Stability Of Positive Steady State Research Articles

Bifurcation analysis of a delayed reaction–diffusion–advection Nicholson’s blowflies equation

In this paper, we investigate the dynamics of a reaction-diffusion Nicholson’s blowflies equation with advection. The stability of positive steady state and existence of Hopf bifurcation are obtained by analyzing the distribution of the eigenvalues. Moreover, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction and stability of the Hopf bifurcation is derived. Meanwhile, we find out that the bifurcation value is increasing with respect to the advection rate. Finally, numerical results demonstrate that the advection term causes the population to move from upstream to downstream, which also indicates that advection term plays a key role in the description and interpretation of some common natural phenomena.

An age-structured tuberculosis model with information and immigration: Stability and simulation study

This paper establishes an age-structured Tuberculosis (TB) model to investigate the joint impacts of information and immigration of population on the spread of TB disease. Mathematically, we show that the model is point dissipative, and the semi-flow generated by the model has the property of asymptotic smoothness, and then study the existence and global stability of positive steady state by the direct Lyapunov functional. Numerically, by using Matlab software, we verify the theoretical results, and further explore the influence of information (including information coverage and disease-related memory delay) and immigration on the final size of TB disease. The simulation results show that both information coverage and immigration are positive correlated with the final size of disease, and disease-related memory delay can affect the arrival time of positive steady state, which implies us that improving information coverage, enlarging disease-related memory, and reducing the immigration of population (especially latent and infected individuals) can effectively control the progression of TB disease.

Dynamics in a Delayed Competition System on a Weighted Network

In this paper, we study the stability of positive steady states in a delayed competition system on a weighted network, which does not satisfy the comparison principle appealing to classical competitive systems. By introducing some auxiliary equations and constructing proper contracting rectangles, we present some sufficient conditions on the stability of the unique positive steady state. Moreover, some numerical examples are given to explore the complex dynamics of this nonmonotone model, which implies the nontrivial roles of weights and time delays.

Global Hopf bifurcation of a general predator-prey system with diffusion and stage structures

In this paper, the dynamics of a general predator-prey system with diffusion and stage structures are investigated. Under some assumptions, the stability of positive steady state and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. Moreover, the global existence of positive periodic solution is established by using the global Hopf bifurcation result of Wu. Finally, an example and some numerical simulations are carried out for illustrating the theoretical results.

Dynamics of a consumer-resource reaction-diffusion model : Homogeneous versus heterogeneous environments.

We study the dynamics of a consumer-resource reaction-diffusion model, proposed recently by Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of time-dependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al. (2017), which says that in consumer-resource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (J Differ Equ 223(2):400-426, 2006. https://doi.org/10.1016/j.jde.2005.05.010 ) for logistic-type models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates.

Dynamics of a predator\u2013prey system with three species

This paper is concerned with the dynamics of a predator–prey system with three species. When the domain is bounded, the global stability of positive steady state is established by contracting rectangles. When the domain is mathbb{R}, we study the traveling wave solutions implying that one predator and one prey invade the habitat of another prey. More precisely, the existence of traveling wave solutions is proved by combining upper and lower solutions with a fixed point theorem, and the asymptotic behavior of traveling wave solutions is obtained by the idea of contracting rectangle. Moreover, we show the nonexistence of traveling wave solutions by applying the theory of the asymptotic spreading.

Global asymptotic stability of positive steady states of a solid avascular tumor growth model with time delays

In this work, global stability of a free boundary problem modeling solid avascular tumor growth is studied. The model is considered with time delays during the proliferation process. We prove that the unique positive constant steady state is globally asymptotically stable under some assumptions. The proof uses the comparison principle and the iteration method.

Dynamics of a Nonlocal Dispersal Model with a Nonlocal Reaction Term

In this paper, we study a class of nonlocal dispersal problem with a nonlocal term arising in population dynamics: [Formula: see text] where [Formula: see text] is a bounded domain, [Formula: see text], [Formula: see text] represents the nonlocal dispersal operator with continuous and non-negative dispersal kernel. The kernel [Formula: see text] is assumed to be non-negative and is allowed to have a degeneracy in a smooth subdomain [Formula: see text] of [Formula: see text]. When [Formula: see text] is either positive or vanishes in a subdomain, we respectively investigate the existence, multiplicity and asymptotical stability of positive steady states under the local/global variation of parameter by means of sub-supersolution method, Lyapunov–Schmidt reduction, and bifurcation theory.

Angiogenesis model with Erlang distributed delays.

We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

Protection zone in a modified Lotka-Volterra model

This paper studies the dynamic behaviorof solutions to a modified Lotka-Volterra reaction-diffusion systemwith homogeneous Neumann boundary conditions, for which a protectionzone should be created to prevent the extinction of the prey only ifthe prey's growth rate is small. We find a critical size of theprotection zone, determined by the ratio of the predation rate andthe refuge ability, to ensure the existence, uniqueness and globalasymptotic stability of positive steady states for generalpredator's growth rate $\mu>0$. Bellow the critical size thedynamics of the model would be similar to the case withoutprotection zones. The known uniqueness results for the protectionproblems with other functional responses, e.g., Holling II model,Leslie model, Beddington-DeAngelis model, were all required that thepredator's growth rate $\mu>0$ is large enough. Such a large $\mu$assumption is not needed for the uniqueness and asymptotic resultsto the modified Lotka-Volterra reaction-diffusion system consideredin this paper.

Multiplicity for a diffusive predator-prey mutualist model

In this paper, we consider a predator–prey mutualist model with diffusion under homogeneous Dirichlet boundary conditions. By making use of the index theory of fixed points, we obtain some sufficient conditions for the existence of positive steady states and derive the multiplicity result when β is suitably large. The effect of γ, which represents the extent of a mutualist deterring predation on a prey, is extensively studied and a good understanding of the existence, uniqueness and stability of positive steady states is gained when γ is sufficiently large. Finally, we discuss some numerical simulations that support the analytic results in one-dimensional cases.

The effect of mutual interference between predators on a predator–prey model with diffusion

We consider a diffusive predator–prey model with Beddington–DeAngelis functional response under homogeneous Dirichlet boundary conditions. The effect of large k which represents the extent of mutual interference between predators is extensively studied. By making use of the fixed point index theory, we obtain a complete understanding of the existence, uniqueness and stability of positive steady-states when k is sufficiently large. Moreover, we present some numerical simulations that supplement the analytic results in one dimension.

Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$

This paper deals with the stability of steady states of thesemilinear heat equation $u_t=$Δ$u+K(x)u^p+f(x)$ under properassumptions on $K(x)$ and $f(x)$. We prove the weak asymptoticstability of positive steady states with respect to weighteduniform norms.

Existence, uniqueness and stability of positive steady states to a prey-predator diffusion system

In the paper, we study the positive solutions of an elliptic system coming from a preypredator model with modified Leslie-Gower and Holling-Type II schemes. We study the existence, non-existence, bifurcation, uniqueness and stability of positive solutions. In particular, we obtain a continuum of positive solutions connecting a semi-trivial solution to the unique positive solution of the limiting system.

Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey–predator model

In this work, we study a ratio-dependent prey–predator model with diffusion and homogeneous Neumann boundary condition. We prove that the unique positive constant steady state is locally and uniformly stable, and is globally asymptotically stable under some assumptions. The proof uses the iteration method.

Uniqueness and stability of steady states for a predator-prey model in heterogeneous environment

In this paper, we deal with a predator-prey model with diffusion in a heterogeneous environment, and we study the uniqueness and stability of positive steady states as the diffusion coefficient of the predator is small enough.

Predator-prey dynamics in models of prey dispersal in two-patch environments

Models are presented for a single species that disperses between two patches of a heterogeneous environment with barriers between patches and a predator for which the dispersal between patches does not involve a barrier. Conditions are established for the existence, uniform persistence, and local and global stability of positive steady states. In particular, an example that demonstrates both the stabilizing and destabilizing effects of dispersion is presented. This example indicates that a stable migrating predator-prey system can be made unstable by changing the amount of migration in both directions.

Spectral and Nonlinear Effects in Certain Elliptic Systems of Three Variables

In this paper, the authors use the method of decomposing operator to prove the existence of positive solutions to certain elliptic biological interacting systems of three species in all possible patterns of interactions in terms of the combinations of competition, symbiosis, and predation under the homogeneous Robin–Dirichlet boundary conditions. The existence of positive solutions on large domains and the asymptotic stability of positive steady states for some cases are also proved. The main idea is that the existence of positive solutions and their properties can be characterized by the spectral properties of certain operators of Schrodinger type and by the equilibria of the system.