Cyclic Predator-prey Models Research Articles

Perturbative field-theoretical analysis of three-species cyclic predator-prey models

We apply a perturbative Doi–Peliti field-theoretical analysis to the stochastic spatially extended symmetric Rock-paper-Scissors (RPS) and May–Leonard (ML) models, in which three species compete cyclically. Compared to the two-species Lotka–Volterra predator-prey (LV) model, according to numerical simulations, these cyclical models appear to be less affected by intrinsic stochastic fluctuations. Indeed, we demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although not yet observed in numerical simulations, we find that the RPS model acquires significant fluctuation-induced renormalizations in the perturbative regime, similar to the LV model. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS system, whereas the spontaneous emergence of spatio-temporal structures features prominently in the LV and the ML models.

Interfaces with internal structures in generalized rock-paper-scissors models.

In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships in the context of cyclic predator-prey models with an even number of species N≥8. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behavior associated with the symmetric or asymmetric evolution of the interface profiles depending on whether N/2 is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive to the internal structure of the interfaces. Although this work focuses on cyclic predator-prey models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.

Phase diagram of a cyclic predator-prey model with neutral-pair exchange

In this paper we obtain the phase diagram of a four-species predator-prey lattice model by using the proposed gradient method. We consider cyclic transitions between consecutive states, representing invasion or predation, and allowed the exchange between neighboring neutral pairs. By applying a gradient in the invasion rate parameter one can see, in the same simulation, the presence of two symmetric absorbing phases, composed by neutral pairs, and an active phase that includes all four species. In this sense, the study of a single-valued interface and its fluctuations give the critical point of the irreversible phase transition and the corresponding universality classes. Also, the consideration of a multivalued interface and its fluctuations bring the percolation threshold. We show that the model presents two lines of irreversible first-order phase transition between the two absorbing phases and the active phase. Depending on the value of the system parameters, these lines can converge into a triple point, which is the beginning of a first-order irreversible line between the two absorbing phases, or end in two critical points belonging to the directed percolation universality class. Standard simulations for some characteristic values of the parameters confirm the order of the transitions as determined by the gradient method. Besides, below the triple point the model presents two standard percolation lines in the active phase and above a first-order percolation transition as already found in other similar models.

Coexistence in the two-dimensional May-Leonard model with random rates

We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-)steady state in two-dimensional stochastic May--Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May--Leonard system (for small system sizes): (1) As the mobility rate exceeds a threshold that separates a species coexistence (quasi-)steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ~ e^{cN} / N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.

Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species

Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations, the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.

Segregation process and phase transition in cyclic predator-prey models with an even number of species

We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) that allows the formation of two defensive alliances consisting of the even and odd label species. The species are distributed on the sites of a square lattice. The evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly: if the sites are occupied by a predator-prey pair then the predator invades the prey's site; otherwise the species exchange their sites with a probability X . For low X values, a self-organizing pattern is maintained by cyclic invasions. If X exceeds a threshold value, then two types of domain grow up that are formed by the odd and even label species, respectively. Monte Carlo simulations indicate the blocking of this segregation process within a range of X for n=8.