A global bifurcation theorem for Darwinian matrix models

Motivated by models from evolutionary population dynamics, we study a general class of nonlinear difference equations called matrix models. Under the assumption that the projection matrix is non-negative and irreducible, we prove a theorem that establishes the global existence of a continuum with positive equilibria that bifurcates from an extinction equilibrium at a value of a model parameter at which the extinction equilibrium destabilizes. We give criteria for the global shape of the continuum, including local direction of bifurcation and its relationship to the local stability of the bifurcating positive equilibria. We discuss a relationship between backward bifurcations and Allee effects. Illustrative examples are given.     

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.     

Modeling and analysis of a prey–predator system with disease in the prey

A three dimensional ecoepidemiological model consisting of susceptible prey, infected prey and predator is proposed and analysed in the present work. The parameter delay is introduced in the model system for considering the time taken by a susceptible prey to become infected. Mathematically we analyze the dynamics of the system such as, boundedness of the solutions, existence of non-negative equilibria, local and global stability of interior equilibrium point. Next we choose delay as a bifurcation parameter to examine the existence of the Hopf bifurcation of the system around its interior equilibrium. Moreover we use the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Some numerical simulations are carried out to support the analytical results.     

NONLINEAR MATRIX MODELS AND POPULATION DYNAMICS

Nonlinear matrix difference equations are studied as models for the discrete time dynamics of a population whose individual members have been categorized into a finite number of classes. The equations are treated with sufficient generality so as to include virtually any type of structuring of the population (the sole constraint is that all newborns lie in the same class) and any types of nonlinearities which arise from the density dependence of fertility rates, survival rates and transition probabilities between classes. The existence and stability of equilibrium class distribution vectors are studied by means of bifurcation theory techniques using a single composite, biologically meaningful quantity as a bifurcation parameter, namely the inherent net reproductive rate r. It is shown that, just as in the case of linear matrix equations, a global continuum of positive equilibria exists which bifurcates as a function of r from the zero equilibrium state at and only at r = 1. Furthermore the zero equilibrium loses stability as r is increased through 1. Unlike the linear case however, for which the bifurcation is “vertical” (i.e., equilibria exist only for r = 1), the nonlinear equation in general has positive equilibria for an interval of r values. Methods for studying the geometry of the continuum based upon the density dependence of the net reproductive rate at equilibrium are developed. With regard to stability, it is shown that in general the positive equilibria near the bifurcation point are stable if the bifurcation is to the right and unstable if it is to the left. Some further results and conjectures concerning stability are also given. The methods are illustrated by several examples involving nonlinear models of various types taken from the literature.     

Hopf bifurcation analysis on an Internet congestion control system of arbitrary dimension with communication delay

This paper carries out a Hopf bifurcation analysis on a model of Internet congestion control system for a network with arbitrary topology. The general form of the rate-based Kelly model for a multi-source multi-link network with a communication delay is considered. Assuming the communication delay as a bifurcation parameter, we find that when the delay parameter passes a critical value, a periodic solution bifurcates from the equilibrium point. The stability and direction of bifurcating periodic solutions are studied by using the center manifold theorem and the normal form theory. We simulate our model for a typical example to show the applicability of the approach.     

Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response

In this paper, a diffusive predator–prey system with Holling III functional response and nonconstant death rate subject to Neumann boundary condition is considered. We study the stability of equilibria, and Turing instability of the positive equilibrium. We also perform a detailed Hopf bifurcation analysis to PDE system, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution. In addition, some numerical simulations are carried out.     

Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a real eigenvalue crosses the imaginary axis. For a model we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, again with respect to spatially localized perturbations.     

Dynamics of a competitive Lotka-Volterra system with three delays

In this paper, a competitive Lotka-Volterra system with three delays is investigated. By choosing the sum τ of three delays as a bifurcation parameter, we show that in the above system, Hopf bifurcation at the positive equilibrium can occur as τ crosses some critical values. And we obtain the formulae determining direction of Hopf bifurcation and stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a real eigenvalue crosses the imaginary axis. For a model we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, again with respect to spatially localized perturbations.     

Effects of awareness program and delay in the epidemic outbreak

In the present paper, an epidemic model has been proposed and analyzed to investigate the impact of awareness program and reporting delay in the epidemic outbreak. Awareness programs induce behavioral changes within the population, and divide the susceptible class into two subclasses, aware susceptible and unaware susceptible. The existence and the stability criteria of the equilibrium points are obtained in terms of the basic reproduction number. Considering time delay as the bifurcating parameter, the Hopf bifurcation analysis has been performed around the endemic equilibrium. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. To verify the analytical results, comprehensive numerical simulations are carried out. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure

In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Hopf bifurcation analysis of a density predator-prey model with crowley-martin functional response and two time delays

In this paper, a delayed density dependent predator-prey model with Crowley-Martin functional response and two time delays for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcation at the coexistence equilibrium is established. With the help of normal form method and center manifold theorem, some explicit formulas determining the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations are given to illustrate the theoretical results.     

Global bifurcation of positive equilibria in nonlinear population models

Existence of nontrivial nonnegative equilibrium solutions for age-structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Bifurcation of a Modified Leslie-Gower System with Discrete and Distributed Del

A modified Leslie-Gower predator-prey system with discrete and distributed delays is introduced. By analyzing the associated characteristic equation, stability and local Hopf bifurcation of the model are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.     

Delay induced oscillation in predator–prey system with Beddington–DeAngelis functional response

The Beddington–DeAngelis predator–prey system with distributed delay is studied in this paper. At first, the positive equilibrium and its local stability are investigated. Then, with the mean delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. The bifurcating periodic solutions are analyzed by means of the normal form and center manifold theorems. Finally, numerical simulations are also given to illustrate the results.     

Stability Analysis of an Enterprise Competitive Model with Time Delay

A three-dimensional enterprise competitive model with time delay is considered. Where the delay is regarded as bifurcation parameters. By analyzing the corresponding characteristic equation of positive equilibrium,the local stability of positive equilibrium is regarded. By using the normal form method and center manifold theorem, we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are shown to illustrate the obtained results.     

A stage-structured predator-prey system with time delay

A stage-structured predator-prey system with time delay is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results. Based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is established.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

A bifurcation analysis of stage-structured density dependent integrodifference equations

There is evidence for density dependent dispersal in many stage-structured species, including flour beetles of the genus Tribolium. We develop a bifurcation theory approach to the existence and stability of (non-extinction) equilibria for a general class of structured integrodifference equation models on finite spatial domains with density dependent kernels, allowing for non-dispersing stages as well as partial dispersal. We show that a continuum of such equilibria bifurcates from the extinction equilibrium when it loses stability as the net reproductive number n increases through 1. Furthermore, the stability of the non-extinction equilibria is determined by the direction of the bifurcation. We provide an example to illustrate the theory.