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Register a new userStabilizing spiral structures in the asymmetric May-Leonard model
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We study the induction and stabilization of spiral structures for the cyclic three-species stochastic May-Leonard model with asymmetric predation rates on a spatially inhomogeneous two-dimensional toroidal lattice using Monte Carlo simulations. In an isolated setting, strongly asymmetric predation rates lead to rapid extinction from coexistence of all three species to a single surviving population. Even for weakly asymmetric predation rates, only a fraction of ecologies in a statistical ensemble manages to maintain full three-species coexistence. However, when the asymmetric competing system is coupled via diffusive proliferation to a fully symmetric May-Leonard patch, the stable spiral patterns from this region induce transient plane-wave fronts and ultimately quasi-stationary spiral patterns in the vulnerable asymmetric region. Thus the endangered ecological subsystem may effectively become stabilized through immigration from even a much smaller stable region. To describe the stabilization of spiral population structures in the asymmetric region, we compare the increase in the robustness of these topological defects at extreme values of the asymmetric predation rates in the spatially coupled system with the corresponding asymmetric May-Leonard model in isolation. We delineate the quasi-stationary nature of coexistence induced in the asymmetric subsystem by its diffusive coupling to a symmetric May-Leonard patch, and propose a (semi-)quantitative criterion for the spiral oscillations to be sustained in the asymmetric region.
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Cyclic dominance of competing species is an intensively used working hypothesis to explain biodiversity in certain living systems, where the evolutionary selection principle would dictate a single victor otherwise. Technically the May--Leonard models offer a mathematical framework to describe the mentioned non-transitive interaction of competing species when individual movement is also considered in a spatial system. Emerging rotating spirals composed by the competing species are frequently observed character of the resulting patterns. But how do these spiraling patterns change when we vary the external environment which affects the general vitality of individuals? Motivated by this question we suggest an off-lattice version of the tradition May--Leonard model which allows us to change the actual state of the environment gradually. This can be done by introducing a local carrying capacity parameter which value can be varied gently in an off-lattice environment. Our results support a previous analysis obtained in a more intricate metapopulation model and we show that the well-known rotating spirals become evident in a benign environment when the general density of the population is high. The accompanying time-dependent oscillation of competing species can also be detected where the amplitude and the frequency show a scaling law of the parameter that characterizes the state of the environment. These observations highlight that the assumed non-transitive interaction alone is insufficient condition to maintain biodiversity safely, but the actual state of the environment, which characterizes the general living conditions, also plays a decisive role on the evolution of related systems.
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.
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