Semi-trivial Solutions Research Articles

Existence and multiplicity of solutions for a class of (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Kirchhoff system with combined nonlinearities on graphs

By using the well-known mountain-pass theorem and Ekeland’s variational principle, we prove that there exist at least two fully nontrivial solutions for a (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Kirchhoff elliptic system with the Dirichlet boundary conditions and perturbation terms on a locally weighted and connected finite graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G=(V,E)$\\end{document}. We also present a necessary condition of the existence of semitrivial solutions for the system. Moreover, by using Ekeland’s variational principle and Clark’s Theorem, respectively, we prove that the system has at least one or multiple semitrivial solutions when the perturbation terms satisfy different assumptions. Finally, we present a nonexistence result of solutions.

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

Dynamics of a size-structured predator–prey model with chemotaxis mechanism

This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect.

Positive steady states in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity

– In this paper, we consider the following stationary two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity (0.1)−d1Δu=λu−u2−buv,x∈Ω,−div[d(w)∇v+χ(w)v∇w]=μv−v2−cuv,x∈Ω,−d3Δw=−κw+τu,x∈Ω,u=v=w=0,x∈∂Ωin a bounded smooth domain Ω⊂RN(N≥1), where λ,μ,b,c,d1,d3,κ,τ are positive constants, and (d(w),χ(w))∈[C1[0,∞)]2 with d(w),χ(w)>0 for all w≥0. Since there does not exist an immediate change variable that transforms (0.1) into a semilinear system when (0.1) is considered with d(w),α(w) being arbitrary functions in w, this makes the analysis of system (0.1) much more difficult. By the W2,p-estimate and the global bifurcation method, we obtain a bounded connected set of positive steady states joining the semitrivial solution of the form (u,0,w) with u,w>0 and the semitrivial solution of the form (0,v,0) with v>0, which provides the sufficient condition for the existence of positive steady states. Combining the eigenvalue theory, homogenization technique and various elliptic estimates, we also derive some sufficient conditions for the nonexistence of positive steady states.

Coexistence of two strongly competitive species in a reaction–advection–diffusion system

The main focus of this article is to investigate the behavior of two strongly competitive species in a spatially heterogeneous environment using a Lotka–Volterra-type reaction–advection–diffusion model. The model assumes that one species diffuses at a constant rate, while the other species moves toward a more favorable environment through a combination of constant diffusion and directional movement. The study finds that no stable coexistence can be guaranteed when both species disperse randomly. In contrast, stable coexistence between the two species is possible when one of the species exhibits advection–diffusion. The study also reveals the existence of unstable coexistence imposed by bistability in a strongly competitive system, regardless of the diffusion type. The results are obtained by analyzing the stability of semitrivial solutions. The study concludes that the species moving toward a better environment has a competitive advantage, allowing them to survive even when their population density is initially low. Finally, the study identifies the unique globally asymptotically stable coexistence steady states of the system at high advection rates, particularly for relatively moderate interspecific competition parameters in species with directional movement. These findings underscore the crucial role of directed movement and interspecific competition coefficients in shaping the dynamics and coexistence of strongly competing species.

Effect of density-dependent diffusion on a diffusive predator–prey model in spatially heterogeneous environment

The paper presents a class of predator–prey model with density-dependent diffusion in the spatially heterogeneous environment. We first provide the global existence and boundedness of the solution for the model. Then, by taking a variable transformation, the difficulty brought by the cross-diffusion can be overcome, and the existence, stability and local bifurcation of semi-trivial steady-state solutions for the equivalent system are further studied. Finally, the existence of positive solutions of the system is also given by using the Leray–Schauder degree theory and the method of principle eigenvalue, especially for the limit cases when the diffusion coefficient tends to zero or infinite.

On a diffusive epidemic model with the tendency to move away from the infectious diseases

This paper is purported to investigate an epidemic system with the tendency of the susceptible to move away from the infectious diseases in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without cross-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are considered for this epidemic system. The results show that there exists spatiotemporal pattern formation when the tendency of susceptible individuals is strong to move away from the infectious diseases with 1<R0<1+min⁡{dSd,βσ1Aσ2}, while there does not exist spatiotemporal pattern formation when the self pressure is big enough with a weak tendency of the susceptible individuals to keep away from the infectious individual. Moreover, there exists positive solution bifurcation from the semi-trivial solution when the basic reproduction number is equal to one, which implies that endemic disease exists locally.

Bifurcations and chaos control in a discrete Rosenzweig-Macarthur prey-predator model.

In this paper, we explore the local dynamics, chaos, and bifurcations of a discrete Rosenzweig-Macarthur prey-predator model. More specifically, we explore local dynamical characteristics at equilibrium solutions of the discrete model. The existence of bifurcations at equilibrium solutions is also studied, and that at semitrivial and trivial equilibrium solutions, the model does not undergo flip bifurcation, but at positive equilibrium solutions, it undergoes flip and Neimark-Sacker bifurcations when parameters go through certain curves. Fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by the center manifold theorem and bifurcation theory. We also studied chaos by the feedback control method. The theoretical results are confirmed numerically.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

Global dynamics of a competitive system with seasonal succession and different harvesting strategies

In this paper, we investigate a two-species competition model with seasonal succession and different harvesting strategies. We aim to comprehend the impact of the proportion of good season on species competition outcomes and harvesting practices. For the case of weak harvest, we establish proportion thresholds for the good season for the species. Based on these threshold settings, we determine conditions for the existence and uniqueness of two semi-trivial periodic solutions and the existence, uniqueness and stability of positive periodic solutions under certain conditions. Numerical simulations and brief discussions are presented to validate our theoretical findings.

Dynamical analysis of a two-dimensional discrete predator–prey model

In this paper, we explore the existence of periodic solutions, local dynamics at equilibrium solutions, chaos and bifurcations of a discrete prey–predator model with Michaelis–Menten type functional response. More specifically, we explore local dynamical characteristics at equilibrium solutions, and existence of periodic solutions for the under consideration model. It is also studied the existence of bifurcations at equilibrium solutions, and investigated that at semitrivial and trivial equilibrium solutions model does not undergo flip bifurcation, but at positive equilibrium solution it undergoes flip and Neimark–Sacker bifurcations when parameters goes through the certain curves. It is also investigated that fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by center manifold theorem and bifurcation theory. We also studied chaos by feedback control method. The theoretical results are confirmed numerically. Furthermore, we use 0–1 chaos test in order to quantify whether chaos exists in the model or not.

Dynamics of a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment

In this paper, we study a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment. The existence and local asymptotic stability of spatially non-homogeneous semi-trivial steady-state solutions and spatially non-homogeneous positive steady-state solutions are obtained by the implicit function theorem and spectral analysis. We show that three scenarios can occur: if random diffusion rates of two species are sufficiently large, both species go extinct; if random diffusion rates of two species are relatively small, two competing species coexist; if one species has a large random diffusion rate and the other has a small random diffusion rate, the species with a large random diffusion rate are driven to extinction. An interesting finding is that a large delay does not lead to Hopf bifurcation at the spatially non-homogeneous steady-state solution, but makes this steady-state solution approach zero. We numerically demonstrate the effects of spatial heterogeneity on spatiotemporal dynamics.

Coexistence solutions for a Lotka–Volterra competition model with density-dependent motion

This paper is concerned with a Lotka–Volterra model with density-dependent motion. Under homogeneous Dirichlet boundary conditions, which is different from the previous article (Liu and Guo, 2021), the corresponding steady-state problem is discussed. Stability properties of the trivial and semitrivial solutions are determined completely. The sufficient conditions for the existence of coexistence solutions are also obtained by using the theory of fixed point index in positive cones. Finally, the limiting behavior of coexistence solutions is studied as some certain parameter tends to infinity.

Predator-prey models with prey-dependent diffusion on predators in spatially heterogeneous habitat

This article considers a predator–prey system with a certain type of prey-dependent diffusion for predators where the source of prey population depends on location in a habitat with spatial heterogeneity distributed within a bounded domain. In particular, it is assumed that the spread rate of predators can change depending on the satisfaction of predators according to the amount of available prey in the vicinity of predators in the habitat. First, how prey-dependent dispersal sensitively affects the migration mechanism of predators is examined. More precisely, it is shown that predators via such prey-dependent diffusion can invade a habitat region by investigating stability analysis of the semitrivial solution of the system where the predator is absent. Additionally, the existence and uniqueness of a positive steady state are studied using the fixed point index theory in a positive cone in a Banach space. The coexistence state is found to be unique if the diffusion rate of the prey is above a certain threshold and the average of the resource function of the prey is within a specific range represented through the equilibrium value of the prey.

Time-periodic traveling waves for a periodic Lotka–Volterra competition system with nonlocal dispersal

This paper is concerned with the time-periodic traveling waves for a periodic Lotka–Volterra competition system with nonlocal dispersal. Under the strong competition assumption, we first prove the existence of periodic traveling fronts connecting two stable semi-trivial periodic solutions. Then we establish the monotonicity and global exponential stability of smooth periodic traveling fronts. Since the standard strong comparison theorem does not hold for the Lotka–Volterra competition system, the monotonicity and global exponential stability are proved by establishing some new strong comparison theorems and constructing some auxiliary monotone systems. The Liapunov stability of the periodic traveling fronts is also proved.

Global dynamics of a Lotka-Volterra competition-diffusion system with nonlinear boundary conditions

In this paper, a Lotka-Volterra competition-diffusion system subject to nonlinear boundary conditions is considered with an aim to understand the nonlinear balance between two competing nonlinear mechanisms (namely, interior reaction and boundary flux). A complete picture on the global dynamics (including the existence, nonexistence, global stability, and steady-state bifurcation of semi-trivial positive steady-state solutions) has been studied in terms of inter-specific competition coefficients, the growth rate functions, and boundary reaction functions. Our results extend largely the relevant results in the existing literature on Lotka-Volterra competition-diffusion systems, even those with homogeneous boundary conditions.

Bifurcation Analysis of a Discrete‐Time Chemostat Model

In this article, a discretized two‐dimensional chemostat model is investigated. By using the algebraic technique, it is proved that the discrete chemostat model has semitrivial equilibrium solution for all involved parametric values, but it has an interior equilibrium solution under certain parametric conditions. We have studied the local dynamics with topological classifications about the semitrivial and interior equilibrium solution on the basis of the theory of the discrete dynamical system. By bifurcation theory, we studied the existence of bifurcations for the understudied discrete chemostat model and proved that at a semitrivial equilibrium solution, it did not undergo flip bifurcation, but it undergoes flip bifurcation at interior equilibrium solution, and no other bifurcation occurs at this point. It has also explored the existence of periodic points and chaos control by the state feedback method. Finally, with the help of MATLAB software, flip bifurcation diagrams and corresponding maximum Lyapunov exponents were also presented. These numerical simulations not only show the correctness of theoretical results but also reveal the complex dynamics such as the orbits of period −7, −8, −13, and −16.

Pattern formation in a ratio-dependent predator-prey model with cross diffusion

&lt;abstract&gt;&lt;p&gt;This paper is focused on a ratio-dependent predator-prey model with cross-diffusion of quasilinear fractional type. By applying the theory of local bifurcation, it can be proved that there exists a positive non-constant steady state emanating from its semi-trivial solution of this problem. Further based on the spectral analysis, such bifurcating steady state is shown to be asymptotically stable when the cross diffusion rate is near some critical value. Finally, numerical simulations and ecological interpretations of our results are presented in the discussion section.&lt;/p&gt;&lt;/abstract&gt;

Bifurcation and Chaos in a Phytoplankton–Zooplankton Model with Holling Type-II Response and Toxicity

Phytoplankton and zooplankton’s interconnection coordinate in various dynamical processes that occur in ecological population and make it a fascinating subject matter to explore. Time delay is an additional factor that plays an imperative part in the dynamical frameworks. For a stable aquatic ecosystem, the growth of both zooplankton and phytoplankton should be in steady state but in previous eras, there have been a universal explosion in destructive plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So in this paper, we studied dynamical characteristics of a discrete-time phytoplankton–zooplankton model with Holling type-II predational functional response and toxic substance distribution. More specifically, we studied local stability with topological classification at equilibrium solutions, periodic points and possible bifurcation analysis of the model under consideration. It is proved that at trivial and semitrivial equilibrium solutions the phytoplankton–zooplankton model does not undergo flip bifurcation but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution. It is also proved that the interior equilibrium solution model does not undergo the fold bifurcation. We also studied chaos in the sense of feedback control strategy. Finally, extensive numerical simulations are provided to demonstrate theoretical results.

Dynamics analysis of a predator–prey model with nonmonotonic functional response and impulsive control

This paper presents the qualitative analysis of a predator–prey model with nonmonotonic functional response and impulsive effects. Different from previous work, by considering two factors of nonmonotonic functional response and impulsive effects, this paper studies the existence and stability of the periodic semi-trivial solution y=0 first. Then, by constructing an appropriate Poincare map and introducing geometric theory, it is shown that the predator–prey model can exhibit a variety of dynamic phenomena, including orbitally asymptotically stable order-1 periodic solution (O1PS), order-2 periodic solution (O2PS) and globally stable equilibrium point under certain conditions. When there exists an O2PS, its appearance and disappearance as well as the appearance of bifurcation phenomenon are discussed in detail with different selections of the initial value of predator. Finally, numerical simulations illustrate the correctness of the results of the theoretical analysis. The theoretical results presented in this paper can be seen as an advancement to the previous related works.