A competition un-stirred chemostat model with virus in an aquatic system

A reaction–diffusion system of two bacteria species competing a single limiting nutrient with the consideration of virus infection is derived and analysed. Firstly, the well-posedness of the system, the existence of the trivial and semi-trivial steady states, and some prior estimations of the steady states are given. Secondly, a single species subsystem with virus is studied. The stability of the trivial and semi-trivial steady states and the uniform persistence of the subsystem are obtained. Further, taking the infective ability of virus as a bifurcation parameter, the global structure of the positive steady states and the effect of virus on the positive steady states are established via bifurcation theory and limiting arguments. It shows that the backward bifurcation may occur. Some sufficient conditions for the existence, uniqueness and stability of the positive steady state are also obtained. Finally, some sufficient conditions on the existence of the positive steady states for the full system are derived by using the fixed point index theory. Some results on persistence or extinction for the full system are also obtained.     

A microbial continuous culture system with diffusion and diversified growth

A reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth. The existence of nonnegative solutions and attractors for the system is obtained, the stability of steady states and the steady state bifurcation are studied under three growth conditions. In the case of no growth inhibition or only product inhibition, the system admits one positive constant steady state which is stable; in the case of growth inhibition only by substrate, the system can have two positive constant steady states, explicit conditions of the stability and the steady state bifurcation are also determined. In addition, numerical simulations are given to exhibit the theoretical results.     

Modeling on an ecological food chain with recycling

We propose two nutrient-phytoplankton models with instantaneous and time delayed recycling, investigate the dynamics and examine the responses to model complexities. Instead of the familiar specific uptake rate and growth rate functions, we assume only that the nutrient uptake and phytoplankton growth rate functions are positive, increasing and bounded above. We use geometrical and analytical methods to find conditions for the existence of none, one, or at most two positive steady states and analyze the stability properties of each of these equilibria. With the variation of parameters, the system may lose its stability and bifurcation may occur. We study the occurrence of Hopf bifurcation and the possibility of stability switching. Numerical simulations illustrate the analytical results and provide further insight into the dynamics of the models, biological interpretations are given.     

Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations

We consider a nonlinear mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting age-structured model is reduced to a system of three differential equations, with three independent delays, and existence of steady states is investigated. The stability of the trivial steady state, describing cells dying out with a saturation of growth factor concentrations is proven to be asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state allows the determination of a stability area, and shows that instability may occur through a Hopf bifurcation, mainly as a destabilization of the proliferative capacity control, when cell cycle durations are very short. Numerical simulations are carried out and result in a stability diagram that stresses the lead role of the introduction rate compared to the apoptosis rate in the system stability.     

The existence and stability of steady states for a prey-predator system with cross diffusion of quasilinear fractional type

This paper is concerned with the existence and stability of steady states for a prey-predator system with cross diffusion of quasilineax fractional type. We obtain a sufficient condition for the existence of positive steady state solutions by applying bifurcation theory and a detailed priori estimate. In virtue of the principle of exchange of stability, we prove the stability of local bifurcating solutions near the bifurcation point.     

The Existence and Stability of Steady States for a Prey-predator System with Cross Difusion of Quasilinear Fractional Type

This paper is concerned with the existence and stability of steady states for a prey-predator system with cross difusion of quasilinear fractional type.We obtain a sufcient condition for the existence of positive steady state solutions by applying bifurcation theory and a detailed priori estimate.In virtue of the principle of exchange of stability,we prove the stability of local bifurcating solutions near the bifurcation point.     

The Coexistence States of a Predator-Prey Model with Nonmonotonic Functional Response and Diffusion

In this paper, a predator-prey model with nonmonotonic functional response is concerned. Using spectrum analysis and bifurcation theory, the bifurcating solution and its stability of the model are investigated. We discuss the bifurcation solution which emanates from the semi-trivial solution by taking the death rate as a bifurcation parameter. Furthermore, by fixed point’s index theory, the result of existence or nonexistence of positive steady states of the model is also obtained.     

The local bifurcation and stability of nontrivial steady states of a logistic type of chemotaxis

Chemotaxis is a type of oriented movement of cells in response to the concentration gradient of chemical substances in their environment. We consider local existence and stability of nontrivial steady states of a logistic type of chemotaxis. We carry out the bifurcation theory to obtain the local existence of the steady state and apply the expansion method on the chemotaxis to investigate the bifurcation direction. Moreover, by applying the bifurcation direction, we obtain the bifurcating steady state is stable when the bifurcation curve turns to right under certain conditions.     

Loops and branches of coexistence states in a Lotka-Volterra competition model

A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.     

Dynamics of three coupled limit cycle oscillators with application to artificial intelligence

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can make decisions by identifying a given initial condition with its associated steady state.     

Allee effects in a Ricker-type predator–prey system

We study a discrete host–parasitoid system where the host population follows the classical Ricker functional form and is also subject to Allee effects. We determine basins of attraction of the local attractors of the single population model when the host intrinsic growth rate is not large. In this situation, existence and local stability of the interior steady states for the host–parasitoid interaction are completely analysed. If the host's intrinsic growth rate is large, then the interaction may support multiple interior steady states. Linear stability of these steady states is provided.     

一个趋化性模型非常数平衡解的存在性

对带两个趋化性参数的趋化性模型平衡解的存在性问题进行研究.在参数满足特定的条件下,应用局部分岔理论得到非常数平衡解的局部分岔结构,从而证明了该趋化性模型存在无穷多个非常数正平衡解.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

Qualitative analysis of stationary Keller–Segel chemotaxis models with logistic growth

We study the stationary Keller–Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing’s instability as the chemotaxis rate \({\chi}\) surpasses a threshold number. By taking the chemotaxis rate as the bifurcation parameter, we carry out bifurcation analysis on the system to obtain the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions. Moreover, we show that solutions stay strictly positive in the continuum of each branch. The stabilities of these steady-state solutions are well studied when the creation and degradation rate of the chemical is assumed to be a linear function. Finally, we investigate the asymptotic behaviors of the monotone steady states. We construct solutions with interesting patterns such as a boundary spike when the chemotaxis rate is large enough and/or the cell motility is small.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Diffusive logistic equation with constant yield harvesting, I: Steady States

We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

     

Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model

We investigate a discrete consumer-resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. Using intrinsic growth rate of the consumer population as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability.     

Stability of steady-state solutions to a prey-predator system with cross-diffusion

This paper is concerned with a cross-diffusion system arising in a prey-predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

On the bifurcation analysis of a food web of four species

This paper is concerned with bifurcations of equilibria and the chaotic dynamics of a food web containing a bottom prey X, two competing predators Y and Z on X, and a super-predator W only on Y. Conditions for the existence of all equilibria and the stability properties of most equilibria are derived. A two-dimensional bifurcation diagram with the aid of a numerical method for identifying bifurcation curves is constructed to show the bifurcations of equilibria. We prove that the dynamical system possesses a line segment of degenerate steady states for the parameter values on a bifurcation line in the bifurcation diagram. Numerical simulations show that these degenerate steady states can help to switch the stabilities between two far away equilibria when the system crosses this bifurcation line. Some observations concerned with chaotic dynamics are also made via numerical simulations. Different routes to chaos are found in the system. Relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.