Non-constant Positive Steady States Research Articles

Dynamics of a single population model with memory and nonlinear boundary condition

In this paper, a heterogeneous single population model with memory effect and nonlinear boundary condition is investigated. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially nonconstant positive steady state solutions appear from two trivial solutions, respectively. Through the distribution of the eigenvalues and bifurcation analysis, sufficient conditions for the occurrence of the Hopf bifurcation associated with one spatially nonconstant positive steady state are obtained. Results about the stability associated with the other one are also yielded. It is found that with the interaction of memory delay, spatial heterogeneity and nonlinear boundary condition, the Hopf bifurcation will happen in such diffusive model. To be specific, the memory delay could induce a single stability switch from stability to instability. As contrast to the diffusive model only with memory effect, the stability switch is independent of memory delay. The obtained results are applied to some specific models, from which we can find that the theoretical analysis is verified and the results enrich the existing ones.

The study of non-constant steady states and pattern formation for an interacting population model in a spatial environment

This manuscript accounts for an investigation of the complex dynamics of a spatial model for interacting populations. We discuss the existence and boundedness of solutions for the proposed spatio-temporal system. The global stability of the co-existing steady state of the proposed system is analyzed with the help of a suitable Lyapunov function. We provide results on the existence and non-existence of positive non-constant solutions of the model. The priori estimate for the positive steady state is obtained for the nonexistence of the non-constant positive steady state by using the maximum principle. The existence of a non-constant positive steady state is studied with the help of Leray–Schauder degree theory. The stability and Hopf bifurcation are briefly revisited for the co-existing steady state in the corresponding temporal model, where a bubble-like structure is observed. The onset of Hopf bifurcation has been analyzed, and different conditions for the formation of the Turing pattern have been established through diffusion-driven instability analysis. Numerical simulations are performed in detail to figure out the effects of saturated harvesting on Turing patterns. The Turing as well as non-Turing patterns in their respective domains are also examined. Finally, the criteria of Turing–Hopf bifurcation is briefly demonstrated with relevant numerical examples and corresponding plots that give a better illustration of this work.

Global existence and steady states of the density-suppressed motility model with strong Allee effect

Abstract This paper considers a density-suppressed motility model with a strong Allee effect under the homogeneous Neumman boundary condition. We first establish the global existence of bounded classical solutions to a parabolic–parabolic system over an $N $-dimensional $\mathbf{(N\le 3)}$ bounded domain $\varOmega $, as well as the global existence of bounded classical solutions to a parabolic–elliptic system over the multidimensional bounded domain $\varOmega $ with smooth boundary. We then investigate the linear stability at the positive equilibria for the full parabolic case and parabolic–elliptic case, respectively, and find the influence of Allee effect on the local stability of the equilibria. By treating the Allee effect as a bifurcation parameter, we focus on the one-dimensional stationary problem and obtain the existence of non-constant positive steady states, which corresponds to small perturbations from the constant equilibrium $(1,1)$. Furthermore, we present some properties through theoretical analysis on pitchfork type and turning direction of the local bifurcations. The stability results provide a stable wave mode selection mechanism for the model considered in this paper. Finally, numerical simulations are performed to demonstrate our theoretical results.

Steady States of a Diffusive Population-Toxicant Model with Negative Toxicant-Taxis

This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4).

Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory

The spatiotemporal dynamics of a three-component model of a food web are considered. The model describes the interactions between populations of resources, prey, and predators that consume both species. It assumes that the predator responds to the spatial change in the resource and prey densities by occupying areas where species density is higher (prey-taxis) and that the prey population avoids areas with a high predator density (predator-taxis). This work studies the conditions for the taxis-driven instability leading to the emergence of stationary patterns resulting from Turing instability and autowaves caused by wave instability. The existence of nonconstant positive steady states for the system is assessed through a rigorous bifurcation analysis. Meanwhile, the conditions for the existence of both types of instabilities are obtained by linear stability analysis. It is shown that the presence of cross-diffusion in the system supports the formation of spatially heterogeneous patterns. For low values of the resource-tactic and predator-tactic coefficients, Turing and wave instabilities coexist. The system undergoes only Turing instability for high levels of these parameters.

Global dynamics of an indirect prey-taxis system with an anti-predation mechanism

In this paper, we shall study the following model:{Nt=dNΔN+λN−N2−NP,x∈Ω,t>0,Pt=Δ(d(S)P)+μP−P2+γNP,x∈Ω,t>0,St=dSΔS+τN−ηS,x∈Ω,t>0,∂N∂ν=∂P∂ν=∂S∂ν=0,x∈∂Ω,(N,P,S)(x,0)=(N0,P0,S0)(x),x∈Ω, where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary. Based on semigroup estimates and Moser iteration, we first establish the existence of the global classical solution with a uniform-in-time bound. Moreover, we build the global stabilization of constant steady states by constructing suitable Lyapunov functionals and give the decay rates of solutions. Furthermore, we find that the anti-predation mechanism will generate steady state bifurcation. We further prove the global existence of non-constant positive steady state solutions under certain conditions. Finally, we numerically demonstrate spatially inhomogeneous coexistence patterns (i.e., non-constant positive steady state solutions) for the predator, the prey and the signal.

Bifurcation and Spatiotemporal Patterns of SI Epidemic Model with Diffusion

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations.

Dynamics of a diffusive two predators- one prey system

This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.

Stationary Patterns of a Cross-Diffusion Prey-Predator Model with Holling Type II Functional Response

In this paper, we consider positive steady-state solutions of a cross-diffusions prey-predator model with Holling type II functional response. We investigate sufficient conditions for the existence and the nonexistence of nonconstant positive steady state solutions. It is observed that nonconstant positive steady states do not exist with small cross-diffusion coefficients, and the constant positive steady state is global asymptotically stable without cross-diffusion. Furthermore, we show that if natural diffusion coefficient or cross-diffusion coefficient of the predator is large enough and other diffusion coefficients are fixed, then under some conditions, at least one nonconstant positive steady state exists.

Spatiotemporal Dynamics and Bifurcation Analysis of a Generalized Two-Prey One-Predator System with Diffusion and Double Prey-Taxes

The presence of a predator can force the mediation of a coexistence state in three-species ordinary differential equation model, where two competing species are preyed on by a common predator. To understand how the addition of diffusion and prey-taxis affects predator-mediated coexistence in such an ecological system, we consider a general two-competing-prey and one-predator model with double prey-taxes under Neumann boundary conditions. We first show that there is a unique global classical solution to this model with ratio-dependent and nonratio-dependent predator functional responses. Then, we demonstrate the emergence of the so-called stationary patterns. Finally, in detail, we give some sufficient conditions for the existence, nonexistence, and stability of nonconstant positive steady states and time-periodic positive solutions. Surprisingly, we find that the combination of a repulsive prey-taxis and an attractive prey-taxis can also induce the emergence of pattern formations. The theoretical results imply that double prey-taxes play an extremely important part in biological control.

Dynamical behaviour of an intraguild predator–prey model with prey refuge and hunting cooperation

ABSTRACT An intraguild predator–prey model including prey refuge and hunting cooperation is investigated in this paper. First, for the corresponding ordinary differential equation model, the existence and stability of all equilibria are given, and the existence of Hopf bifurcation, direction and stability of bifurcating periodic solutions are investigated. Then, for partial differential equation model, the diffusion-driven Turing instability is obtained. What is more, the existence and non-existence of the non-constant positive steady state of the reaction–diffusion model are established by the Leray–Schauder degree theory and some priori estimates. Next, some numerical simulations are performed to support analytical results. The results showed that prey refuge can change the stability of model and even have a stabilizing effect on this model, meanwhile the hunting cooperation can make such model without diffusion unstable, but make such model with diffusion stable. Lastly, a brief conclusion is concluded in the last section.

Boundedness, stability and pattern formation for a predator-prey model with Sigmoid functional response and prey-taxis

This article concerns the structure of the nonconstant steady states for a predator-prey model of Leslie-Gower type with Sigmoid functional and prey-taxis subject to the homogeneous Neumann boundary condition. The existence of bounded classical global solutions is discussed in bounded domains with arbitrary spatial dimension and any prey-taxis sensitivity coefficient. The local stability of the homogeneous steady state is analyzed to show that the prey-taxis sensitivity coefficient destabilizes the stability of the homogeneous steady state when prey defends. Then we study the existence and stability of the nonconstant positive steady state of the system over 1D domain by applying the bifurcation theory and present properties of local branches such as pitchfork and turning direction. Moreover, we discuss global bifurcation, homogeneous steady state solutions, nonconstant steady states solutions, spatio-temporal periodic solutions and spatio-temporal irregular solutions which demonstrate the coexistence and spatial distribution of prey and predator species. Finally, we perform numerical simulations to illustrate and support our theoretical analysis.

Dynamics of an aquatic diffusive predator–prey model with double Allee effect and [formula omitted]-dependent capture rate

To investigate ocean acidification and Allee effects on the dynamics of a marine predator–prey system, an aquatic diffusive predator–prey model with double Allee effect on prey and pH-dependent capture rate is considered. First, we study the stability of constant steady state solutions using linearized theory. Second, the nonexistence of nonconstant positive steady state solutions is shown for appropriate ranges of parameters. Furthermore, we show the existence of a Hopf bifurcation and derive the direction and stability of the bifurcating periodic solutions. Both theoretical analysis and numerical simulation show that changing predator–prey interaction strengths, due to changing environmental conditions, can fundamentally change the system dynamics, even for apparently small changes in interaction strength. As the interaction strength decreases due to decreasing ocean pH, the system dynamics transition from persistent fluctuations in species abundances (periodic solutions), to stable coexistence, to predator extinct (with stable non-zero prey abundance), suggesting the potential for ocean acidification to decrease the abundance and diversity of marine species by weakening predation rates. Moreover, double Allee effect parameters together determine the stability of periodic solutions when the spatially homogeneous bifurcating periodic solutions exist, and the wavelength becomes longer as the Allee effect increases.

Stability of a delayed diffusion–advection vector-disease model with spatial heterogeneity

Existence of nonconstant positive steady state and its locally asymptotical stability of a delayed diffusion–advection vector-disease model in the heterogeneous environment are considered, by using the Rabinowitz local bifurcation theory and the distribution of associated eigenvalues. Ecologically, in the spatially heterogeneous environment, the incubation period cannot change the stability when the diffusion is large enough.

Nonconstant steady states and pattern formations of generalized 1D cross‐diffusion systems with prey‐taxis

Cross-diffusion effects and tactic interactions are the processes that preys move away from the highest density of predators preferentially, or vice versa. It is renowned that these effects have played significant roles in ecology and biology, which are also essential to the maintenance of diversity of species. To simulate the stability of systems and illustrate their spatial distributions, we consider positive nonconstant steady states of a generalized cross-diffusion model with prey-taxis and general functional responses in one dimension. By applying linear stability theory, we analyze the stability of the interior equilibrium and show that even in the case of negative cross-diffusion rate, which appeared in many models, the corresponding cross-diffusion model has opportunity to achieve its stability. Meanwhile, in addition to the cross-diffusion effect, tactic interactions can also destabilize the homogeneity of predator–prey systems if the tactic interaction coefficient is negative. Otherwise, taxis effects can stabilize the homogeneity.

Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

<abstract><p>In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.</p></abstract>

Hopf bifurcation in a Lotka-Volterra competition-diffusion-advection model with time delay

In this paper, we mainly investigate a Lotka-Volterra competition-diffusion-advection system with time delay, where the diffusion and advection rates of two competitors are different. By employing the Lyapunov-Schmidt reduction method, we obtain the existence of steady state solution. A weighted inner product has been introduced to study stability and Hopf bifurcation at the spatially nonhomogeneous steady-state. Our results imply that the infinitesimal generator associated with the linearized system have two pairs of purely imaginary eigenvalues, and time delay can make the spatially nonconstant positive steady state unstable for a reaction-diffusion-advection model. In addition, the bifurcation direction and stability of Hopf bifurcating periodic orbits was obtained by means of the center manifold reduction and the normal form theory.

Bifurcation analysis for a single population model with advection.

In this paper, the dynamics of a single population model with a general growth function is investigated in an advective environment. We show the existence of a nonconstant positive steady state, and give sufficient conditions for the occurrence of a Hopf bifurcation at the positive steady state. Moreover, the theoretical results are applied to the diffusive Nicholson's blowflies and Mackey-Glass's models with advection and delay, respectively. We numerically show that the population density decreases as the increase of advection rate or death rate, and a delay-induced Hopf bifurcation is more likely to occur with small advection or low mortality rate.

Bifurcation structures of a Leslie–Gower model with diffusion and advection

This paper is devoted to a modified Leslie–Gower model with diffusion and advection. By regarding the advection coefficient as a bifurcation parameter, we obtain the local and global structure of nonconstant positive steady states bifurcating from the homogeneous steady states in multi-dimension by the asymptotic analysis and bifurcation theory.

Bifurcation dynamics of a reaction–diffusion predator–prey model with fear effect in a predator‐poisoned environment

In this paper, we propose a reaction–diffusion predator–prey model with fear effect under a predator‐poisoned environment. First, we analyze the existence and stability of constant positive steady states. Second, taking the cost of minimum fear as the bifurcation parameter, the direction and stability of spatially homogeneous/inhomogeneous periodic solutions are investigated. Then, some properties of nonconstant positive steady states are investigated, such as nonexistence, existence, and steady state bifurcation. Applying the fixed point index theory, the existence of the nonconstant positive steady state is discussed, indicating that the proper diffusion rate of prey and larger diffusion rate of predator are beneficial to survival of populations. Moreover, taking the diffusion rate of predator as the bifurcation parameter, the steady state bifurcations from simple and double eigenvalues are intensively established. At last, the validity of the theoretical analysis is verified by numerical simulations. Biological interpretation reveals that the fear effect and the diffusion rate of populations can lead to the emergence of Hopf and steady state bifurcation, thereby destroying the stability of populations and promoting the benign evolution of populations.