Bifurcation Analysis of Two Predator-Prey Models

Bifurcation analysis is used to systematically detect unrealistic dynamic behaviors of two predator-prey models that have recently received support and attention. Comparisons with other existing models are also made, in order to show that the singularities of the two examined models are due to some biological weakness of their functional forms

A Universal Bifurcation Diagram for Seasonally Perturbed Predator-Prey Models

The bifurcations of a periodically forced predator-prey model (the chemostat model), with a prey feeding on a limiting nutrient, are numerically detected with a continuation technique. Eight bifurcation diagrams are produced (one for each parameter in the model) and shown to be topologically equivalent. These diagrams are also equivalent to those of the most commonly used predator-prey model (the Rosenzweig-McArthur model). Thus, all basic modes of behavior of the two main predator-prey models can be explained by means of a single bifurcation diagram

Acid Rain and the Fate of the Forest Ecosystem

In this paper a sixth-order nonlinear dynamical system modelling forest-pests interactions is presented. Forest growth is described by mechanisms of the exchange of carbon and nitrogen between soil and vegetation, and by microbial mineralization. The influence of acid deposition, an increase of which can cause either the decline or the collapse of the forest ecosystem, is discussed. The analysis is carried out by numerically finding the bifurcations of the model

Love Dynamics Between Secure Individuals: A Modeling Approach

A mathematical model that qualitatively describes the dynamics of love between secure individuals is presented in this paper with two goals. The general goal is to show how dynamic phenomena in the field of social psychology can be analyzed following the modelling approach, traditionally used in all other fields of sciences. The specific goal is to derive, from very general assumptions on the behavior of secure individuals, a series of rather detailed properties of the dynamics of their feelings. The analysis shows, in particular, why couples can be partitioned into fragile and robust couples, how the quality of romantic relationships is influenced by behavioral parameters and in which sense individual appeal can create order in a community structure

Pollution Control Policies and Natural Resource Dynamics: A Theoretical Analysis

Policies for the management of natural resources and the resources themselves interact to form complex systems. In this paper, we present a highly simplified model that can be used to study the general features of those systems. The model has three state variables, the abundance of the resource, environmental pollution, and the capital devoted to pollution control. We analyze it graphically using the singular perturbation approach. Two modes of behavior are possible, stationary and cyclic. When the abundance of the resources varies cyclically, the length of the period of resource scarcity depends on population size, economic activity, pollution per unit of output and on policy constraints. We distinguish between two classes of policies, one in which decision-makers base their investments in pollution control capital on the abundance of the resource, and another in which those decisions depend on the amount of pollution. We show that policies based on the observation of pollution are safer than those based on resource abundance, because in the latter case, small changes in policy variables can much more easily lead to a collapse of the resource. Increases in population size, even when accompanied by an equiproportional increase in the pollution control budget can lead to a change from stationary to cyclic behavior, especially where policies are based on resource abundance

Complex Dynamics in Romantic Relationships

Minimal models composed of two ordinary differential equations are considered in this paper to mimic the dynamics of the feelings between two persons. In accordance with attachment theory, individuals are divided into secure and non-secure individuals, and in synergic and non-synergic individuals, for a total of four different classes. Then, it is shown that couples composed of secure individuals, as well as couples composed of nonsynergic individuals can only have stationary modes of behavior. By contrast, couples composed of a secure and synergic individual and a non-secure and non-synergic individual can experience cyclic dynamics. In other words, the coexistence of insecurity and synergism in the couple is the minimum ingredient for complex love dynamics. The result is obtained through a detailed local and global bifurcation analysis of the model. Supercitical Hopf, fold and homoclinic bifurcation curves are numerically detected around a Bogdanov-Takens codimension-2 bifurcation point. The existence of a codimension-2 homoclinic bifurcation is also ascertained. The bifurcation structure allows one to identify the role played by individual synergism and reactiveness to partner's love and appeal. It also explains why aging has a stabilizing effect on the dynamics of the feelings. All results are in agreement with common wisdom on the argument. Possible extensions are briefly discussed at the end of the paper

Dynamics of Drug Consumption: a Theoretical Model

A continuous time model is proposed to describe the dynamics of drug consumption in a given country. The model has two state variables, addicts and dealers, and eleven parameters, including the total effort exerted by the State, which is considered as control parameter. The model is highly nonlinear and the analysis shows that it is characterized by a transcritical and a fold bifurcation. This implies that for intermediate values of the State's effort the model has two stable equilibria, one trivial, corresponding to the absence of drugs, and one positive, corresponding to drug consumption. On the contrary, for low and high values of the effort only one of the two equilibria is stable. This suggests a two-step control policy. First, exert a very high effort for a few years, so that the system has the time to approach the trivial equilibrium, and then reduce the effort but maintain it sufficiently high so that drug consumption cannot rise anymore. Interesting results on the role played by the price of the drug and the severity of the punishment inflicted to dealers, as well as on the allocation of the effort between therapy and police, have also been obtained

Bifurcation Analysis of Piecewise Smooth Ecological Models

The aim of this paper is the study of the long-term behavior of population communities described by piecewise smooth models (known as Filippov systems). Models of this kind are often used to describe populations with selective switching between alternative habitats or diets or to mimic the evolution of an exploited resource where harvesting is forbidden when the resource is below a prescribed threshold. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations

Persistent Unstable Equilibria in Wonderland

Models of the interactions between population, economy, and environment often contain nonlinear functional relationships and variables that vary at different speeds. These properties foster apparent unpredictabilities in system behavior. Using a simple deterministic model of demographic, economic and environmental interactions, we illustrate the usefulness of geometric singular perturbation theory and local bifurcation theory. In particular we show how it is possible to obtain analytic expressions for: (1) the level of emissions above which environmental deterioration begins, (2) the time it takes from reaching the critical level of emissions to the beginning of rapid environmental deterioration, and (3) the level of emissions at the time that rapid deterioration begins. Because our results are analytic, they make the outcomes of demographic, economic, and environmental interactions more predictable, and, therefore, potentially more manageable

Bifurcation Analysis of Filippov's Ecological Models

The aim of this paper is the study of the long-term behavior of population communities described by a class of discontinuous models known as Filippov systems. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations