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Register a new userSome elementary mechanisms for critical transitions and hysteresis in simple predator prey models
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Authors
John Vandermeer
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Trait-mediated indirect effects are increasingly acknowledged as important components in the dynamics of ecological systems. The hamiltonian form of the LV equations is traditionally modified by adding density dependence to the prey variable and functional response to the predator variable. Enriching these non-linear elements with a trait-mediation added to the carrying capacity of the prey creates the dynamics of critical transitions and hysteretic zones.
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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 equation for stochastic particle interaction models to (i) map the ensuing action in the vicinity of the absorbing state phase transition to Reggeon field theory, and (ii) to quantitatively address fluctuation-induced renormalizations of the population oscillation frequency, damping, and diffusion coefficients in the species coexistence phase.
We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.
We simulate an individual-based model that represents both the phenotype and genome of digital organisms with predator-prey interactions. We show how open-ended growth of complexity arises from the invariance of genetic evolution operators with respect to changes in the complexity, and that the dynamics which emerges is controlled by a non-equilibrium critical point. The mechanism is analogous to the development of the cascade in fluid turbulence.
We perform individual-based Monte Carlo simulations in a community consisting of two predator species competing for a single prey species, with the purpose of studying biodiversity stabilization in this simple model system. Predators are characterized with predation efficiency and death rates, to which Darwinian evolutionary adaptation is introduced. Competition for limited prey abundance drives the populations optimization with respect to predation efficiency and death rates. We study the influence of various ecological elements on the final state, finding that both indirect competition and evolutionary adaptation are insufficient to yield a stable ecosystem. However, stable three-species coexistence is observed when direct interaction between the two predator species is implemented.
Mathematical modelling and numerical simulations of interaction populations are crucial topics in systems biology. The interactions of ecological models may occur among individuals of the same species or individuals of different species. Describing the dynamics of such models occasionally requires some techniques of model analysis. Choosing appropriate techniques of model analysis is often a difficult task. We define a prey (mouse) and predator (cat) model. The system is modelled by a pair of non-linear ordinary differential equations using mass action law, under constant rates. A proper scaling is suggested to minimize the number of parameters. More interestingly, we propose a homotopy technique with n expanding parame- ters for finding some analytical approximate solutions. Furthermore, using the local sensitivity method is another important step forward in this study because it helps to identify critical model parameters. Numerical simulations are provided using Matlab for different parameters and initial conditions.
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