Takens Bifurcation Research Articles

Dynamics of a Prey–Predator System with Foraging Facilitation in Predators

The dynamics of a prey–predator system with foraging facilitation among predators are investigated. The analysis involves the computation of many semi-algebraic systems of large degrees. We apply the pseudo-division reduction, real-root isolation technique and complete discrimination system of polynomial to obtain the parameter conditions for the exact number of equilibria and their qualitative properties as well as do a complete investigation of bifurcations including saddle-node, transcritical, pitchfork, Hopf and Bogdanov–Takens bifurcations. Moreover, numerical simulations are presented to support our theoretical results.

The bifurcation analysis of an SIRS epidemic model with immunity age and constant treatment

ABSTRACT In this paper, an SIRS epidemic model with immunity age is investigated, where the constant treatment rate and the loss of the acquired immunity are incorporated. The well-posedness of the model is verified by changing it into an abstract non-densely defined Cauchy problem, and the conditions for the existence of disease-free equilibrium and the endemic equilibria are found. The theoretic analysis showed that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number is less than unity, and the numerical simulation illustrated that it is asymptotically stable as the number is greater than unity. Combining numerical simulations, the instability and the local stability of different endemic equilibrium, and the existence of saddle-node bifurcation, and Hopf bifurcation are analyzed. Again, we think it is possible that the Bogdanov–Takens bifurcation may occur for the model under some conditions. Both non-periodic and periodic behaviors are shown when the disease persists in population, where the duration that the recovered individual stays in the recovery class plays an important role in the spread of the disease.

A Holling–Tanner Predator–Prey Model with Strong Allee Effect

We analyze a modified Holling–Tanner predator–prey model where the predation functional response is of Holling type II and we incorporate a strong Allee effect associated with the prey species production. The analysis complements the results of previous articles by Saez and González-Olivares [1999] and Arancibia-Ibarra and González-Olivares [2015] discussing Holling–Tanner models which incorporate a weak Allee effect. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to coexistence and extinction of the species. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogdanov–Takens bifurcations.

Bogdanov–Takens singularity in the simple memristive time-delay system

In this paper, a simple memristive system is considered. We obtain existence of Bogdanov–Takens (B–T) bifurcation at the equilibrium in the 2D memristive time-delay system. With the change of two bifurcation parameters, in particular, it will lead to different bifurcations when the delay passes a certain critical value. Based on center manifold and classic normal form method, Hopf, pitchfork, homoclinic, and double limit cycle bifurcation are derived.

Multiple bifurcations analysis in a delayed predator–prey system with disease in prey and stage structure for predator

In this paper, a delayed eco-epidemiological model with disease in prey, anti-predator factor for the prey and stage structure for the predator is proposed. The conditions for the stability and coexistence of the populations have been derived. Positivity and boundedness of all solutions with positive initial conditions has been proved. Sufficient conditions for the occurrence of several Hopf and Bogdanov–Takens bifurcations about the endemic steady state are derived. By taking the delay as the bifurcation parameter, it is shown that Hopf and Bogdanov–Takens bifurcations occur at positive steady state for some critical values of the delay. The normal form of the Bogdanov–Takens singularity is obtained by performing a center manifold reduction and by using the normal form theory of FDEs. Some numerical simulations of solutions in different parametric regions illustrating Hopf and homoclinic bifutcations are given to support the analytical results.

Unfolding Symmetric Bogdanov–Takens Bifurcations for Front Dynamics in a Reaction–Diffusion System

This manuscript extends the analysis of a much studied singularly perturbed three-component reaction-diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to stable periodic front motion, including stable traveling breathers, and these results are illustrated by numerical computations.

Automatic Detection of Saddle-Node–Transcritical Interactions

Dynamical systems with special structure can exhibit transcritical bifurcations of codimension one. In such systems, the interactions of transcritical bifurcations of codimension two can act as organizing centers. We consider saddle-node–transcritical interactions with either one or two zero eigenvalues and show that, using default test functions, the widely used continuation packages MatCont and AUTO classify these interactions as cusp and Bogdanov–Takens bifurcations, respectively. We propose a new test function that distinguishes these singularities and demonstrate its use in the analysis of a predator–prey-nutrient model strained by a toxicant. The details of the implementation are provided, along with test codes for MatCont.

Bifurcation of Multiple Limit Cycles in an Epidemic Model on Adaptive Networks

Very recently, Zhang et al. considered an epidemic model on adaptive networks [Zhang et al., 2019], in which Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation are studied. Degenerate Hopf bifurcation is investigated via simulation and a numerical example is given to show the existence of two limit cycles. However, whether the codimension of the Hopf bifurcation is two is still open. In this paper, we will rigorously prove that the codimension of the Hopf bifurcation is two. That is, the maximal two limit cycles can bifurcate from the Hopf critical point. Moreover, the conditions for the existence of two limit cycles are derived.

Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment

• A delayed epidemic model with logistic growth and saturated treatment is studied. • Backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation may occur. • Effects of varying parameters are studied in numerical simulations. • Presence of logistic growth and high treatment rate can decrease infection levels. In this paper, we incorporate a nonlinear incidence rate and a logistic growth rate into a SIR epidemic model for a vector-borne disease with incubation time delay and Holling type II saturated treatment. We compute the basic reproduction number and show that it completely determines the local stability of the disease-free equilibrium. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also established. Furthermore, we determine the direction and stability of the Hopf bifurcation around the endemic equilibrium by means of the center manifold theory. Our study reveals that the model admits a Bogdanov–Takens bifurcation when the time delay and the maximal disease transmission rate are varied. Numerical simulations are presented to illustrate the dynamics of the model and to study the effects caused by varying the treatment rate and delay parameters.

Approximations of the Heteroclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two. Application to a Liénard Equation

A dynamical system possessing an equilibrium point with two zero eigenvalues is considered. We assume that a degenerate Bogdanov–Takens bifurcation with symmetry of order two is present and, in the parameter space, a curve of heteroclinic bifurcation values emerges from the codimension two bifurcation point. Using a blow-up transformation and a perturbation method, we obtain second order approximations both for the heteroclinic orbits and for the curve of heteroclinic bifurcation values. Applications of our results for the truncated normal form and for a Liénard equation are presented. Some numerical simulations illustrating the accuracy of our results are performed.

Bifurcation Analysis of an Enzyme Reaction System with General Power of Autocatalysis

In this paper, we study the local bifurcations of an enzyme reaction system with positive parameters [Formula: see text], [Formula: see text], [Formula: see text] and integer [Formula: see text]. This system is orbitally equivalent to a polynomial differential system of order [Formula: see text]. Although not all coordinates of its equilibria can be calculated because of the high order polynomial, parameter conditions for the existence of each equilibrium are given. Moreover, qualitative properties of each equilibrium are determined. It shows that various bifurcations, including saddle-node bifurcation, Bogdanov–Takens bifurcation and Hopf bifurcation, may occur in this system as the parameters are varied. With Lyapunov quantities, the maximum order of the weak focus in this system is proved to be at most two by resultant elimination and real-root isolation. Furthermore, parameter conditions of its exact order are obtained. Finally, numerical simulations demonstrate the theoretical results.

Complex Dynamics in a Unified SIR and HIV Disease Model: A Bifurcation Theory Approach

This paper is concerned with complex dynamical behaviors of a simple unified SIR and HIV disease model with a convex incidence and four real parameters. Due to the complex nature of the disease dynamics, our goal is to explore bifurcations including multistable states, limit cycles, and homoclinic loops in the whole parameter space. The first contribution is the proof of the existence of multiple limit cycles giving rise from Hopf bifurcation, which further induces bistable or tristable states because of the coexistence of stable equilibria and periodic motion. Next, we propose that the existence of Bogdanov–Takens (BT) bifurcation yields the bifurcation of homoclinic loops, which provides a new mechanism for generating disease recurrence, for example, the relapse–remission, viral blip cycles in HIV infection. Last, we present a novel method for the derivation of the normal forms of codimension two and three BT bifurcations. The method is based on the simplest normal form theory from Yu’s previous works.

Bifurcation manifolds in predator–prey models computed by Gröbner basis method

Many natural processes studied in population biology, systems biology, biochemistry, chemistry or physics are modeled by dynamical systems with polynomial or rational right-hand sides in state and parameter variables. The problem of finding bifurcation manifolds of such discrete or continuous dynamical systems leads to a problem of finding solutions to a system of non-linear algebraic equations. This approach often fails since it is not possible to express equilibria explicitly. Here we describe an algebraic procedure based on the Gröbner basis computation that finds bifurcation manifolds without computing equilibria. Our method provides formulas for bifurcation manifolds in commonly studied cases in applied research – for the fold, transcritical, cusp, Hopf and Bogdanov–Takens bifurcations. The method returns bifurcation manifolds as implicitly defined functions or parametric functions in full parameter space. The approach can be implemented in any computer algebra system; therefore it can be used in applied research as a supporting autonomous computation even by non-experts in bifurcation theory. This paper demonstrates our new approach on the recently published Rosenzweig–MacArthur predator–prey model generalizations in order to highlight the simplicity of our method compared to the published analysis.

Global Dynamics in a Beddington–DeAngelis Prey–Predator Model with Density Dependent Death Rate of Predator

The article aims to investigate a prey–predator model which includes density dependent death rate for predators and Beddington–DeAangelis type functional response. We observe the changes in the existence and stability of the equilibrium points and investigate the complete global dynamics of the model. A two-parametric bifurcation diagram has been described here which shows the effect of density dependent death rate parameter of predator. We have also examined all possible local and global bifurcations that the system could go through, namely transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation and homoclinic bifurcation.

Qualitative analysis of additional food provided predator–prey system with anti-predator behaviour in prey

Predator–prey interactions are the most common phenomena in the natural population, which are widely exploited for control of economically damaging pest species in an eco-friendly manner. It is proven that the pest adopts several mechanisms for overcoming the predation pressure. Two such techniques which are not well studied in the literature are group defence, anti-predator behaviour in prey. Anti-predator behaviour of prey is a counterattacking technique, in which adult prey attacks the juvenile predators to reduce the future predation pressure. From the existing experimental studies, it is observed that the anti-predator behaviour of the pest can have adverse effects on the biological control programmes. One of the ways to overcome the loss due to anti-predator behaviour is to cater the predators with some alternative food. In this current work, we have investigated this aspect in a dynamical system framework. This study uncovers many fascinating phenomena. It is found that the considered system displays rich dynamics and demonstrates various bifurcations such as saddle-node bifurcation, (supercritical and subcritical) Hopf bifurcation, homoclinic bifurcation and a Bogdanov–Takens bifurcation of co-dimension 2 are studied. Treating the anti-predator behaviour in prey as one of the control parameters, and by characterising the additional food supplied to the predators, the strategies for achieving the successful biological control are derived.

The basins of attraction in a modified May–Holling–Tanner predator–prey model with Allee affect

I analyse a modified May–Holling–Tanner predator–prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator–prey population. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle–node, Hopf, and Bogdanov–Takens bifurcations. We use simulations to illustrate the behaviour of the model.

Regularization of saddle-fold singularity for nonsmooth differential systems

This paper is concerned with the saddle-fold singularity that is a codimension two singularity for piecewise smooth planar differential systems. The approach consists in apply the Regularization's Method to the unfolding of the saddle-fold singularity in order to obtain a 2-parameter family of smooth vector fields. Our main result says that the regularized family undergoes thought a classical Bogdanov–Takens bifurcation.

Bifurcations and simulations of two predator–prey models with nonlinear harvesting

Abstract We propose and investigate a predator-prey model with selective nonlinear harvesting for the prey and predator, such harvesting increases smoothly to a limit value when the density of harvested population is large enough. The existence of nonlinear harvesting makes the dynamics of the model more complicated, including multiple equilibria, limit cycle, heteroclinic and homoclinic orbit, bistability, saddle-node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation and Bogdanov–Takens bifurcation. Furthermore, some numerical simulations are given to illustrate these results.

The impact of provision of additional food to predator in predator–prey model with combined harvesting in the presence of toxicity

In this article, the global dynamics of a predator–prey system incorporating combined harvesting has been investigated. The additional food is being provided to the predator, accordingly, modified Holling type-II functional response is considered in the model. The presence of toxicants affect the quality of food for both the species, reducing their growth. The steady states of the system are obtained under some suitable conditions. The local and global dynamics are explored. The conditions for permanence and existence for bionomic equilibrium of the system have been investigated. It is also observed that the system exhibits local bifurcations i.e., transcritical, Hopf, saddle-node as well as global bifurcations i.e., Bogdanov–Takens bifurcation and generalized Hopf bifurcation with respect to the suitable set of parameters. Optimal harvesting policy is discussed with the help of Pontryagin’s maximum principle to preserve both the species from extinction and maintain a sustainable fishery. Numerical simulations are carried out for the suitable choice of parameters.

Bifurcation Analysis of a Mosquito Population Model with a Saturated Release Rate of Sterile Mosquitoes

Releasing sterile mosquitoes is a method of mosquito control that uses area-wide inundative releases of sterile male mosquitoes to reduce reproduction in a field population of wild mosquitoes. In this paper, we consider a mosquito population model with a nonlinear saturated release rate of sterile mosquitoes and study the complex dynamics and bifurcations of the model. It is shown that there are a weak focus of multiplicity 3 and a nilpotent cusp of codimension 4 for various parameter values and the model exhibits Hopf bifurcation of codimension 3 and Bogdanov--Takens bifurcation of codimension 2 as the parameter values vary. Our analysis also shows that there exists a critical release rate coefficient of sterile mosquitoes, above which the mosquito population can be eliminated and below which the interacting sterile and wild mosquitoes coexist in the form of multiple periodic oscillations and steady states for some initial populations. Numerical simulations are presented to demonstrate the coexistence of a homoclinic loop and a limit cycle, the existence of two limit cycles, and the existence of three limit cycles, respectively.