Existence Of Nonconstant Steady States Research Articles

Steady states and spatiotemporal dynamics of a diffusive predator-prey system with predator harvesting

<p>From the perspective of ecological control, harvesting behavior plays a crucial role in the ecosystem natural cycle. This paper proposes a diffusive predator-prey system with predator harvesting to explore the impact of harvesting on predatory ecological relationships. First, the existence and boundedness of system solutions were investigated and the non-existence and existence of non-constant steady states were obtained. Second, the conditions for Turing instability were given to further investigate the Turing patterns. Based on these conditions, the amplitude equations at the threshold of instability were established using weakly nonlinear analysis. Finally, the existence, direction, and stability of Hopf bifurcation were proven. Furthermore, numerical simulations were used to confirm the correctness of the theoretical analysis and show that harvesting has a strong influence on the dynamical behaviors of the predator-prey systems. In summary, the results of this study contribute to promoting the research and development of predatory ecosystems.</p>

Global bifurcation for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis

The aim of this paper is to establish a precise illustration for the structure of the nonconstant steady states for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis. We treat the nonlinear prey‐taxis as a bifurcation parameter to analyze the bifurcation structure of the system. Furthermore, the exported global bifurcation theorem, under a rather natural condition, offers the existence of nonconstant steady states. In the proof, a priori estimates of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given. Finally, some conclusions including biological meanings are performed to summarize our main analytic results and future investigations.

Dynamics of two-species Holling type-II predator-prey system with cross-diffusion

In this paper, we are devoted to studying a Holling type-II predator-prey system with cross diffusion. First of all, by a series of estimates, we establish the global existence of generalized solutions under some proper assumptions. By analyzing the distribution of eigenvalues, we investigate the local and global stability of the constant solution and also consider the steady state bifurcation and Hopf bifurcation near the constant steady state in details. In addition, the nonexistence of non-constant steady states is also investigated when diffusion rate d is large enough. Finally, sufficient conditions ensuring the existence of non-constant steady states are obtained by using Leray-Schauder degree theory.

Steady states and spatiotemporal evolution of a diffusive predator–prey model

The existence of steady states, bifurcations and the spatiotemporal patterns are presented for the diffusive predator–prey model. First, the boundedness and positivity of solutions are justified, respectively. By employing the priori estimates, Poincare inequalities and Leray–Schauder degree, nonexistence and existence of nonconstant steady states are established, respectively. To further explore the pattern dynamics, the Hopf bifurcation and Turing instability are analyzed, the weakly nonlinear analysis is employed to establish the amplitude equations. It is found that the various complex pattern solutions can be identified from amplitude equations. The numerical results are in agreement with the theoretical analysis. We also find that the predator–prey model with Beddington–Deangelis (BD)-type functional response can admit various spatiotemporal patterns, such as labyrinthine-like patterns, the wave patterns near the Hopf–Turing bifurcation threshold, and so on. Such complex spatiotemporal patterns may be useful to help us understand the interaction among species.

Steady states of a diffusive predator-prey model with prey-taxis and fear effect

In this paper, a diffusive predator-prey system with a prey-taxis response subject to Neumann boundary conditions is considered. The stability, the Hopf bifurcation, the existence of nonconstant steady states, and the stability of the bifurcation solutions of the system are analyzed. It is proved that a high level of prey-taxis can stabilize the system, the stability of the positive equilibrium is changed when χ crosses chi _{0}, and the Hopf bifurcation occurs for the small s. The system admits nonconstant positive solutions around (bar{u}, bar{v}, chi _{i} ), the stability of bifurcating solutions are controlled by int _{Omega} Phi _{i}^{3} ,mathrm{d}x and int _{Omega} Phi _{i}^{4} ,mathrm{d}x. Finally, numerical simulation results are carried out to verify the theoretical findings.

Bifurcation and Pattern Formation in an Activator-Inhibitor Model with Non-local Dispersal.

In this paper, by approximating the non-local spatial dispersal equation by an associated reaction-diffusion system, an activator-inhibitor model with non-local dispersal is transformed into a reaction-diffusion system coupled with one ordinary differential equation. We prove that, to some extent, the non-locality-induced instability of the non-local system can be regarded as diffusion-driven instability of the reaction-diffusion system for sufficiently small perturbation. We study the structure of the spectrum of the corresponding linearized operator, and we use linear stability analysis and steady-state bifurcations to show the existence of non-constant steady states which generates non-homogeneous spatial patterns. As an example of our results, we study the bifurcation and pattern formation of a modified Klausmeier-Gray-Scott model of water-plant interaction.

Non-constant steady states and Hopf bifurcation of a species interaction model

In this paper, we are concerned with the species interaction model with the ratio-dependent Holling III functional response and strong Allee effect. To explore nonhomogeneous solutions of the model, we consider the existence and non-existence of non-constant steady states and temporal bifurcation. Then we show the boundedness of the global positive solutions for the parabolic system and present the upper and lower bounds of positive solutions for the associated elliptic system. The non-existence and existence of the non-constant steady states of the elliptic system with the homogeneous Neumann boundary conditions are obtained by using the priori estimates, maximum principle and index theory. Furthermore, the existence and the direction of Hopf bifurcation are investigated via the stability analysis, center manifold theory and normal form reduction. Numerical simulations are carried out to verify our theoretical analysis and to illustrate that the ratio-dependent Holling III functional response and strong Allee effect have strong impact on dynamical behaviors of the species interaction systems.

Global bifurcation and pattern formation for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall–Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.

Drivers of pattern formation in a predator\u2013prey model with defense in fearful prey

Existence of predator is routinely used to induce fear and anxiety in prey which is well known for shaping entire ecosystem. Fear of predation restricts the development of prey and promotes inducible defense in prey communities for the survival. Motivated by this fact, we investigate the dynamics of a Leslie–Gower predator prey model with group defense in a fearful prey. We obtain conditions under which system possess unique global-in-time solutions and determine all the biological feasible states of the system. Local stability is analyzed by linearization technique and Lyapunov direct method has been applied for global stability analysis of steady states. We show the occurrence of Hopf bifurcation and its direction at the vicinity of coexisting equilibrium point for temporal model. We consider random movement in species and establish conditions for the stability of the system in the presence of diffusion. We derive conditions for existence of non-constant steady states and Turing instability at coexisting population state of diffusive system. Incorporating indirect prey taxis with the assumption that the predator moves toward the smell of prey rather than random movement gives rise to taxis-driven inhomogeneous Hopf bifurcation in predator–prey model. Numerical simulations are intended to demonstrate the role of biological as well as physical drivers on pattern formation that go beyond analytical conclusions.

Steady-state solutions of the templator model in chemical self-replication

In this paper, we study the dynamics of the diffusive templator model in chemical self-replication. Firstly, the stability of nonnegative constant equilibrium solution and the stationary patterns induced by diffusion are investigated. Next, we study the properties and existence of nonconstant steady states. The global structures of bifurcation emitting from the positive constant solution are also established. Finally, numerical simulations are done to complemented theoretical results.

Optimality for a diffusive predator-prey system in a spatially heterogeneous environment incorporating a prey refuge

Abstract In this paper, we consider the optimal harvesting controls for a diffusive predator-prey system, which is formulated by prey refuge and modified Leslie-Gower functional response. The locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous Neumann boundary condition. The nonexistence and existence of nonconstant steady states are obtained by Leray-Schauder degree theory. Also, the optimal harvesting strategies to get maximum sustainable yield (MSY) and maximum economic yield (MEY) are correspondingly given.

Patterns and dynamics in the diffusive model of a nutrient–microorganism system in the sediment

This paper is devoted to study patterns and dynamics in the diffusive minimal sediment model. Firstly, we analyze nonnegative constant equilibrium solutions and investigate patterns induced by diffusions. Then we study the properties and existence of nonconstant steady states. Moreover, we describe the local bifurcation structure and global bifurcation structure from two positive constant solutions, respectively. The main tools used here include the stability theory, degree theory, bifurcation theory. From extensive numerical simulations, the theoretical results are confirmed and complemented, and the influence of parameters δ1,δ2 on these patterns is depicted.

Global bifurcation for a Holling–Tanner predator–prey model with prey-taxis

The local existence and stability of non-constant positive steady states for the Holling–Tanner predator–prey model with prey-taxis was discussed by Wang et al. (2017) through local bifurcation analysis. It is our purpose in this paper to make a detailed description for the structure of the set of the non-constant steady states. We will prove a global bifurcation theorem by treating prey-taxis as a bifurcation parameter, which gives the existence of non-constant steady states under a rather natural condition. A priori estimates for steady state solutions will play a key role in the proof. In addition, the stability results for the homogeneous equilibrium will extend to the case that the derivative of prey’s functional response with prey is positive, and it is found that attractive prey-taxis can stabilize homogeneous equilibrium even diffusion-driven instability has occurred.

Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity

In this paper, we develop a diffusivepredator–prey model with toxins under the homogeneous Neumann boundary condition. First, the persistence property and global asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, we derive explicit conditions for the existence of nonconstant steady states that emerge through steady-state bifurcation from related constant steady states. Furthermore, the existence and nonexistence of nonconstant positive steady states of this model are studied by considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. These explicit conditions are numerically verified in detail and further compared to those conditions ensuring Turing instability. It is shown that the numerically observed behaviors are in good agreement with the theoretically proposed results. All theoretical analyses and numerical simulations show that toxic substances have a perceptible effect on the system.

WITHDRAWN: Existence of non-constant steady states to a Brusselator model

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Hopf and steady state bifurcation analysis in a ratio-dependent predator–prey model

In this paper, we perform spatiotemporal bifurcation analysis in a ratio-dependent predator–prey model and derive explicit conditions for the existence of non-constant steady states that emerge through steady state bifurcation from related constant steady states. These explicit conditions are numerically verified in details and further compared to those conditions ensuring Turing instability. We find that (1) Turing domain is identical to the parametric domain where there exists only steady state bifurcation, which implies that Turing patterns are stable non-constant steady states, but the opposite is not necessarily true; (2) In non-Turing domain, steady state bifurcation and Hopf bifurcation act in concert to determine the emergent spatial patterns, that is, non-constant steady state emerges through steady state bifurcation but it may be unstable if the destabilising effect of Hopf bifurcation counteracts the stabilising effect of diffusion, leading to non-stationary spatial patterns; (3) Coupling diffusion into an ODE model can significantly enrich population dynamics by inducing alternative non-constant steady states (four different states are observed, two stable and two unstable), in particular when diffusion interacts with different types of bifurcation; (4) Diffusion can promote species coexistence by saving species which otherwise goes to extinction in the absence of diffusion.

Global existence and steady states of a two competing species Keller--Segel chemotaxis model

We study an one{dimensional quasilinear system proposed by J. Tello and M. Winkler [19] which models the population dynamics of two competing species attracted by the same chemical. The kinetics terms of the interacting species are chosen to be the Lotka{Volterra type. We prove the existence of global bounded and classical solutions for all chemoattraction rates. Under homogeneous Neumann boundary conditions, we establish the existence of nonconstant steady states by local bifurcation theory. The stability of the bifurcating solutions is also obtained when the diffusivity of both species is large. Finally, we perform extensive numerical studies to demonstrate the formation of stable positive steady states with various interesting spatial structures.

Analysis of Spatiotemporal Dynamic and Bifurcation in a Wetland Ecosystem

A wetland ecosystem is studied theoretically and numerically to reveal the rules of dynamics which can be quite accurate to better describe the observed spatial regularity of tussock vegetation. Mathematical theoretical works mainly investigate the stability of constant steady states, the existence of nonconstant steady states, and bifurcation, which can deduce a standard parameter control relation and in return can provide a theoretical basis for the numerical simulation. Numerical analysis indicates that the theoretical works are correct and the wetland ecosystem can show rich dynamical behaviors not only regular spatial patterns. Our results further deepen and expand the study of dynamics in the wetland ecosystem. In addition, it is successful to display tussock formation in the wetland ecosystem may have important consequences for aquatic community structure, especially for species interactions and biodiversity. All these results are expected to be useful in the study of the dynamic complexity of wetland ecosystems.

Steady states of a predator–prey model with prey-taxis

We study the steady states of a predator–prey model with prey-taxis incorporating Holling type II functional response under the homogeneous Neumann boundary condition. The stability of equilibrium points and the existence of non-constant steady states are investigated. We obtain that the prey-tactic sensitivity coefficient delays the stability of the unique positive constant solution, but for other equilibrium points’ stability, the prey-tactic sensitivity coefficient does not influence on it. Furthermore, we derive some sufficient conditions relative to the prey-tactic sensitivity coefficient which confines the existence of steady states and find that even if the interaction coefficient is sufficiently large, there also exist non-constant positive steady states under some conditions.

Turing patterns in the Lengyel-Epstein system for the CIMA reaction

The first experimental evidence of the Turing pattern was observed by De Kepper and her associates (1990) on the CIMA reaction in an open unstirred gel reactor, almost 40 years after Turing’s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. In this paper we report some fundamental analytic properties of the Lengyel-Epstein system. Our result also indicates that if either of the initial concentrations of the reactants, the size of the reactor, or the effective diffusion rate, are not large enough, then the system does not admit nonconstant steady states. A priori estimates are fundamental to our approach for this nonexistence result. The degree theory was combined with the a priori estimates to derive existence of nonconstant steady states.