Explicit normal form coefficients for all codim 2 bifurcations of equilibria in ODEs

In this paper, explicit formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous ODEs are derived. They include second-order coefficients for the Bogdanov-Takens bifurcation, third-order coefficients for the cusp and fold-Hopf bifurcations, and coefficients of the fifth-order terms for the generalized Hopf (Bautin) and double Hopf bifurcations. The formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the Jacobian matrix of the right-hand sides and its transpose, as well as multilinear functions from the Taylor expansion of the right-hand sides at the critical equilibrium

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

Multiple Attractors, Catastrophes and Chaos in Seasonally Perturbed Predator-Prey Communities

The 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 analyzed 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 population dynamics

The Influence of Pests on Forest Age Structure Dynamics: The Simplest Mathematical Models

This paper is devoted to the investigation of the simplest mathematical models of non-even-age forests affected by insect pests. Two extremely simple situations are considered: 1) the pest feeds only on young trees; 2) the pest feeds only on old trees. It is shown that an invasion of a small number of pests into a steady-state forest ecosystem could result in intensive oscillations of its age structure. Possible implications of environmental changes on forest ecosystems are also considered

Spatial-Temporal Structure of Mixed-Age Forest Boundary: The Simplest Mathematical Model

The modeling of forest ecosystems is one of IIASA's continuous research activities in the Environment Program. There are two main approaches to this modeling: a) simulation and b) qualitative (analytical). This paper belongs to the latter. Analytical models allow the prediction of the behavior of key variables of ecosystems and can be used to organize and analyze data produced by simulation models or obtained by observations. This paper is devoted to the study of a simple mathematical model of spatially distributed non-even-age forests. The main tools used in the paper are new methods of qualitative theory of non-linear differential equations. This work is a continuation of the cooperation in forest modeling at IIASA started in 1986-1989 by W. Clark, H. Shugart, R. Fleming and the authors of this paper

Belyakov homoclinic bifurcations in a tritrophic food chain model

Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper. First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node). Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed. The overall bifurcation scenario explains why stable limit cycles and strange attractors with dierent geometries can coexist. The analysis is conducted by combining numerical continuation techniques with theoretical arguments

Forest-Pest Interaction Dynamics: The Simplest Mathematical Models

This paper is devoted to the investigation of the simplest mathematical models of non-even-aged forests affected by insect pests. Two extremely simple situations are considered: (1) the pest feeds only on young trees; (2) the pest feeds only on old trees. The parameter values of the second model are estimated for the case of balsam fir forests and the eastern spruce budworm. It is shown that an invasion of a small number of pests into a steady-state forest ecosystem could result in intensive oscillations of its age structure. Possible implications of environmental changes on forest ecosystems are also considered

The Response of the Balsam Fir Forest to a Spruce Budworm Invasion: A Simple Dynamical Model

The parameter values of a simple dynamical model of a non-even age forest-insect ecosystem are estimated for the case of balsam fir forests and the eastern spruce budworm. It is shown that, despite its extreme simplicity, the model can reproduce time series of a real budworm outbreak and can be considered a compact presentation of available forest data. Strengths and weaknesses of the model are discussed and some directions for further research proposed

On the Stability Analysis of the Standing Forest Boundary

It is well known that many existing models of forest age structure dynamics describe time dynamics for a local area. However, it is known that real forest areas have age structures that vary from one gap to another. Local gaps are integrated into a joint forest ecosystem by various seed dispersion mechanisms and by the penetration of roots. The work of Antonovsky et al. (1989) was used as a base model for a qualitative description of a spatially distributed monospecies mixed age forest. This phenomenological model employs a diffusion term corresponding to various processes of "young" tree dispersion. The model allows one to predict the possibility of the existence of a stationary or traveling forest boundary. From one side of the boundary, the modeled forest demonstrates an equilibrium state with non-zero age class densities, while from the other side, there are no trees of the studied type. As was shown, the model analyzes the changes in the behavior of the forest boundary caused by an increase in the tree mortality rate due to anthropogenic impacts (from acid rain, for example). The present paper is devoted to studying the ability of the forest to resist the internal "negative" forces of forest ecosystems and external impacts on changes in the standing forest boundary (on a given model level)

Forest-Pest Interaction Dynamics: The Simplest Mathematical Models

This report is devoted to the investigation of the simplest mathematical models of non-even-aged forests affected by insect pests. Two extremely simple situations are considered: (1) the pest feeds only on young trees; (2) the pest feeds only on old trees. The parameter values of the second model are estimated for the case of balsam fir forests and the eastern spruce budworm. It is shown that an invasion of a small number of pests into a steady-state forest ecosystem could result in intensive oscillations of its age structure. Possible implications of environmental changes in forest ecosystems are also considered