Existence Of Positive Steady States Research Articles

Dynamical analysis of a diffusive population-toxicant model with toxicant-taxis in polluted aquatic environments

This paper deals with a diffusive population-toxicant model in polluted aquatic environments, with a toxicant-taxis term describing a toxicant-induced behavior change, that is, the population tends to move away from locations with high-level toxicants. The global existence of solutions is established by the techniques of the semigroup estimation and Moser iteration. Based on a detailed study on the properties of the principal eigenvalue for non-self-adjoint eigenvalue problems, we investigated the local and global stability of the toxin-only steady-state solution and the existence of positive steady state, which yields sufficient conditions that lead to population persistence or extinction. Finally, by numerical simulations, we studied the effects of some key parameters, such as toxicant-taxis coefficient, advection rate, and effect coefficient of the toxicant on population growth, on population persistence. Both numerical and analytical results show that a weak chemotaxis effect, a small advection rate of the population, and a weak effect of the toxicant on population growth are favorable for population persistence.

On a limiting system of the triangular SKT competition model with large cross-diffusion and random-diffusion

This paper mainly focus on the limiting system which arises from the study of SKT competition model when both the cross-diffusion rate and the random-diffusion rate tend to infinity. For multi-dimensional domains, the existence of positive steady states bifurcating from a double eigenvalue can be proved by applying the bifurcation argument with some special transformation when the birth rate of one species is near some critical value. Further by virtue of the spectral perturbation argument based on the Lyapunov–Schmidt decomposition method, we prove the spectral instability of such nontrivial positive steady states for the limiting system.

Dynamics of a diffusive competition model with memory effect and spatial heterogeneity

In this paper, a diffusive competition model is formulated by considering the memory-based diffusions of species in their movement and spatial heterogeneity of the resource. The global existence of solution for the proposed model is proved. Under weak competition, the existence and stability of the positive steady state are obtained, when the random diffusion rates are large enough, by using implicit function theorem and some priori estimations. However, when memory-based diffusions dominate the dynamics, spatial-temporal patterns can be observed. We also consider the case that both competition coefficients equal to one, showing that the stabilities of semi-trivial steady states, existence of positive steady state by the method of upper and lower solutions, and bistable dynamics of the model. The results reveal that the memory-based diffusion plays an important role on the competition outcomes. Specifically, the species with slower random diffusion may not help itself to wipe out its competitor because of disadvantageous memory-based diffusion, while the species with faster random diffusion but with advantageous memory-based diffusion may win the competition.

Asymptotic profiles of positive steady states for a diffusive predator–prey model with predator interference

This paper is concerned with positive steady states for a diffusive predator–prey model with predator interference in a spatially heterogeneous environment. We first establish the necessary and sufficient conditions for the existence of positive steady states. In order to get a better understanding of the structure of positive steady states, we further investigate the asymptotic profiles of positive steady states as some parameter tends to zero or infinity.

Single Phytoplankton Species Growth with Nonlocal Crowding Effect Caused by Chemosensory Aggregation in a Water Column

In this paper, we study the existence of positive steady states for a nonlocal reaction‐diffusion model which describes the growth of a single phytoplankton species with nonlocal crowding effect caused by chemosensory aggregation in a water column. Our result shows that under some conditions, the nonlocal crowding effect is effective to the existence of steady states of phytoplankton populations. Based on a key boundedness Lemma, the main result is proved by fixed point index theory and some analytical techniques.

Diffusive predator-prey models with fear effect in spatially heterogeneous environment

This article concerns diffusive predator-prey models incorporating the cost of fear and environmental heterogeneity. Under homogeneous Neumann boundary conditions, we establish the uniform boundedness of global solutions and global stability of the trivial and semi-trivial solutions for the parabolic system. For the corresponding steady-state problem, we obtain sufficient conditions for the existence of positive steady states, and then study the effects of functional responses and the cost of fear on the existence, stability and number of positive steady states. We also discuss the effects of spatial heterogeneity and spatial diffusion on the dynamic behavior and establish asymptotic profiles of positive steady states as the diffusion rate of prey or predator individuals approaches zero or infinity. Our theoretical results suggest that fear plays a very important role in determining the dynamic behavior of the models, and it is necessary to revisit existing predator-prey models by incorporating the cost of fear.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/70/abstr.html

Existence of the positive steady states of a reaction–diffusion–advection competition model

In this paper, we investigate a reaction–diffusion–advection two-species competition biological model by employing the methods of Lyapunov–Schmidt reduction, perturbation theory and the implicit function theorem and so on. For any given diffusion rates we obtain the existence of positive steady states by using the advection rate η and the competitive coefficients c and b as the bifurcation parameters.

A Diffusive One-Prey and Two-Cooperative-Predators Model with C–M Functional Response

This paper is dedicated to a study of a diffusive one-prey and two-cooperative-predators model with C–M functional response subject to Dirichlet boundary conditions. We first discuss the existence of positive steady states by the fixed point index theory and the degree theory. In the meantime, we analyze the uniqueness and stability of coexistence states under conditions that one predator’s consumer rate is small and the effect of interference intensity of another predator is large. Then, steady-state bifurcations from two strong semi-trivial steady states (provided that they uniquely exist under some conditions) and from one weak semi-trivial steady state are investigated in detail by the Crandall–Rabinowitz bifurcation theorem, the technique of space decomposition and the implicit function theorem. In addition, we study the asymptotic behaviors including the extinction and permanence of the time-dependent system by the comparison principle, upper-lower solution method and monotone iteration scheme. Finally, numerical simulations are done not only to validate the theoretical conclusions, but also to further clarify the impacts of parameters on the three species.

Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment

In this paper, we investigate a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. We first prove that the solution of the model is bounded when the susceptible and infected individuals have same or distinct dispersal rates. The global stability of the constant endemic equilibrium is proved by constructing suitable Lyapunov functionals when all parameters are positive constants. We employ the topological degree argument to show the existence of positive steady state. Most importantly, we have also investigated the asymptotic profiles of the positive steady state as the dispersal rate of susceptible or infected individuals tends to zero or infinity. Our result reveals that a linear source and spontaneous infection can significantly enhance disease persistence no matter what dispersal rate of the susceptible or infected population is small or large, which leads to the situation that when total population number allows to vary, disease becomes more difficult to control.

Analysis of leukaemic cells dynamics with multi‐stage maturation process using a new non‐linear positive model with distributed time‐delay

This study proposes a new non-linear positive model with distributed time-delay describing the dynamics of leukaemic cells lineages and their levels of maturity. The dynamic model is first reformulated into coupled non-linear equations with distributed time-delay using an appropriate transformation of the age-structured transport partial differential equations that govern the dynamics of leukaemic cell populations. After assessing the positiveness of the proposed model, the conditions for the existence of positive steady-states are established. Furthermore, the necessary and sufficient conditions for boundedness of the solutions are derived. Finally, the convergence of the proposed model to the origin is studied as this has an important biological interpretation which is the extinction of all generations of leukaemic cells following anti-leukaemia treatments. To investigate the global asymptotic stability of this important equilibrium point, both model properties (i.e. positiveness and boundedness) are used to construct a suitable Lyapunov function for the characterisation of leukaemic cells dynamics. The results demonstrate a significant improvement for the treatment of leukaemia. Some simulations are presented to illustrate the effectiveness of the proposed model.

Existence of Positive Steady States for Weakly Reversible Mass-Action Systems

We prove the following. For each weakly reversible mass-action system, there exists a positive steady state in each positive stoichiometric class.

Net reproduction functions for nonlinear structured population models

The goal of this note is to present a general approach to define the net reproduction function for a large class of nonlinear physiologically structured population models. In particular, we are going to show that this can be achieved in a natural way by reformulating a nonlinear problem as a family of linear ones; each of the linear problems describing the evolution of the population in a different, but constant environment. The reformulation of a nonlinear population model as a family of linear ones is a new approach, and provides an elegant way to study qualitative questions, for example the existence of positive steady states. To define the net reproduction number for any fixed (constant) environment, i.e. for the linear models, we use a fairly recent spectral theoretic result, which characterizes the connection between the spectral bound of an unbounded operator and the spectral radius of a corresponding bounded operator. For nonlinear models, varying the environment naturally leads to a net reproduction function.

Positive steady states of reaction–diffusion–advection competition models in periodic environment

In this paper, we consider the positive steady states for reaction–diffusion–advection competition models in the whole space with a spatially periodic structure. Under the spatially periodic setting, we establish sufficient conditions for the existence of positive steady states of this model, respectively, by investigating the sign of the principal eigenvalue for some linearized eigenvalue problems. As an application, a Lotka–Volterra reaction–diffusion–advection model for two competing species in a spatially periodic environment is considered. Finally, some numerical simulations are presented to seek dynamical behaviors.

Existence of positive solution for a cross-diffusion predator-prey system with Holling type-II functional response

This paper is purported to investigate a cross-diffusion system arising in a predator-prey population model including Holling type-II functional response in a bounded domain with Dirichlet boundary condition. The asymptotical stabilities are investigated to this system by using the method of eigenvalue. Moreover, the existence of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function.

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.

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.

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].

Mathematical description of the interactions of CycE/Cdk2, Cdc25A, and P27Kip1 in a core cancer subnetwork

Recent studies have shown that the initiation of human cancer is due to the malfunction of some genes (such as E2F, CycE, CycD, Cdc25A, P27Kip1, and Rb) at the R‐checkpoint during the G1‐to‐S transition of the cell cycle. Identifying and modeling the dynamics of these genes provide new insight into the initiation and progression of many types of cancers. In this study, a cancer subnetwork that has a mutual activation between phosphatase Cdc25A and the CycE/Cdk2 complex and a mutual inhibition between the Cdk inhibitor P27Kip1 and the CycE/Cdk2 complex is identified. A new mathematical model for the dynamics of this cancer subnetwork is developed. Positive steady states are determined and rigorously analyzed. We have found a condition for the existence of positive steady states from the activation, inhibition, and degradation parameter values of the dynamical system. We also found a robust condition that needs to be satisfied for the steady states to be asymptotically stable. We determine the parameter value(s) under which the system exhibits a saddle–node bifurcation. We also identify the condition for which the system exhibits damped oscillation solutions. We further explore the possibility of Hopf and homoclinic bifurcations from the saddle–focus steady state of the system. Our analytic and numerical results confirm experimental results in the literature, thus validating our model. Copyright © 2016 John Wiley & Sons, Ltd.

Spatial dynamics of a diffusive predator-prey model with stage structure

In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.

Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities

We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a fixed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new fixed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any finite dimension $n$. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial differential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile-adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process.