Interior Steady State Research Articles

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.

Continuous-time predator–prey systems with Allee effects in the prey

A continuous-time predator–prey model without spatial variation is proposed to study the impact of Allee effects upon the predator–prey interaction when the prey population is subject to Allee effects. The system exhibits a threshold below which both populations become extinct. The model undergoes a Hopf bifurcation when the unique interior steady state loses its stability. We also examine the notion of Turing instability of the homogeneous interior steady state when both populations can disperse randomly. We prove that the solutions are square integrable and the system does not exhibit Turing instability. It is demonstrated via numerical simulations that the system may exhibit spatial nonhomogeneous steady states.

Intellectual property rights and foreign direct investment: A welfare analysis

This paper examines how intellectual property rights (IPR) protection affects innovation and foreign direct investment (FDI) using a North–South quality-ladder model incorporating the exogenous and costless imitation of technology and subsidy policies for both R&D and FDI. We show that for the interior steady state to be stable, either R&D or FDI subsidy rates must be positive. Our findings also indicate that strengthening IPR protection promotes both innovation and FDI. Moreover, a strengthening of IPR protection can also improve welfare if the initial IPR protection in the South is weak and the R&D subsidy rate is not too high.

A Dynamic Analysis of Schelling’s Binary Corruption Model: A Competitive Equilibrium Approach

Schelling (in Micromotives and Macrobehavior, Norton, New York, 1978) suggested a simple binary choice model to explain the variation of corruption levels across societies. His basic idea was that the expected profitability of engaging in corruption depends on its prevalence. The key result of the so-called Schelling diagram is the existence of multiple equilibria and a tipping point. The present paper puts Schelling's essentially static approach into an intertemporal setting. We show how the existence of an unstable interior steady state leads to thresholds such that history alone or history in addition to expectations (or coordination) is necessary to determine the long-run outcome. In contrast to the related literature, which classifies these two cases according to whether the unstable equilibrium is a node or a focus, the actual differentiation is more subtle because even a node can lead to an overlap of solution paths such that the initial conditions alone are insufficient to uniquely determine the competitive equilibrium. Another insight is that a (transiently) cycling competitive equilibrium can dominate the direct and monotonic route to a steady state, even if the direct route is feasible.

Multiple attractors in a Leslie–Gower competition system with Allee effects

We investigate asymptotic dynamics of the classical Leslie–Gower competition model when both competing populations are subject to Allee effects. The system may possess four interior steady states. It is proved that for certain parameter regimes both competing populations may either go extinct, coexist or one population drives the other population to extinction depending on initial conditions.

A model for the impact of contaminants on fish population dynamics

Mathematical models have been widely applied to perform chemical risk assessments on biological populations for a variety of ecotoxicological processes. In this paper, by introducing a dose-dependent mortality rate function, we formulate a toxin-dependent aquatic population model that integrates mortality as toxin effect in addition to considering the effects of toxin on growth and recruitment. The model describes the direct effect of toxin on population by treating the concentration of toxin in the environment as a parameter. The model is more convenient to connect with data than traditional differential equation models that describe the interaction between toxin and population. We analyze the positive invariant region and the stability of boundary and interior steady states. The model is connected to experimental data via model parametrization. In particular, we consider the toxic effects of mercury on rainbow trout (Oncorhynchus mykiss) and obtain an appropriate range for each model parameter. The parameter estimates are then used to illustrate the long-time behavior of the population under investigation. The numerical results provide threshold values of toxin concentration in the environment to keep the population from extirpation. The findings are consistent with surface water quality guidelines. It may be appropriate to apply our model to other species and other chemicals of interest to consider guideline development.

Migration modelling in the New Economic Geography

It is our aim to study some of the migration laws utilized in Economic Geography, their dynamic properties and how their long-run predictions and stability change with the specificities of the economic models under consideration. After a thorough description of Fujita and Thisse (2002), we introduce a different migration law à la Krugman (1991a). Although individuals do not foresee price changes, the steady state outcome does not vary qualitatively: the unique steady state is a symmetric distribution of skilled labour across regions. We can prove that this interior steady state is asymptotically stable, which represents a net improvement in the dynamic analysis of the long run with respect to Fujita and Thisse. When we model the economy using the Romer (1990) model applied to two regions and allowing for inter-regional skilled migration, then there exists a solution path that converges to an asymmetric steady state. In effect, the new steady state depends on technology, fixed costs, knowledge spillovers and transportation costs.

On the Lotka–Volterra competition system with Allee effects

We study asymptotic dynamics of the classical Lotka–Volterra competition model when both competing populations are subject to Allee effects. The resulting system can have up to four interior steady states. In such case, it is proved that both competing populations may either go extinct, coexist, or one population drives the other population to extinction depending on initial conditions.

Analysis of a model for the effects of an external toxin on anaerobic digestion

Anaerobic digestion has been modeled as a two-stage process using coupled chemostat models with non-monotone growth functions, [9]. This study incorporates the effects of an external toxin. After reducing the model to a 3-dimensional system, global stability of boundary and interior equilibria is proved using differential inequalities and comparisons to the corresponding toxin-free model. Conditions are given under which the behavior of the toxin-free model is preserved. Introduction of the toxin results in additional patterns such as bistabilities of coexistence steady states or of a periodic orbit and an interior steady state.

An analysis of the world's environment and population dynamics with varying carrying capacity, concerns and skepticism

Because of the open-access nature of the environment we consider an ad hoc adjustment of people's environmental footprint to the quality of the environment. The adjustment is motivated by environmental concern, but hindered by skepticism about announced changes in the state of the environment. Changes in the quality of the environment affect Earth's carrying capacity. By expanding the Lotka–Volterra predator–prey model to include these features we show that, despite skepticism, the environment does not deteriorate to a state in which humans cannot exist. We also show that in the ideal case of no skepticism, the interplay between the non-optimally changing environmental concerns and carrying capacity leads the world's environment and human population to a unique interior steady state along an oscillating course. These results require no further technological, social or international progress.

AGGLOMERATION AND GROWTH WITH ENDOGENOUS EXPENDITURE SHARES*

ABSTRACT We develop a New Economic Geography and Growth model which, by using a CES utility function in the second‐stage optimization problem, allows for expenditure shares in industrial goods to be endogenously determined. The implications of our generalization are quite relevant. In particular, we obtain the following novel results: (1) two additional nonsymmetric interior steady states emerge for some intermediate values of trade costs. These steady‐states are stable if the industrial and the traditional goods are either good or very poor substitutes, while they are unstable for intermediate (yet lower than one) values of the intersectoral elasticity of substitution. In the latter case, the model displays three interior steady states—the symmetric and the core‐periphery allocations—which are stable at the same time; (2) agglomeration processes may always take place, whatever the degree of market integration, provided that the traditional and the industrial goods are sufficiently good substitutes; (3) the regional rate of growth is affected by the interregional allocation of economic activities even in the absence of localized spillovers, so that geography always matters for growth and (4) the regional rate of growth is affected by the degree of market openness: in particular, depending on whether the traditional and the industrial goods are good or poor substitutes, economic integration may be, respectively, growth‐enhancing or growth‐detrimental.

Cannibalism in discrete-time predator–prey systems

In this study, we propose and investigate a two-stage population model with cannibalism. It is shown that cannibalism can destabilize and lower the magnitude of the interior steady state. However, it is proved that cannibalism has no effect on the persistence of the population. Based on this model, we study two systems of predator–prey interactions where the prey population is cannibalistic. A sufficient condition based on the nontrivial boundary steady state for which both populations can coexist is derived. It is found via numerical simulations that introduction of the predator population may either stabilize or destabilize the prey dynamics, depending on cannibalism coefficients and other vital parameters.

Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics

This note shows, by means of two simple, three-strategy games, the existence of stable periodic orbits and of multiple, interior steady states in a smooth version of the Best-Response Dynamics, the Logit Dynamics. The main finding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, occur under the Logit Dynamics, even for three-strategy games. We also show that the Logit Dynamics displays another bifurcation which cannot occur under the Replicator Dynamics: the fold bifurcation, with non-monotonic creation and disappearance of steady states.

RANDOM DISPERSAL IN A PREDATOR-PREY-PARASITE SYSTEM

We propose predator-prey-parasite models to study the effects of parasites upon the predator-prey interaction. There are two parameters that are used to model the effectiveness of the infected prey and infected predator. For the spatial homogeneous system, the asymptotic dynamics depend on the reproductive number of the parasite. The parasite can persist in the population if this reproductive number is larger than one. Numerical simulations suggest that less competitiveness of the infected predator can make the predator-prey interaction less stable. The dynamics may move from coexisting steady state to oscillations. For the spatial heterogeneous system, diffusion may destabilize the homogeneous interior steady state for a particular set of diffusion coefficients. However, both systems do not exhibit complicated dynamical behavior.

Sustainable capitalism: Full-employment flexicurity growth with real wage rigidities

Abstract In this paper we present a model of flexicurity capitalism that exhibits a second labor market with the government as an employer of first resort, where all workers not employed by firms in the private sector find meaningful employment. We show that the model exhibits a unique interior steady state which is asymptotically stable under real wage adjustment dynamics of the type considered in Blanchard and Katz (1999), and under a type of Okun’s Law that links the level of utilization of firms to their hiring and firing decision. The introduction of a company pension fund can be shown to contribute to the viability of the analyzed economic system. However, when credit is incorporated in the model, in place of savings-driven supply side fluctuations in economic activity, investment-driven demand side business cycle fluctuations (of a probably much more volatile type) can take place.

The dynamics of environmentalism and the environment

We study the relationship between environmental preferences and the environment. Preferences are transmitted intergenerationally and through social interactions, where we assume that agents are more likely to adopt environmental preferences the larger the amount of pollution. In the basic setting we find that both converge non-monotonically towards an interior steady state. When including technical change we notice that there will be no change in the steady state level of the environment unless technical change is sufficiently strong, which stands in stark contrast to the literature. Upon introducing environmental laws we find that these may lead to a virtually pollution-free environment. This happens if environmental laws are implemented when public support is strong enough.

Models of plant quality and larch budmoth interaction

We present and investigate two discrete-time models of leaf quality and larch budmoth interaction. In particular, existence and stability of equilibria are studied. For one model, the moth population may grow unboundedly large and the interior steady state is always locally asymptotically stable when it exists. A Hopf bifurcation may occur for the other model when the interior steady state becomes unstable.

Happiness due to Consumption and its Increases, Wealth and Status

This paper departs from the standard open-economy Ramsey model and introduces additional concerns for wealth, status and Easterlin's (2001) hypothesis that consumption changes, in particular increases, are important and not only the level. These extensions induce first of all interior steady states, which are lacking in the standard model, multiple steady states (separated by thresholds leading to history dependence) and limit cycles. The existence of cyclical consumption patterns could provide a so far ignored source for real business cycles. Surprisingly, introducing status conferred by private wealth or conspicuous consumption has no effect despite the involved externalities (the outcomes are observationally equivalent) as long as the social influence associated with the status externalities remains moderate.

A Gravito‐electrostatic Sheath Model for Surface Origin of Subsonic Solar Wind Plasma

Parker's model prediction of an unbounded solution for the supersonic solar wind plasma (SWP) requires a subsonic origin at the solar surface. The transition to supersonic SWP flow is well understood by analogy with the de Laval nozzle. The basic physics, however, of the self-consistent solar surface emission of subsonic SWP is still not very clear. We propose a theoretical model based on a gravito-electrostatic plasma sheath (GES) to investigate the surface emission mechanism of the quasi-neutral SWP. The basic equations for the interior steady state description are solved numerically as an initial-value problem under spherical geometry. A bounded solution for the solar self-gravity distribution is found to exist, meaning that a global quasi-hydrostatic equilibrium is formed at some distance from the heliocentric origin. This defines the solar surface boundary in our model, which lies at radius ξ ~ 3.5 (in units of the Jeans length). A minimum speed of ~3.0 cm s-1, corresponding to Mach number ~10-7, is obtained at this location. Consequently, a subsonic origin of the solar interior plasma flowing radially outward becomes a physical reality. Our model requires a mean electron temperature Te ~ 107 K (=1 keV) for the numerical results to match the standard solar values. The boundary is found to be negatively biased, with a normalized value of the electrostatic potential θs ~ -1 (=1 kV). It is conjectured that our model provides an interesting new physical basis for understanding the collective dynamical coupling processes of solar interior and exterior (heliosphere) regions through local, as well as global, modes of GES-induced collective waves, instabilities, and oscillations.

Lateral interactions in the adsorption layer and critical phenomena in the Langmuir-Hinshelwood reaction

The report deals with the effect of the lateral interactions between adsorbed species on the steady state multiplicity area and on the possible rate self-oscillations for a reaction proceeding via the Langmuir-Hinshelwood mechanism. The adsorption layer is modeled as a lattice gas on a square lattice. For irreversible adsorption, the maximum possible number of interior steady states is at least ten. Reaction rate self-oscillations resulting from Andronov-Hopf bifurcation take place for some lateral interaction energy sets. These self-oscillations are associated with a checkerboard-type ordered dense phase existing owing to the attraction between different adsorbed species.