Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

Stability Analysis of Diffusive Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

Stability of an age-structured epidemiological model for hepatitis C

This article introduces an age-structured epidemiological model for the disease transmission dynamics of hepatitis C. We first show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number ? ₀ is below one, in this case, the disease always dies out, then we prove that at least one endemic steady state exists when the reproductive number ? ₀ is above one, the stability conditions for the endemic steady states are also given.     

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.     

Consequences of refuge and diffusion in a spatiotemporal predator–prey model

In this investigation, we offer and examine a predator–prey interacting model with prey refuge in proportion to both the species and Beddington–DeAngelis functional response. We first prove the well-posedness of the temporal and spatiotemporal models which are restricted in a positive invariant region. Then for the temporal model, we analyse its temporal dynamics including uniform boundedness, permanence, stability of all feasible non-negative equilibria and show that refugia can induce periodic oscillation via Hopf bifurcation around the unique positive equilibrium; for the spatiotemporal model, we not only investigate its permanence, stability of non-negative constant steady states and Turing instability but also study the existence and non-existence of non-constant positive steady states by Leray–Schauder degree theory. The key observation is that the coefficient of refuge cooperates a significant part in modifying the dynamics of the current system and mediates the population permanence, stability of coexisting equilibrium and even the Turing instability parameter space. Finally, general numerical simulation consequences are given to illustrate the validity of the theoretical results. Through numerical simulations, one observes that the model dynamics shows prey refugia and self-diffusion control spatiotemporal pattern growth to spots, stripe–spot mixtures and stripes reproduction. The outcomes assign that the dynamics of the model with prey refuge is not simple, but rich and complex. Additionally, numerical simulations show that the other model parameters have an important effect on species’ spatially inhomogeneous distribution, which results in the formation of spots pattern, mixture of spots and stripes pattern, mixture of spots, stripes and rings pattern and anti-spot pattern. This may improve the model dynamics of the prey refuge on the reaction–diffusion predator–prey system.     

VOLE POPULATION DYNAMICS UNDER THE INFLUENCE OF SPECIALIST AND GENERALIST PREDATION

Abstract To understand the impact of predation by different types of predators on the vole population dynamics, we formulate a three differential equation model describing the population dynamics of voles, the “specialist predator” and the “generalist predator.” First we perform a local stability study of the different steady states of the basic model and deduce that the predation rates of the “specialist” as well as the “generalist” predator are the main parameters controlling the existence/extinction criteria of the concerned populations. Next we analyze the model from a thermodynamic perspective and study the thermodynamic stability of the different equilibria. Finally using stochastic driving forces, we incorporate the exogenous factor of environmental forcing and investigate the stochastic stability of the system. We compare the stability criteria of the different steady states under deterministic, thermodynamic and stochastic situations. The analysis reveals that when the “specialist” and the “generalist” predator are modeled separately, the system exhibits rich dynamics and the predation rates of both types of predators play a major role in controlling vole oscillation and/or stability. These findings are also seen to resemble closely with the observed behavior of voles in the natural setting. Numerical simulations are carried out to illustrate analytical findings.     

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.     

Steady state analysis of a nonlinear renewal equation

In the analysis of a nonlinear renewal equation it is natural to anticipate the existence of nonzero steady states and deal with the question of their stability. Sufficient conditions for existence and uniqueness for these steady states are given. The study of the linearized version of the renewal equation around the steady state helps to a great extent to have insight into some complicated dynamics of the full problem. At this stage the first eigenvalue of the steady state plays a vital role. The characteristic equation, a functional equation whose roots are the eigenvalues, is derived. We give various structures showing that the steady state may be stable or unstable (though the fertility rate is decreasing with competition). A similar study is carried out on a nonlinear model motivated by neuroscience in which the total population is conserved.     

Existence and instability of steady states for a triangular cross-diffusion system: A computer-assisted proof

In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton–Kantorovich theorem. It allows to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we obtain as many as 13 coexisting steady states. We also apply the a posteriori validation procedure to study the linear stability of the obtained steady states, proving that many of them are in fact unstable.     

Positive Steady States of a Prey-predator Model with Diffusion and Non-monotone Conversion Rate

In this paper, we study the positive steady states of a prey-predator model with diffusion throughout and a non-monotone conversion rate under the homogeneous Dirichlet boundary condition. We obtain some results of the existence and non-existence of positive steady states. The stability and uniqueness of positive steady states are also discussed.     

Chemical Reactor Dynamics: Stability of Steady States

The equations describing the dynamics and steady states of a tubular reactor are studied in the region where an arbitrary number of steady-state solutions exist. The stability of these solutions is examined and their persistence for increases in the parameter values is determined. The physical implications of these solutions are examined.     

Dynamics of a spatiotemporal model on populations in a polluted river

In this paper, we investigate the dynamics for a reaction–diffusion–advection system which models populations in a polluted river. More precisely, we study the stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Furthermore, some dependence of the stability of the toxicant-only steady state and the population-toxicant coexistence steady state on the model parameters are given.     

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.     

Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence

In this paper, an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence is proposed. In the model, we consider the infection age of infectious individuals and the biological age of pathogen in the environment. It is verified that the global dynamics of the model is completely determined by the basic reproduction number. Asymptotic smoothness is verified as a necessary argument. By analyzing corresponding characteristic equations, we discuss the local stability of each of feasible steady states. Uniform persistence is shown by using the persistence theory for infinite dimensional dynamical system. The global stability of each of feasible steady states is established by using suitable Lyapunov functionals and LaSalle’s invariance principle. Numerical simulations are carried out to illustrate the theoretical results.     

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.     

Stability results for a nonlinear two-species competition model with size-structure

We formulate a system of integro-differential equations to model the dynamics of competition in a two-species community, in which the mortality, fertility and growth are sizedependent. Existence and uniqueness of nonnegative solutions to the system are analyzed. The existence of the stationary size distributions is discussed, and the linear stability is investigated by means of the semigroup theory of operators and the characteristic equation technique. Some sufficient conditions for asymptotical stability/instability of steady states are obtained. The resulting conclusion extends some existing results involving age-independent and age-dependent population models.     

Equilibria of a predator prey model of phenotype evolution

A selection mutation predator prey model for the distribution of individuals with respect to an evolutionary trait is considered. Existence of steady states is proved using the theory of positive semigroups. We show that these steady states tend to concentrate, when the mutation rate is small, at the so-called evolutionary stable value of the trait in the sense of adaptive dynamics of the corresponding ecological model.     

A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.     

Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.     

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.