Species Food Chain Model Research Articles

Dynamical behavior of a three species food chain model with Beddington–DeAngelis functional response

Abstract A three species food chain model with Beddington–DeAngelis functional response is investigated. The local stability analysis is carried out and global behavior is simulated numerically for a biologically feasible choice of parameters. The persistence conditions of a food chain model are established. The bifurcation diagrams are obtained for different parameters of the model after intensive numerical simulations. The results of simulations show that the model could exhibit chaotic dynamics for realistic and biologically feasible parametric values. Finally, the effect of immigration within prey species is investigated. It is observed that adding small amount of constant immigration to prey species stabilize the system.

Chaos to Order: Role of Toxin Producing Phytoplankton in Aquatic Systems

Toxin producing phytoplankton (TPP) plays an important role in aquatic systems. To observe the role of TPP, we consider a three species food chain model consisting of TPP-zooplankton-fish population. The similar type of model considered by Upadhyay et al. [1] for terrestrial ecosystem and obtained chaotic dynamics in some region of parametric space. We modify their models by taking into account the toxin liberation process of TPP population and represented as aquatic systems. We consider Holling type I, type II and type III functional forms for this process. We observe that increasing the strength of toxic substance change the state from chaos to order. Our conclusion is that TPP has a stabilizing contribution in aquatic systems and may be used as a bio-control mechanism.

Coexistence in the three species predator–prey model with diffusion

The three species food chain model is discussed, in which the third species is the predator of the second one and the second species is the predator of the first one. We consider coexistence states of the associated weakly-coupled elliptic problem under the homogeneous Neumann boundary conditions. It is shown that there are no non-constant solutions if the diffusion rates of species are strong or if the intrinsic growth rate of a prey is slow and the intrinsic drop rates of predators are fast. It is also shown that the weakly-coupled parabolic system has a unique global solution for any non-negative initial data.

Order and chaos in predator to prey ratio-dependent food chain

In this article, we investigate the dynamical behavior and chaos of a realistic three species food chain model considering predator to prey ratio-dependence for the interaction together with type II functional response. The model, for biologically reasonable parameter values, exhibits stable, periodic and chaotic dynamics in long-time behavior. The bifurcation diagrams have been obtained; Lyapunov exponents and dimensions have been computed. The model shows the rich dynamics in the positive octant. The dynamics behavior is found to be very sensitive to parameter values and initial data.

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a 1=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.

THE ROLE OF RAPID DIFFUSION IN A SIMPLE FOOD CHAIN

This paper deals with the stabilizing effect of diffusion on a four species food chain model. Conditions for local stability and instability of the model without diffusion are derived in terms of the threshold value of the functional responses of the predators at the steady state. It is also shown that an unstable equilibrium of the food chain without diffusion can be made stable for the model with diffusion, provided diffusion coefficients and predator functional responses satisfy certain threshold properties at the steady state. Lastly conditions for global stability of the model with diffusion are derived.

Permanence effect in a three—species food chain model

This paper treats a reaction–diffusion system which models the dynamics of three—species food chain interactions in ecology. A sufficient condition is given to ensure the existence of a positive steady—state solution in terms of the natural growth rates of the three species. Under the same circumstance, the reaction—diffusion sysem has a positive global attractor which indicates the permanence effect in the ecological model.

Modelling pelagic biogeography

Various combinations of physical and biological models are used to explore factors that determine the distribution of organisms in the world's oceans. The physical models examined include simple box models with parameterized inter-box exchanges that take into account variable box geometries, and specified continuous flows either in the Eulerian frame as stream-functions or as Lagrangian trajectories. A 1-dimensional mixed-layer model and a primitive equation channel model are introduced as examples of dynamical models depicting ocean physics. Biological models are discussed starting with a simple nitrogen (N), phytoplankton (P), zooplankton (Z) and detritus (D), NPZD formulation. The equilibria of this model is explored analytically as an example of computing steady state solutions, and then considering where in parameter space extinction occurs. Nonlinearities and expansion of NPZD to multi-species models are also treated. This is followed by the introduction of a nonlinear three-component food chain model, multi-species Lotka-Voltera competition models, and finally a discussion of structured population models including a derivation of a genetics model written in terms of genotypes. The physical models are then coupled with the biological ones in a series of examples. Both the box model with Lotka-Voltera multi-species population dynamics, and the 1-dimensional mixed-layer model with NPZD are used to demonstrate how the existence of spatial and temporal niches can allow a large number of species to coexist within biogeographic domains even though conditions at most sites and times are not conducive to supporting such diversity. These models recreate the basic diversity patterns observed in the pelagic ecosystem at various latitudes. The box model simulations also demonstrate the tendency for diffusive models to overestimate the dispersion of a species. In order to explore the dynamics of the edges of biogeographic domains a three species food chain model is combined with a Lagrangian trajectory calculation that specifies the dynamics of populations in a meandering jet environment. This model shows that the interaction of biological and physical dynamics can produce complex species distribution patterns that would be difficult to interpret from field observations of species abundance alone. Finally, results from a population genetics box model show that even when there is significant exchange of organisms between domains and natural selection affects allele and genotype proportions, genotype frequencies are in approximate Hardy-Weinberg equilibrium. This demonstrates the overly restrictive nature of some of the Hardy-Weinberg assumptions and provides a means for exploring population dynamics in a biogeographic context.

Environmental effects on the linear stability of a three species food chain model

The stability of the equilibrium state of a system of three interacting species forming a chain and dispersing along finite and semifinite habitats has been studied under reservoir and flux boundary conditions. It has been shown that if an equilibrium state is stable without dispersion, it is always stable with dispersion for equal dispersion coefficients, but for different dispersion coefficients dispersive instability may arise. However, a food chain system which is stable without dispersion continues to be stable with dispersion even for different dispersion coefficients. If an equilibrium state is unstable without dispersion, it may be stable with dispersion under reservoir conditions, but it can never be stable under flux conditions. The effect of convective migration of the species is to stabilize the system in all cases.

Convergence of Group Size Strategies by Mammalian Social Carnivores

Convergence of Group Size Strategies by Mammalian Social Carnivores