Oscillations and patterns in interacting populations of two species (original) (raw)
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We present a spatial host-parasitoid model where individuals move on a square lattice of patches. Local interactions between hosts and parasitoids within patches are described by the Nicholson-Bailey model. Dispersal between patches is represented by a series of movement events from a patch to neighbouring patches. We study the effect of the number of movement events on the stability of the host-parasitoid system. The aim of this work is to determine conditions on this number for using a reduced model (called aggregated model) to predict the total host and parasitoid population dynamics. When the number of movement events is small, the system is usually persistent and spatial patterns are observed, such as spiral waves or chaotic dynamics. We show that when this number is larger than a critical value, spatial homogeneity is observed after some transient dynamics and the system does not persist; in that case the reduced model can be used. Our results show that the critical value is relatively small and that the reduced model can be used in realistic situations.
Journal of theoretical biology, 2006
A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered. r
Stochastic population oscillations in spatial predator-prey models
2011
It is well-established that including spatial structure and stochastic noise in models for predator-prey interactions invalidates the classical deterministic Lotka-Volterra picture of neutral population cycles. In contrast, stochastic models yield long-lived, but ultimately decaying erratic population oscillations, which can be understood through a resonant amplification mechanism for density fluctuations. In Monte Carlo simulations of spatial stochastic predator-prey systems, one observes striking complex spatio-temporal structures. These spreading activity fronts induce persistent correlations between predators and prey. In the presence of local particle density restrictions (finite prey carrying capacity), there exists an extinction threshold for the predator population. The accompanying continuous non-equilibrium phase transition is governed by the directed-percolation universality class. We employ field-theoretic methods based on the Doi-Peliti representation of the master equa...
Spatiotemporal pattern formation of Beddington-DeAngelis-type predator-prey model
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PE ] 2 1 M ay 2 01 1 Stochastic population oscillations in spatial predator-prey models
2011
It is well-established that including spatial structure and stochastic noise in models for predator-prey interactions invalidates the classical deterministic Lotka–Volterra picture of neutral population cycles. In contrast, stochastic models yield long-lived, but ultimately decaying erratic population oscillations, which can be understood through a resonant amplification mechanism for density fluctuations. In Monte Carlo simulations of spatial stochastic predator-prey systems, one observes striking complex spatio-temporal structures. These spreading activity fronts induce persistent correlations between predators and prey. In the presence of local particle density restrictions (finite prey carrying capacity), there exists an extinction threshold for the predator population. The accompanying continuous non-equilibrium phase transition is governed by the directed-percolation universality class. We employ fieldtheoretic methods based on the Doi–Peliti representation of the master equat...
Role of the Colored Noise in Spatio-Temporal Behavior of Two Competing Species
Fluctuation and Noise Letters, 2005
We study the spatial distributions of two randomly interacting species, in the presence of an external multiplicative colored noise. The dynamics of the ecosystem is described by a coupled map lattice model. We find a nonmonotonic behavior in the formation of large scale spatial correlations as a function of the multiplicative colored noise intensity. This behavior is shifted towards higher values of the noise intensity for increasing correlation time of the noise.
Pattern dynamics of a spatial predator–prey model with noise
Nonlinear Dynamics, 2012
A spatial predator-prey model with colored noise is investigated in this paper. We find that the number of the spotted pattern is increased as the noise intensity is increased. When the noise intensity and temporal correlation are in appropriate levels, the model exhibits phase transition from spotted to stripe pattern. Moreover, we show the number of the spotted and stripe pattern, with respect to both noise intensity and temporal correlation. These studies raise important questions on the role of noise in the pattern formation of the populations, which may well explain some data obtained in the ecosystems.
Theoretical population biology, 2014
Cooperative interactions, their stability and evolution, provide an interesting context in which to study the interface between cellular and population levels of organization. Here we study a public goods model relevant to microorganism populations actively extracting a growth resource from their environment. Cells can display one of two phenotypes - a productive phenotype that extracts the resources at a cost, and a non-productive phenotype that only consumes the same resource. Both proliferate and are free to move by diffusion; growth rate and diffusion coefficient depend only weakly phenotype. We analyze the continuous differential equation model as well as simulate stochastically the full dynamics. We find that the two sub-populations, which cannot coexist in a well-mixed environment, develop spatio-temporal patterns that enable long-term coexistence in the shared environment. These patterns are purely fluctuation-driven, as the corresponding continuous spatial system does not d...
Oscillatory dynamics and spatial scale: the role of noise and unresolved pattern (2001)
Ecology, 2001
Predator-prey and other nonlinear ecological interactions often lead to oscillatory dynamics in temporal systems and in spatial systems when the rates of movement are large, so that individuals are effectively well mixed and space becomes unimportant. When individuals are not well mixed, however, properties of fluctuations in population densities, and in particular their amplitudes, are known to vary with the spatial scale at which the system is observed. We investigate the relationship among dynamics at different spatial scales with an individual-based predator-prey model that is stochastic and nonlinear. Results elucidate the role of spatial pattern and individual variability in the dynamics of densities. We show that spatial patterns in this system reduce the per capita rates of predation and prey growth but preserve functional forms. The functional forms remain those one would expect in a well-mixed system in which individuals interact according to mean population densities, but with modified parameters. This similarity of the functional forms allows us to approximate accurately the long-term dynamics of the spatial system at large scales with a temporal predator-prey model with only two variables, a simple system of ordinary differential equations of the type ecologists have been using for a long time. This approximation provides an explanation for the stabilizing role of space, the decrease in the amplitude of fluctuations from the well-mixed to the limited-movement case.