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Discrete & Continuous Dynamical Systems - S

November 2020 , Volume 13 , Issue 11

Issue on recent advances in bifurcation theory and application

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Preface: Recent advances in bifurcation theory and application
Jingli Ren and Gail S. K. Wolkowicz
2020, 13(11): i-ii doi: 10.3934/dcdss.2020417 +[Abstract](121) +[HTML](37) +[PDF](91.57KB)
Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator
Dingyong Bai, Jianshe Yu and Yun Kang
2020, 13(11): 2949-2973 doi: 10.3934/dcdss.2020132 +[Abstract](720) +[HTML](310) +[PDF](1203.75KB)

In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value \begin{document}$ \rho $\end{document} of the ratio \begin{document}$ \frac{d_2}{d_1} $\end{document} of diffusions of predator to prey is obtained, such that if \begin{document}$ \frac{d_2}{d_1}>\rho $\end{document}, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if \begin{document}$ \frac{d_2}{d_1}<\rho $\end{document}, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio \begin{document}$ \frac{d_2}{d_1} $\end{document}. As \begin{document}$ \frac{d_2}{d_1} $\end{document} increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.

A nutrient-prey-predator model: Stability and bifurcations
Mary Ballyk, Ross Staffeldt and Ibrahim Jawarneh
2020, 13(11): 2975-3004 doi: 10.3934/dcdss.2020192 +[Abstract](587) +[HTML](265) +[PDF](778.71KB)

We model a nutrient-prey-predator system in a chemostat with general functional responses, using the input concentration of nutrient as the bifurcation parameter. We study changes in the existence and the stability of isolated equilibria, as well as changes in the global dynamics, as the nutrient concentration varies. The bifurcations of the system are analytically verified and we identify conditions under which an equilibrium undergoes a Hopf bifurcation and a limit cycle appears. Numerical simulations for specific functional responses illustrate the general results.

Stability analysis of an equation with two delays and application to the production of platelets
Loïs Boullu, Laurent Pujo-Menjouet and Jacques Bélair
2020, 13(11): 3005-3027 doi: 10.3934/dcdss.2020131 +[Abstract](625) +[HTML](326) +[PDF](1069.93KB)

We analyze the stability of a differential equation with two delays originating from a model for a population divided into two subpopulations, immature and mature, and we apply this analysis to a model for platelet production. The dynamics of mature individuals is described by the following nonlinear differential equation with two delays: \begin{document}$ x'(t) = -\gamma x(t) + g(x(t-\tau_1)) - g(x(t-\tau_1 - \tau_2)) e^{-\gamma \tau_2} $\end{document}. The method of D-decomposition is used to compute the stability regions for a given equilibrium. The centre manifold theory is used to investigate the steady-state bifurcation and the Hopf bifurcation. Similarly, analysis of the centre manifold associated with a double bifurcation is used to identify a set of parameters such that the solution is a torus in the pseudo-phase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate \begin{document}$ \gamma $\end{document} or of a decrease of the survival time \begin{document}$ \tau_2 $\end{document} of platelets on the onset of oscillations. We show that the stability is lost through a small decrease of survival time (from 8.4 to 7 days), or through an important increase of the death rate (from 0.05 to 0.625 days\begin{document}$ ^{-1} $\end{document}).

Dynamical stabilization and traveling waves in integrodifference equations
Adèle Bourgeois, Victor LeBlanc and Frithjof Lutscher
2020, 13(11): 3029-3045 doi: 10.3934/dcdss.2020117 +[Abstract](623) +[HTML](340) +[PDF](1253.04KB)

Integrodifference equations are discrete-time analogues of reaction-diffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.

Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes
Guowei Dai, Alfonso Romero and Pedro J. Torres
2020, 13(11): 3047-3071 doi: 10.3934/dcdss.2020118 +[Abstract](579) +[HTML](303) +[PDF](470.44KB)

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime

with \begin{document}$ 0 $\end{document}-Dirichlet boundary condition on the unit ball. According to the behavior of \begin{document}$ H $\end{document} near \begin{document}$ 0 $\end{document}, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs.

Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects
Juping Ji and Lin Wang
2020, 13(11): 3073-3081 doi: 10.3934/dcdss.2020135 +[Abstract](540) +[HTML](298) +[PDF](777.16KB)

In this paper, we analyze a nutrient-phytoplankton model with toxic effects governed by a Holling-type Ⅲ functional. We show the model can undergo two saddle-node bifurcations and a Hopf bifurcation. This results in very interesting dynamics: the model can have at most three positive equilibria and can exhibit relaxation oscillations. Our results provide some insights on understanding the occurrence and control of phytoplankton blooms.

Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity
Jibin Li and Yan Zhou
2020, 13(11): 3083-3097 doi: 10.3934/dcdss.2020113 +[Abstract](711) +[HTML](387) +[PDF](1161.57KB)

For the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity, by using the method of dynamical systems, we investigate the bifurcations and exact traveling wave solutions. Because obtained traveling wave system is an integrable singular traveling wave system having a singular straight line and the origin in the phase plane is a high-order equilibrium point. We need to use the theory of singular systems to analyze the dynamics and bifurcation behavior of solutions of system. For \begin{document}$ m>1 $\end{document} and \begin{document}$ 0<m = \frac1n<\frac12 $\end{document}, corresponding to the level curves given by \begin{document}$ H(\psi, y) = 0 $\end{document}, the exact explicit bounded traveling wave solutions can be given. For \begin{document}$ m = 1 $\end{document}, corresponding all bounded phase orbits and depending on the changes of system's parameters, all exact traveling wave solutions of the equation can be obtain.

Global analysis of SIRI knowledge dissemination model with recalling rate
Fang Liu, Zhen Jin and Cai-Yun Wang
2020, 13(11): 3099-3114 doi: 10.3934/dcdss.2020116 +[Abstract](568) +[HTML](311) +[PDF](2915.0KB)

In order to study the dissemination mechanism of knowledge, a SIRI dynamics model with the learning rate, the forgetting rate and the recalling rate is constructed in this paper. Stability of equilibria and global dynamics of the SIRI model are analyzed. Two thresholds that determine whether knowledge is disseminated are given. We describe the stability of the equilibria for the SIRI model in which there are an equilibrium and a line of equilibria. In particular, we find the dividing curve function which is used to partition invariant set in order to discuss the local stability, and obtain the equation of the wave peak value or wave trough value in the process of knowledge dissemination. Numerical simulations are provided to support the theoretical results. The complicated dynamics properties exhibit that the model is very sensitive to variation of parameters, which play an important role on controlling and administering the knowledge dissemination.

Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory
Shanshan Liu and Maoan Han
2020, 13(11): 3115-3124 doi: 10.3934/dcdss.2020133 +[Abstract](591) +[HTML](319) +[PDF](329.21KB)

In this paper we study the maximal number of limit cycles for a class of piecewise smooth near-Hamiltonian systems under polynomial perturbations. Using the second order averaging method, we obtain the maximal number of limit cycles of two systems respectively. We also present an application.

Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate
Min Lu, Chuang Xiang and Jicai Huang
2020, 13(11): 3125-3138 doi: 10.3934/dcdss.2020115 +[Abstract](616) +[HTML](330) +[PDF](680.26KB)

In this paper, we study a SIRS epidemic model with a generalized nonmonotone incidence rate. It is shown that the model undergoes two different topological types of Bogdanov-Takens bifurcations, i.e., repelling and attracting Bogdanov-Takens bifurcations, for general parameter conditions. The approximate expressions for saddle-node, Homoclinic and Hopf bifurcation curves are calculated up to second order. Furthermore, some numerical simulations, including bifurcations diagrams and corresponding phase portraits, are given to illustrate the theoretical results.

Dirichlet problem for a diffusive logistic population model with two delays
Xuejun Pan, Hongying Shu and Yuming Chen
2020, 13(11): 3139-3155 doi: 10.3934/dcdss.2020134 +[Abstract](512) +[HTML](306) +[PDF](482.35KB)

In this paper, we investigate a diffusive logistic equation with non-zero Dirichlet boundary condition and two delays. We first exclude the existence of positive heterogeneous steady states, which implies the uniqueness of constant positive steady state. Then, we analyze the local stability and local Hopf bifurcation at the positive steady state. We show that multiple delays can induce multiple stability switches. Furthermore, we prove global stability of the positive steady state under certain conditions and obtain global Hopf bifurcation results. Our theoretical results are illustrated with numerical simulations.

Pest control by generalist parasitoids: A bifurcation theory approach
Gunog Seo and Gail S. K. Wolkowicz
2020, 13(11): 3157-3187 doi: 10.3934/dcdss.2020163 +[Abstract](668) +[HTML](401) +[PDF](801.99KB)

Magal et al. [13] studied both spatial and non-spatial host-parasitoid models motivated by biological control of horse-chestnut leaf miners that have spread through Europe. In the non-spatial model, they considered pest control by predation of leaf miners by a generalist parasitoid with a Holling type II functional (Monod) response. They showed that there can be at most six equilibrium points and discussed local stability. We revisit their model in the non-spatial case, identify cases missed in their investigation and discuss consequences for possible pest control strategies. Both the local stability of equilibria and global properties are considered. We use a bifurcation theoretical approach and provide analytical expressions for fold and Hopf bifurcations and for the criticality of the Hopf bifurcations. Our numerical results show very interesting dynamics resulting from codimension one bifurcations including: Hopf, fold, transcritical, cyclic-fold, and homoclinic bifurcations as well as codimension two bifurcations including: Bautin and Bogdanov-Takens bifurcations, and a codimension three Bogdanov-Takens bifurcation.

Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation
Yixia Shi and Maoan Han
2020, 13(11): 3189-3204 doi: 10.3934/dcdss.2020114 +[Abstract](648) +[HTML](328) +[PDF](382.29KB)

In this paper, we investigate the modified steady Swift-Hohenberg equation

where \begin{document}$ k>0 $\end{document}, \begin{document}$ \alpha $\end{document} and \begin{document}$ \varepsilon $\end{document} are constants. We obtain a homoclinic solution about the dominant system which will be proved to deform a reversible homoclinic solution approaching to a periodic solution of the whole equation with the aid of the Fourier series expansion method, the fixed point theorem, the reversibility and adjusting the phase shift. And the homoclinic solution approaching to a periodic solution of the equation are called generalized homoclinic solution.

Rich dynamics in some generalized difference equations
Xiujuan Wang and Mingshu Peng
2020, 13(11): 3205-3212 doi: 10.3934/dcdss.2020191 +[Abstract](493) +[HTML](263) +[PDF](4911.4KB)

There has been an increasing interest in the study of fractional discrete difference since Miller and Ross introduced the \begin{document}$ v $\end{document}-th fractional sum and the fractional integral was given as a fractional sum in 1989. It is known that fractional discrete difference equations hold discrete memory effects and can describe the long interaction of all the last states during evolution. Therefore the QR factorization algorithm described by Eckmann et al. in 1986 can not be directly applied to determine chaotic or nonchaotic behaviour in such a system, which becomes an interesting problem. Motivated by this, in this study, we propose a direct way to calculate the finite-time local largest Lyapunov exponent. Compared with those in the literature, we find that the test for determining the presence of chaos is reliable. Moreover, bifurcation diagrams which depends on the given fractional order parameter are given in Captuto like discrete Hénon maps and Logistic maps, which was not discussed in the literature. A transient behaviour in chaotic fractional Logistic maps is also discovered.

Bifurcations in a pollination-mutualism system with nectarless flowers
Hong Wu, Shan Sun and Yuanshi Wang
2020, 13(11): 3213-3229 doi: 10.3934/dcdss.2020162 +[Abstract](539) +[HTML](315) +[PDF](364.92KB)

This paper considers a pollination-mutualism system in which flowering plants have strategies of secreting and cheating: secretors produce a substantial volume of nectar in flowers but cheaters produce none. Accordingly, floral visitors have strategies of neglecting and selecting: neglectors enter any flower encountered but selectors only enter full flowers since they can discriminate between secretors and cheaters. By combination of replicator equations and two-species dynamical systems, the games are described by a mathematical model in this paper. Dynamics of the model demonstrate mechanisms by which nectarless flowers can invade the secretor-pollinator system and by which a cyclic game between nectarless flowers and pollinators could occur. Criteria for the persistence of nectarless flowers are derived in terms of the given parameters (factors), including the nectar-producing cost and cheaters' efficiency. Numerical simulations show that when parameters vary, cheaters would vary among extinction, persistence in periodic oscillations, and persistence without secretors (i.e., cheaters spread widely). We also consider the evolution of plants in a constant state of pollinator population, and the evolution of pollinators in a constant state of plant population. Dynamics of the models demonstrate conditions under which nectarless flowers (resp. selectors) could persist.

How interest rate influences a business cycle model
Qigang Yuan, Yutong Sun and Jingli Ren
2020, 13(11): 3231-3251 doi: 10.3934/dcdss.2020190 +[Abstract](566) +[HTML](288) +[PDF](2469.15KB)

We study the effect of interest rate on phenomenon of business cycle in a Kaldor-Kalecki model. From the information of the People's Bank of China and the Federal Reserve System, we know the interest rate is not a constant but with remarkable periodic volatility. Therefore, we consider periodically forced interest rate in the model and study its dynamics. It is found that, both limit cycle through Hopf bifurcation in unforced system and periodic solutions generated by period doubling bifurcation or resonance in periodically forced system, can lead to cyclical economic fluctuations. Our analysis reveals that the cyclical fluctuation of interest rate is one of a key formation mechanism of business cycle, which agrees well with the pure monetary theory on business cycle. Moreover, this fluctuation can cause chaos in a business cycle system.

Bogdanov-Takens bifurcation in predator-prey systems
Bing Zeng, Shengfu Deng and Pei Yu
2020, 13(11): 3253-3269 doi: 10.3934/dcdss.2020130 +[Abstract](918) +[HTML](309) +[PDF](350.67KB)

In this paper, we consider Bogdanov-Takens bifurcation in two predator-prey systems. It is shown that in the full parameter space, Bogdanov-Talens bifurcation can be codimension \begin{document}$ 2 $\end{document}, \begin{document}$ 3 $\end{document} or \begin{document}$ 4 $\end{document}. First, the simplest normal form theory is applied to determine the codimension of the systems as well as the unfolding terms. Then, bifurcation analysis is carried out to describe the dynamical behaviour and bifurcation property of the systems around the critical point.

Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system
Hongyong Zhao and Daiyong Wu
2020, 13(11): 3271-3284 doi: 10.3934/dcdss.2020129 +[Abstract](663) +[HTML](291) +[PDF](319.04KB)

In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.

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