Unique Steady State Research Articles

Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior

• A predator-prey model with diffusion and herd behavior is investigated. • The local and global stability are determined. • The steady-state bifurcations from both simple and double eigenvalues are studied. • Numerical simulations illustrate and supplement our theoretical analysis. This paper is concerned with a diffusive predator-prey model with herd behavior. The local and global stability of the unique homogeneous positive steady state U * is obtained. Treating the conversion or consumption rate γ as the bifurcation parameter, the steady-state bifurcations both from simple and double eigenvalues are studied near U *. The techniques include the Lyapunov function, the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.

Cross-diffusion-driven Turing instability and weakly nonlinear analysis of Turing patterns in a uni-directional consumer-resource system

Spatiotemporal patterns driven by cross-diffusion of a uni-directional consumer-resource (C-R) system with Holling-II type functional response are investigated in this paper. The existence of a unique positive steady state of the considered system is studied first. The linear stability analysis shows that the cross-diffusion is the key mechanism for the formation of spatiotemporal patterns through Turing bifurcation. We choose the cross-diffusion coefficient as bifurcation parameter and discuss three different types of Turing bifurcations, corresponding to simple, double non-resonant and double resonant cases. Based on weakly nonlinear analysis with the multiple scale method and the adjoint system theory, we derive the amplitude equations of the Turing patterns near the Turing bifurcation point and obtain the analytical approximation solutions of the patterns for each case. Specially, some qualitative results of amplitude equations of the resonant case are given in detail. Finally, numerical simulations are performed to illustrate the weakly nonlinear theoretical predictions and through these simulations some patterns (single mode pattern, mixed-mode pattern, super-squares pattern, roll pattern, hexagonal pattern) are found. Simultaneously, numerical simulations show that the resource supplying rate has an important impact on the direction of Turing bifurcation.

Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model

We investigate a discrete consumer-resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. Using intrinsic growth rate of the consumer population as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability.

A stoichiometric organic matter decomposition model in a chemostat culture.

Biodegradation, the disintegration of organic matter by microorganism, is essential for the cycling of environmental organic matter. Understanding and predicting the dynamics of this biodegradation have increasingly gained attention from the industries and government regulators. Since changes in environmental organic matter are strenuous to measure, mathematical models are essential in understanding and predicting the dynamics of organic matters. Empirical evidence suggests that grazers' preying activity on microorganism helps to facilitate biodegradation. In this paper, we formulate and investigate a stoichiometry-based organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. We determine the criteria for the uniform persistence and extinction of the species and chemicals. Our results show that (1) if at the unique internal steady state, the per capita growth rate of bacteria is greater than the sum of the bacteria's death and dilution rates, then the bacteria will persist uniformly; (2) if in addition to this, (a) the grazers' per capita growth rate is greater than the sum of the dilution rate and grazers' death rate, and (b) the death rate of bacteria is less than some threshold, then the grazers will persist uniformly. These conditions can be achieved simultaneously if there are sufficient resources in the feed bottle. As opposed to the microcosm decomposition models' results, in a chemostat culture, chemicals always persist. Besides the transcritical bifurcation observed in microcosm models, our chemostat model exhibits Hopf bifurcation and Rosenzweig's paradox of enrichment phenomenon. Our sensitivity analysis suggests that the most effective way to facilitate degradation is to decrease the dilution rate.

Metastability for nonlinear convection–diffusion equations

We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity $$\varepsilon $$ . In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple $$(\xi ,v)$$ , where $$\xi $$ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in $$\varepsilon $$ . As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion.

An evolutive financial market model with animal spirits: imitation and endogenous beliefs

We propose a financial market model with optimistic and pessimistic fundamentalists who, respectively, overestimate and underestimate the true fundamental value due to ambiguity in the stock market. We assume that agents form their beliefs about the fundamental value through an imitative process, considering the relative ability shown by optimists and pessimists in guessing the realized stock price. We also introduce an endogenous switching mechanism, allowing agents to switch to the other group of speculators if they performed better in terms of relative profits. Moreover, the stock price is determined by a nonlinear mechanism. We study, via analytical and numerical tools, the stability of the unique steady state, its bifurcations and the emergence of complex behaviors, with possible multistability phenomena. To quantify the global propensity to optimism/pessimism of the market, we introduce an index, depending on pessimists’ and optimists’ beliefs and shares, thanks to which we are able to show that the occurrence of the waves of optimism and pessimism are due to the joint effect of imitation and switching mechanism. Finally, we perform a statistical analysis of a stochastically perturbed version of the model, which high lights fat tails and excess volatility in the returns distributions, as well as bubbles and crashes for stock prices, in agreement with the empirical literature.

Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal

We consider an integro-PDE model from evolutionary biology. The solution \(u_\epsilon (x,\alpha )\) is structured by two variables \(x\in D \subset {\mathbb {R}}^k\) and \(\alpha \in (\underline{\alpha },{\overline{\alpha }})\subset \subset {\mathbb {R}}_+\). The diffusion coefficient in the x direction depends on \(\alpha \) and the diffusion coefficient in the \(\alpha \) direction is a constant \(\epsilon ^2\). A special feature of this model is the appearance of the integral \({\hat{u}}_\epsilon (x)\) of the solution in the \(\alpha \) variable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Dirac-concentration in one of the variables yet remains regular in the other variables was proved independently by Lam and Lou (J Funct Anal 272:1755–1790, 2017) and by Perthame and Souganidis (Math Model Nat Phenom 11:154–166, 2016). In this paper, we tackle the long-time dynamics of solutions. When the environment function is non-constant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is a constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by \(\alpha \). As with the competition model for three or more species, the integro-PDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability.

Qualitative Analysis of a Predator-prey Model with Weak Allee Effect

In the past decades, the research of reaction-diffusion equations has been one of the dominant themes due to its universal existence and importance. The applications of reaction-diffusion equations play an active role in physics, chemistry, biological populations and medicine. In various ecological systems, the predator- prey model is an important branch. In this paper, a predator-prey reaction-diffusion system with weak Allee effect and homogeneous Neumann boundary condition is considered. First, it is proved that the unique positive constant steady state is stable, and the methods to study local stability are based on local linearization techniques. Second, a prior-estimate of positive steady state is given by Harnack inequality and Maximum principle. Also, we found that system has no non-constant positive solution when the diffusion coefficients meeting certain conditions. Finally, we establish the existence of positive solutions to the system mainly using the fixed point index theory in cone. Through the research, we find out the conditions for the existence of positive steady state, which provide theoretical basis for the improvement of yield predators.

Non-constant Positive Steady State of the Predator-Prey-Mutualist System with Diffusion

The predator-prey model is one of the most famous models in ecology. The research of the predator- prey model has been one of the dominant themes due to its universal existence and importance. And the applications of the predator-prey model play an active role in physics, chemistry, biological populations and medicine. In this paper, a reaction-diffusion predator-prey-mutualist system with the Holling II functional response and homogeneous Neumann boundary condition is considered. First, it is proved that the unique positive constant steady state is stable. The methods to study local stability are based on local linearization techniques. Second, a prior-estimate of positive steady state is given by Harnack inequality and Maximum principle. Finally, the non-existence and the existence of non-constant positive steady state are studied by the degree theorem and bifurcation technique. Through the research, we find out the conditions for the existence of positive steady state, which provide theoretical basis for the improvement of yield predators.

Endogenous Cycles in a Simple Model with Lagged Externalities of Human Capital

This paper presents a simple model of endogenous cycles. In the model, human capital has a positive externality on labor productivity but it takes long time before the externality comes into effect. In addition, individuals have preferences with a subsistence consumption level. The lagged externality of human capital and the requirement of subsistence consumption make human capital investments countercyclical, which serves as the driving force for generating cycles endogenously. The model analysis shows that the dynamics of human capital may converge to a unique steady state through cyclical fluctuations or may converge to a period-2 cycle.

Some Global Results for the Degn–Harrison System with Diffusion

This paper considers the Degn–Harrison reaction–diffusion system subject to homogeneous Neumann boundary conditions in a smooth and bounded domain. Using the presence of contracting rectangles and the method of Lyapunov, we establish sufficient conditions for the global asymptotic stability of the unique constant steady state.

Numerical integration of an erythropoiesis model with explicit growth factor dynamics

Erythropoiesis, the red blood cell production process, involves interactions between cell populations with different differentiation states, mainly immature progenitor cells and mature erythrocytes, and growth factors such as erythropoietin and glucocorticoids, known to respectively inhibit cell apoptosis, stimulate proliferation and differentiation, and stimulate self-renewal. The feedback regulation of this process allows a very fast and efficient recovery in the case of a severe anemia. We consider an age-structured model of red blood cell production accounting for these feedback regulations and the dynamics of growth factors. We theoretically show the existence of a unique positive steady state for the model and we propose a numerical method to obtain an approximation to its solution. Experiments are reported to show numerically, on one hand, the optimal convergence order of the numerical scheme and, on the other hand, a fine approximation to real experimental data, with a suitable selection of the parameters involved.

Asymptotic dynamics in a two-species chemotaxis model with non-local terms

In this study, we consider an extended attraction two species chemotaxis system of parabolic-parabolic-elliptic type with nonlocal terms under homogeneous Neuman boundary conditions in a bounded domain $Ω \subset \mathbb{R}^n(n≥1)$ with smooth boundary. We first prove the global existence of non-negative classical solutions for various explicit parameter regions. Next, under some further explicit conditions on the coefficients and on the chemotaxis sensitivities, we show that the system has a unique positive constant steady state solution which is globally asymptotically stable. Finally, we also find some explicit conditions on the coefficient and on the chemotaxis sensitivities for which the phenomenon of competitive exclusion occurs in the sense that as time goes to infinity, one of the species dies out and the other reaches its carrying capacity. The method of eventual comparison is used to study the asymptotic behavior.

Dynamical behavior for a Lotka-Volterra weak competition system in advective homogeneous environment

We consider a two-species Lotka-Volterra weak competition model in a one-dimensional advective homogeneous environment, where individuals are exposed to unidirectional flow. It is assumed that two species have the same population dynamics but different diffusion rates, advection rates and intensities of competition. We study the following useful scenarios: (1) if one species disperses by random diffusion only and the other assumes both random and unidirectional movements, two species will coexist; (2) if two species are drifting along the different direction, two species will coexist; (3) if the intensities of inter-specific competition are small enough, two species will coexist; (4) if the intensities of inter-specific competition are close to 1, the competitive exclusion principle holds. These results provide a new mechanism for the coexistence of competing species. Finally, we apply a perturbation argument to illustrate that two species will converge to the unique coexistence steady state.

Poor People's Beliefs and the Dynamics of Clientelism

Political science literature on clientelism has tended to focus on vote-buying, viewed primarily from the perspective of parties/brokers. The motives that drive clients to engage in clientelism and the different forms of clientelism that result remain relatively unexplored. This paper proposes a formal model of clientelism that focuses primarily on the client’s side, setting clientelism against the possibility of supporting a redistributive platform and incorporating insights from social psychology theories. We consider two key perceptions: political efficacy and inequality legitimation, and make them endogenous. In the model, perceptions of inefficacy and the legitimacy of inequality reinforce each other leading to multiple steady states. One of these resembles a “traditional” form of clientelism with disempowered clients that legitimize social inequalities. Community connectivity breaks this reinforcement mechanism and leads to a unique steady state, where clientelism and programmatic redistribution coexist, that resembles “modern” clientelism characterized by vote-buying.

Stability analysis of a chemostat model with maintenance energy

In this paper, we dedicate ourselves to the study of a diffusive model for unstirred membrane reactors with maintenance energy and subject to a homogeneous Neumann boundary condition. It shows that the unique constant steady state is globally asymptotically stable when it exists. This result further implies the non-existence of any spatial patterns.

On the global asymptotic stability of solutions to a generalised Lengyel–Epstein system

This paper presents a Lengyel–Epstein system with a general reaction term. It is concerned with the global dynamics and asymptotic stability of the solutions. Sufficient conditions for the stability of the system’s unique steady state solution are derived and confirmed through numerical analysis in Matlab.

Monopoly pricing and diffusion of social network goods

I present a model of dynamic pricing and diffusion of a network good sold by a monopolist. In the model, the network good is a subscription social network good. This means that in each period, each consumer has to pay a subscription price to use the good, and the utility derived from subscribing to the good increases as more of her neighboring consumers subscribe. Consumers myopically optimize their subscription decisions, and the monopolist chooses a sequence of subscription prices that maximizes his discounted sum of per-period profits. Three main results emerge. First, I characterize a unique steady state of the monopoly market. Second, I find that optimal sequences of subscription prices oscillate around the subscription price at the steady state. Third, I analyze how changes in the monopolist's discount factor and the density of the social network affect the subscription price, subscription rate, and deadweight loss at the steady state.

Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response.

In any reaction-diffusion system of predator-prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To save the endangered prey species, some human interference is useful, such as creating a protection zone where the prey could cross the boundary freely but the predator is prohibited from entering. This paper studies the existence of positive steady states to a predator-prey model with reaction-diffusion terms, Beddington-DeAngelis type functional response and non-flux boundary conditions. It is shown that there is a threshold value [Formula: see text] which characterizes the refuge ability of prey such that the positivity of prey population can be ensured if either the prey's birth rate satisfies [Formula: see text] (no matter how large the predator's growth rate is) or the predator's growth rate satisfies [Formula: see text], while a protection zone [Formula: see text] is necessary for such positive solutions if [Formula: see text] with [Formula: see text] properly large. The more interesting finding is that there is another threshold value [Formula: see text], such that the positive solutions do exist for all [Formula: see text]. Letting [Formula: see text], we get the third threshold value [Formula: see text] such that if [Formula: see text], prey species could survive no matter how large the predator's growth rate is. In addition, we get the fourth threshold value [Formula: see text] for negative [Formula: see text] such that the system admits positive steady states if and only if [Formula: see text]. All these results match well with the mechanistic derivation for the B-D type functional response recently given by Geritz and Gyllenberg (J Theoret Biol 314:106-108, 2012). Finally, we obtain the uniqueness of positive steady states for [Formula: see text] properly large, as well as the asymptotic behavior of the unique positive steady state as [Formula: see text].

A note on the global properties of an age-structured viral dynamic model with multiple target cell populations.

Some viruses can infect different classes of cells. The age of infection can affect the dynamics of infected cells and viral production. Here we develop a viral dynamic model with the age of infection and multiple target cell populations. Using the methods of semigroup and Lyapunov function, we study the global asymptotic property of the steady states of the model. The results show that when the basic reproductive number falls below 1, the infection is predicted to die out. When the basic reproductive number exceeds 1, there exists a unique infected steady state which is globally asymptotically stable. The model can be extended to study virus dynamics with multiple compartments or coinfection by multiple types of viruses. We also show that under some scenarios the age-structured model can be reduced to an ordinary differential equation system with or without time delays.