Semi-trivial Solutions Research Articles

Competition for two essential resources with internal storage and periodic input

We study a mathematical model of two species competing in a chemostat for two internally stored essential nutrients, where the nutrients are added to the culture vessel by way of periodic forcing functions. Persistence of a single species happens if the nutrient supply is sufficient to allow it to acquire a threshold of average stored nutrient quota required for growth to balance dilution. More precisely, the population is washed out if a sub-threshold criterion holds, while there is a globally stable positive periodic solution, if a super-threshold criterion holds. When there is mutual invasibility of both semitrivial periodic solutions of the two-species model, both uniform persistence and the existence of periodic coexistence state are established.

Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with d ≥ 3 equations

In this work we consider the weakly coupled Schrödinger cubic system{−Δui+λiui=μiui3+ui∑j≠ibijuj2ui∈H1(RN;R),i=1,…,d, where 1≤N≤3, λi,μi>0 and bij=bji>0 for i≠j. This system admits semitrivial solutions, that is solutions u=(u1,…,ud) with null components. We provide optimal qualitative conditions on the parameters λi, μi and bij under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial.This question had been clarified only in the d=2 equations case. For d≥3 equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case λi≡λ and bij≡b. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d=2 case.

Crowding effects on coexistence solutions in the unstirred chemostat

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.

Qualitative analysis of a producer–scrounger model

In this paper, we consider a producer–scrounger model proposed in [6]. We prove that when the trivial steady state solution or the semi-trivial steady state solution to the system is stable, then the system admits no positive steady state solution, while if both the trial steady state solution and the semi-trivial steady state solution are unstable, then the system admits at least one positive steady state solution. Furthermore, in the one dimension case, we prove that the positive steady state solution, if exists, is unique.

A Feedback Control Model of Comprehensive Therapy for Treating Immunogenic Tumours

Surgery is the traditional method for treating cancers, but it often fails to cure patients for complex reasons so new therapeutic approaches that include both surgery and immunotherapy have recently been proposed. These have been shown to be effective, clinically, in inhibiting cancer cells while allowing retention of immunologic memory. This comprehensive strategy is guided by whether a population of tumour cells has or has not exceeded a threshold density. Conditions for successful control of tumours in an immune tumour system were modeled and the related dynamics were addressed. A mathematical model with state-dependent impulsive interventions is formulated to describe combinations of surgery with immunotherapy. By analyzing the properties of the Poincaré map, we examine the global dynamics of the immune tumour system with state-dependent feedback control, including the existence and stability of the semi-trivial order-1 periodic solution and the positive order-[Formula: see text] periodic solution. The main results showed that surgery alone can only control the tumour size below a certain level while there is no immunologic memory. If comprehensive therapy involving combining surgery with immunotherapy is considered, then not only can the cancers be controlled below a certain level, but the immune system can also retain its activity. The existence of positive order-[Formula: see text] periodic solutions implies that periodical therapy is needed to control the cancers. However, choosing the treatment frequency and the strength of the therapy remains challenging, and hence a strategy of individual-based therapy is suggested.

Dynamic behaviors of a modified predator-prey model with state-dependent impulsive effects

In this paper, we formulate a two-dimensional autonomous predator-prey model with state-dependent impulsive effects and square root response function. The square root response function indicates that the prey population gather together for self-defense purposes. Firstly, we prove that the system has semitrivial periodic solution under some conditions. Furthermore, by the Poincare map and the theory of impulsive differential equations, the existence and stability of positive order-1 or order-2 periodic solution of the system are investigated in detail. The validity of all results is illustrated by numerical simulations.

Competition between two similar species in the unstirred chemostat

This paper deals with the competition between two similar speciesin the unstirred chemostat. Due to the strict competition of the unstirred chemostat model,the global dynamics of the system is attained by analyzing the equilibria and their stability.It turns out that the dynamics of the system essentially depends upon certain function of the growth rate.Moreover, one of the semi-trivial stationary solutions or the unique coexistence steady state is a global attractor under certain conditions.Biologically, the results indicate that it is possible for the mutant to force the extinction of resident species or to coexist with it.

Analysis on Bifurcation for a Predator-Prey Model with Beddington-DeAngelis Functional Response and Non-Selective Harvesting

The biomathematical model is an important tool in exploring the dynamic behavior of different species. In this paper, we deal with a predator-prey model with Beddington-DeAngelis functional response and non-selective harvesting. We discuss the existence and stability of the local bifurcation solution, which emanates from the semi-trivial solution. Moreover, by using the boundedness of positive solutions and the global bifurcation theory, we find that the local bifurcation branch can be extended to the global bifurcation.

Global analysis of a delayed Monod type chemostat model with impulsive input on two substrates

In this paper, a new Monod type chemostat model with delay and impulsive input on two substrates is considered. By using the global attractivity of a k times periodically pulsed input chemostat model, we obtain the bound of the system. By the means of a fixed point in a Poincare map for the discrete dynamical system, we obtain a semi-trivial periodic solution; further, we establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. Using the theory on delay functional and impulsive differential equations, we obtain a sufficient condition with time delay for the permanence of the system.

A predator-prey model of Holling-type II with state dependent impulsive effects

We investigate a predator-prey model with state dependent impulsive effects, which is based on a modified version of the Leslie-Gower scheme and on the Holling-type II scheme. Using topological methods, we present some sufficient conditions to guarantee the existence and asymptotical stability of semi-trivial periodic solutions and positive periodic solutions, respectively.

Multiple non semi-trivial solutions of systems of Kirchhoff-type equations with discontinuous nonlinearities inRN

In the present paper, we study the problem of multiple non semi-trivial solutions for the following systems of Kirchhoff-type equations with discontinuous nonlinearities (1.1) where F∈C1(RN×R+×R+,R),V∈C(RN,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi-trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.

A PDE system modeling the competition and inhibition of harmful algae with seasonal variations

In this paper, we study a reaction–diffusion–advection system modeling the competition of harmful algae with seasonal variations in a flowing water habitat. We assume that harmful algae produce toxins, which have inhibitory effects on their algal competitors, that is, the produced toxins can inhibit the growth of its competitor. For the single population model, we prove that the algae will be washed out eventually if the trivial periodic state is locally asymptotically stable, while there exists a unique positive periodic state which is globally attractive if the trivial periodic state is unstable. When there is mutual invasibility of both semitrivial periodic solutions of the two-species model, we are able to prove the existence of periodic coexistence state.

Dynamics of an impulsive predator–prey logistic population model with state-dependent

In this paper, the dynamics for a class of state-dependent impulsive predator–prey models with the logistic growth for the predator and prey species are analyzed. By a direct calculation, the existence of a semi-trivial periodic solution is obtained. Based on the geometrical analysis and biological background, the strict threshold value conditions for the existence of positive periodic solutions are depicted. The stabilities of the semi-trivial periodic solution and positive order-1 periodic solutions are proved due to the analogue of Poincaré criterion. Numerical results are carried out to illustrate the feasibility of our main results.

On the steady-state solutions of a nonlinear photonic lattice model

In this paper, we consider the steady-state solutions of the following equation related with nonlinear photonic lattice model Δu=Pu1+u2+v2+λu, Δv=Qv1+u2+v2+λv, where u, v are real-value function defined on R/(τ1Z) × R/(τ2Z). The existence and non-existence of non-constant semi-trivial (with only one component zero) solutions are considered.

Dynamics analysis of a two-species competitive model with state-dependent impulsive effects

In this paper, we present two novel mathematical models of two-species competitive model with state-dependent impulsive control strategies, one is incorporating the density of one species as control threshold value, the other determines the control strategies by monitoring the densities of the two species. By the Poincaré map, analogue of Poincaré criterion and qualitative analysis method, some sufficient conditions on the existence and orbitally asymptotical stability of the semi-trivial periodic solution, positive order-1 or order-2 periodic solution of two models are presented. Furthermore, bifurcation diagrams and phase diagrams are investigated by means of numerical simulations, which illustrate the feasibility of our main results presented here.

A reaction–advection–diffusion system modeling the competition for two complementary resources with seasonality in a flowing habitat

In this current paper, we investigate a periodic bio-reactor model where microorganisms compete for two essential nutrient resource(s) needed for growth. For the single population growth model: we show that when the trivial solution is (locally) asymptotically stable, then the single population will be washed out; when the trivial solution is unstable, there is a unique periodic positive solution which attracts all solutions with nonzero initial data. For the two species model, we prove that the existence of a periodic coexistence state is possible if each species can invade the semi-trivial periodic state established by the other species. More precisely, if the semi-trivial periodic solutions are both unstable, there exists at least one periodic coexistence solution. Numerical work indicates conditions for persistence depend on the flow characteristics (advection and diffusivity). From our numerical simulations, competitive exclusion, bistability, and coexistence are all observed.

Holling type II predator–prey model with nonlinear pulse as state-dependent feedback control

We propose a Holling II predator–prey model with nonlinear pulse as state-dependent feedback control strategy and then provide a comprehensively qualitative analysis by using the theories of impulsive semi-dynamical systems. First, the Poincaré map is constructed based on the domains of impulsive and phase sets which are defined according to the phase portraits of the model. Second, the threshold conditions for the existence and stability of the semi-trivial periodic solution are given, and subsequently an order-1 periodic solution is generated through the transcritical bifurcation. Furthermore, the different parameter spaces for the existence and stability of an order-1 periodic solution are investigated. In addition, the existence and nonexistence of order-k(k≥2) periodic solutions have been studied theoretically. Moreover, the numerical investigations are presented in order to substantiate our theoretical results and show the complex dynamics of proposed model. Finally, some biological implications of the mathematical results are discussed in the conclusion section.

Существование нижних и верхних решений в обратном порядке по отношению к переменной в модели ацидогенеза для анаэробного сбраживания

We prove existence of upper and lower solutions in reverse order with respect a part of the variables in a system of nonlinear ordinary di erential equations modelling acidogenesis in anaerobic digestion. The corresponding existence theorems are established. The upper and lower solutions are constructed analytically, by de ning semi-trivial solutions for each of the variables in the model. We introduce the concept of indicator semi-trivial solutions. Finally, we numerically solve the system supported by the Matlab software and matching the graphs of the numerical solutions with analytical solutions is found.

Positive Coexistence of Steady States for a Diffusive Ratio-Dependent Predator-Prey Model with an Infected Prey

We examine a diffusive ratio-dependent predator-prey system with disease in the prey under homogeneous Dirichlet boundary conditions with a hostile environment at its boundary. We investigate the positive coexistence of three interacting species (susceptible prey, infected prey, and predator) and provide nonexistence conditions of positive solutions to the system. In addition, the global stability of the trivial and semitrivial solutions to the system is studied. Furthermore, the biological interpretation based on the result is also presented. The methods are employed from a comparison argument for the elliptic problem as well as the fixed-point theory as applied to a positive cone on a Banach space.

Effects of interface delay in real-time dynamic substructuring tests on a cable for cable-stayed bridge

Real-time dynamic substructuring tests have been conducted on a cable-deck system. The cable is representative of a full scale cable for a cable-stayed bridge and it interacts with a deck, numerically modelled as a single-degree-of-freedom system. The purpose of exciting the inclined cable at the bottom is to identify its nonlinear dynamics and to mark the stability boundary of the semi-trivial solution. The latter physically corresponds to the point at which the cable starts to have an out-of-plane response when both input and previous response were in-plane. The numerical and the physical parts of the system interact through a transfer system, which is an actuator, and the input signal generated by the numerical model is assumed to interact instantaneously with the system. However, only an ideal system manifests a perfect correspondence between the desired signal and the applied signal. In fact, the transfer system introduces into the desired input signal a delay, which considerably affects the feedback force that, in turn, is processed to generate a new input. The effectiveness of the control algorithm is measured by using the synchronization technique, while the online adaptive forward prediction algorithm is used to compensate for the delay error, which is present in the performed tests. The response of the cable interacting with the deck has been experimentally observed, both in the presence of delay and when delay is compensated for, and it has been compared with the analytical model. The effects of the interface delay in real-time dynamic substructuring tests conducted on the cable-deck system are extensively discussed.