Simona Muratori

Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities

A classical predator-prey model is considered in this paper with reference to the case of periodically varying parameters. Six elementary seasonality mechanisms are identified and analysed in detail by means of a continuation technique producing complete bifurcation diagrams. The results show that each elementary mechanism can give rise to multiple attractors and that catastrophic transitions can occur when suitable parameters are slightly changed. Moreover, the two classical routes to chaos, namely, torus destruction and cascade of period doublings, are numerically detected. Since in the case of constant parameters the model cannot have multiple attractors, catastrophes and chaos, the results support the conjecture that seasons can very easily give rise to complex populations dynamics.

Low- and high-frequency oscillations in three-dimensional food chain systems

The dynamic behavior of a three-trophic-level food chain (prey-predator-superpredator) is analyzed in this paper. Prey is logistic, while predator and superpredator have functional response of the Holling type. Moreover, the trophic levels are assumed to be characterized by increasing and quite diversified time responses. Using a singular perturbation approach, explicit conditions for the persistence of the three populations are derived, and the structure of the corresponding attractors is noted, as well as the nature of transients. The analysis shows that the system can have low-frequency cycles due to the interactions between predator and superpredator. Nevertheless, the interactions between prey and predator can give rise to high-frequency oscillations, which can arise during the transients toward the attractor or during the low-frequency cycle. This periodic burst of high-frequency oscillations develops, in particular, when the two predators are fairly efficient. These results note in a very concise w...

BIFURCATIONS AND CHAOS IN A PERIODIC PREDATOR-PREY MODEL

The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected.

Slow-fast limit cycles in predator-prey models☆

Abstract This paper is devoted to the analysis of the cyclic behavior of predator-prey systems under the assumption that the time responses of prey and predator are quite diversified. In such a case, a geometric approach (separation principle) based on the singular perturbation method can be applied to detect slow-fast limit cycles. The main characteristic of these cycles is that the fast component of the ecosystem is present at significantly high densities only during a fraction of the cycle. At the start and at the end points of this period the slow component of the ecosystem reaches its minimum and maximum densities, which are related to each other through an integral equation. This equation is specialized for the classical predator-prey model and is later used to study the cyclic behavior of more complex predator-prey systems, some of which were not fully understood up to now. The conclusion of the analysis is that the existence of slow-fast limit cycles can be ascertained by means of the separation principle, while the geometry of the cycle can be fully specified only by using the integral equation discussed in this paper.

Remarks on competitive coexistence

The asymptotic behavior of two predators competing exploitatively for the same prey in a constant environment is analyzed in this paper. Assuming that the prey have fast dynamics, it is proved, through a singular perturbation argument, that the two predators can coexist if the prey and predators’parameters lie within specified ranges. It is also shown that coexistence occurs through a simple periodic behavior: a poor season, characterized by an almost endemic presence of the prey, alternates with a rich season, during which prey are abundant and predators regenerate. This property follows from the geometric nature of the equilibrium manifold and is therefore generic. The results are in agreement with some conjectures supported by computer simulation and with the most recent theoretical findings on competitive coexistence.

Limit cycles in slow-fast forest-pest models

Abstract A simple age-structured forest model is considered in this paper to prove that, in contrast with some recent findings, a forest can exhibit periodic behavior even in the case where the insect pest is adapted only to mature trees. The insect pest is assumed to have a very fast dynamics with respect to trees and the analysis is carried out through singular perturbation arguments. The method is based only upon simple geometric characteristics of the equilibrium manifolds of the fast, intermediate, and slow variables of the system and allows one to derive explicit conditions on the parameters that guarantee the existence of a limit cycle in the extreme case of very fast-very slow dynamics. Nevertheless, simulation shows that the limit cycle disappears through a Hopf bifurcation only when the dynamics of the different components of the system become almost comparable.

Excitability, stability, and sign of equilibria in positive linear systems

Abstract The relationship between sign and stability of equilibria in positive linear systems is investigated in this paper. In particular it is proved that the well-known result “stability and irreducibility imply strict positivity” can be better specified by introducing the notion of excitability. In fact stability, excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others.

Decision support systems for environmental impact assessment of transport infrastructures

Abstract Practical applications of decision support systems for environmental impact assessment of transport infrastructures are described. Silvia interactive software, one of such systems, covering the four phases (preparatory/orientation, land analysis, impact estimation, and decision) of an environmental impact assessment study, is presented. Two directions for further development: sensitivity analysis and group decisions are suggested.

A separation condition for the existence of limit cycles in slow-fast systems

Abstract A condition for the existence of limit cycles in slow-fast dynamical systems is presented in this paper. The condition, used up to now to analyze relaxation oscillations in second-order systems, is extended to higher-dimensional systems and applied to models of particulat interest. The analysis is purely geometric and based on singular perturbation arguments, and the limit cycles that the condition identifies are composed of alternate slow and fast transitions.

An application of the separation principle for detecting slow-fast limit cycles in a three-dimensional system

A geometric method (called the separation principle) is used in this paper for detecting limit cycles in a slow-fast dynamical system describing a food chain with trophic levels with diversified time responses. The analysis is very simple and based on singular-perturbation arguments. The limit cycle resulting from the method is composed of a concatenation of catastrophic transitions occuring at different speeds.