Breaking unidirectional invasions jeopardizes biodiversity in spatial May-Leonard systems

https://doi.org/10.1016/j.chaos.2020.110356Get rights and content

Highlights

  • We study the consequence of reversed invasions in spatial May-Leonard models.

  • It breaks the “survival of the weakest” effect and jeopardizes coexistence of species.

  • These behaviors are in stark contrast to those observed in Lotka-Volterra systems.

  • The extinction to a uniform state is characterized by a non-monotonous probability function.

Abstract

Non-transitive dominance and the resulting cyclic loop of three or more competing species provide a fundamental mechanism to explain biodiversity in biological and ecological systems. Both Lotka-Volterra and May-Leonard type model approaches agree that heterogeneity of invasion rates within this loop does not hazard the coexistence of competing species. While the resulting abundances of species become heterogeneous, the species who has the smallest invasion power benefits the most from unequal invasions. Nevertheless, the effective invasion rate in a predator and prey interaction can also be modified by breaking the direction of dominance and allowing reversed invasion with a smaller probability. While this alteration has no particular consequence on the behavior within the framework of Lotka-Volterra models, the reactions of May-Leonard systems are highly different. In the latter case, not just the mentioned “survival of the weakest” effect vanishes, but also the coexistence of the loop cannot be maintained if the reversed invasion exceeds a threshold value. Interestingly, the extinction to a uniform state is characterized by a non-monotonous probability function. While the presence of reversed invasion does not fully diminish the evolutionary advantage of the original predator species, but this weakened effective invasion rate helps the related prey species to collect larger initial area for the final battle between them. The competition of these processes determines the likelihood in which uniform state the system terminates.

Introduction

Preserving biodiversity has a paramount importance in all ecosystems especially nowadays when climate change causes rapidly altering living environments for species and their adaptations to the new conditions are hardly predictable [1], [2], [3], [4]. In general, biodiversity can be considered as a delicate balance between different processes including speciation, extinction, migration and others. Several theoretical theories have been suggested to understand its origin, and a surprisingly simple and powerful tool is offered by non-transitive dominance among competing members [5], [6], [7]. The latter situation is modeled by the well-known rock-scissors-paper game where every participant dominates another one and is dominated simultaneously by a third one [8], [9], [10], [11].

The two mainstream microscopic mathematical models which capture the essence of these relations are the so-called Lotka-Volterra (LV) and May-Leonard (ML) systems [12], [13], [14], [15], [16], [17]. While in the former LV approach the particle conservation is maintained because a dominant species occupies the empty space of dominated species immediately, there is no such constraint in ML models. In the latter case the invasion is split into a selection and a probabilistic reproduction step which makes the sum of all species a non-conserved quantity.

Previous works highlighted that behaviors of the spatial cyclic dominant systems are remarkably robust with respect to model variations and both LV and ML models predict some universal features [18], [19], [20]. One of these is when varying reaction rates have little effect on the dynamical evolution [21], or quenched spatial disorder has only minor effect on species coexistence [22]. A particularly interesting observation is the so-called “survival of the weakest” paradox which emerges when the invasion rates within the loop are heterogeneous. Counter-intuitively, in this case the “weakest species”, who has the smallest invasion power, gains more and has the largest population in the stationary state [23], [24]. It was recently demonstrated that despite the different population dynamics and spatial patterns, both LV and ML formulations lead to qualitatively similar results and confirm the robustness of this effect [25].

On the other hand, mean-field calculations warn us that the details of microscopic interactions between competing species can be important and careful studies are necessary to explore the frontiers of robustness. A well-known example is when we leave cyclic LV model by separating selection and reproduction processes the resulting ML model modifies the nonlinear dynamics from neutral orbits of LV model to an unstable spiral [26].

In our present work, motivated by the possible importance of microscopical details, we explore how the breaking of unidirectional invasions influences the evolutionary outcome. When we break the direction of dominance between a predator and a prey species and allow a reversed process with a certain probability then it conceptually may result in similar effect as was observed for heterogeneous invasion rates. Namely, the simultaneous usage of direct and reversed invasions will result in a decreased effective invasion rate, hence a weakened power of predator species. This picture is confirmed by a previous study of a LV model where direct invasion was applied with probability 1q and the related indirect process was executed with probability q [23]. Such an intervention to the original model does not change the coexistence of species because all three species survive for all q < 0.5 values, but the above mentioned “survival of the weakest” effect can be still observed.

The situation, however, is strikingly different when invasion in the reversed direction is allowed in the framework of ML systems. As we will demonstrate, here not just the coexistence of species, hence biodiversity, is jeopardized, but also “the survival of the weakest” effect vanishes. These observations highlight that we should be careful when we estimate the robustness of some effect based solely on a single theoretical approach.

Our paper is organized as follows: in the next Section 2 we describe the model details, the applied microscopic rules and specify how extinction to a homogeneous state is evaluated. In Section 3 we present our main results and provide detailed explanation to the observed behaviors. Finally, in Section 4 we conclude with some discussion and potential issues for future investigations.

Section snippets

Breaking unidirectional invasions

In the present work, we start from the classical spatial rock-scissors-paper game where three species, red “1”, blue “2”, and green “3” dominate each other. The lattice sites are occupied by one of these species, or remain empty which is marked by “0”. According to the ML approach the microscopic rules contain a mobility step (with probability m), a competition or predation (with probability p), and a reproduction step (with probability r), where the m+p+r=1 constraint is used [8]. In most

Results

Our first key observations are summarized in Fig. 1, where we plot the snapshots of spatial distribution of species for some representative P values. At P=0, shown in panel (a), we detect the well-known rotating spiral pattern of standard model. For a moderate value of P, shown in panel (b), the pattern of the stationary state changes significantly. The above mentioned spirals vanish and larger homogeneous spots emerge, but the coexistence of competing species is still stable. Increasing P

Discussion

Cyclic dominance is always the source of counter-intuitive phenomenon in population dynamics [31], [32], [33], [34], [35], [36], [37], [38], [39]. Hence designing such ecosystems could be an intellectual challenge because naive intervention into these systems may result in undesired consequences [40], [41]. We should also note that the mentioned non-transitive relations are not restricted to ecological systems, but can occur in social dilemmas, too [42], [43], [44], [45], [46], [47], [48], [49]

CRediT authorship contribution statement

D. Bazeia: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. B.F. de Oliveira: Conceptualization, Methodology, Software, Validation. J.V.O. Silva: Software, Validation. A. Szolnoki: Conceptualization, Methodology, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

D.B. acknowledges Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Grants nos. 303469/2019-6 and 404913/2018-0) and Paraíba State Research Foundation (FAPESQ-PB, Grant no. 0015/2019) (Grant no. 2014/50983-3) for financial support. B.F.O. and J.V.O.S. thank CAPES - Finance Code 001, Fundação Araucária, and INCT-FCx (CNPq/FAPESP) for financial and computational support. A.S. is grateful to the Hungarian National Research Fund (Grant K-120785).

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