rules [8,[10][11][12][28][29][30][31][32][33][34]. Specifically, for the snowdrift game, it was shown that accounting for spatial extension can change population ratios compared to well-mixed systems [29,35].…”

Cooperation in Microbial Populations: Theory and Experimental Model Systems

“…In such a situation cooperators can benefit from other cooperators, and they also run a lower risk to be exploited by non-cooperating free-riders. Possible ecological scenarios promoting such an assortment include spatially extended systems [263,137,298,112,19,21,20], populations structured into distinct sub-populations [376,317,88,195,159,177,236,335,336,222] or more complex interactions between different individuals within a population (networks) [215,305,269,274,5].…”

“…This suggests that in many biological scenarios, the spatial clustering of individuals cannot explain stable cooperation but additional mechanisms are necessary for spatial structure to promote the evolution of cooperation. For example, it has recently been shown that delayed adjustment to changes in the environments can affect the long-term behaviour of microbial communities and promote robust cooperation [20,21,19]. Active motility is an essential part of biological reality.…”

“…Importantly, the transient increase in cooperator fraction is a genuine stochastic effect which is not captured by the deterministic equations, Eq. (20). It is caused by the amplification of stochastic fluctuations during the initial phase of the population dynamics where the population size is still small.…”