Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.

Center Bifurcation in the Lozi Map

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number [Formula: see text]. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form.

Detection of Synergistic Combinatorial Perturbations by a Bifurcation-Based Approach

Drug combination has become an attractive strategy against complex diseases, despite the challenges in handling a large number of possible combinations among candidate drugs. How to detect effective drug combinations and determine the dosage of each drug in the combination is still a challenging task. When regarding a drug as a perturbation, we propose a bifurcation-based approach to detect synergistic combinatorial perturbations. In the approach, parameters of a dynamical system are divided into two groups according to their responses to perturbations. By combining two parameters chosen from two groups, three types of combinations can be obtained. Synergism for different perturbation combinations can be detected by relative positions of the bifurcation curve and the isobole. The bifurcation-based approach can be used not only to detect combinatorial perturbations but also to determine their perturbation quantities. To demonstrate the effectiveness of the approach, we apply it to the epithelial-to-mesenchymal transition (EMT) network. The approach has implications for the rational design of drug combinations and other combinatorial control, e.g. combinatorial regulation of gene expression.

Predicting Limit Cycle Boundaries Deep Inside Parameter Space of a 2D Biochemical Nonlinear Oscillator Using Renormalization Group

Formation and study of periodic orbits in phase space in the case of nonlinear oscillators have been a topic of much interest in the recent past. In the current work, a method to go deep inside the limit cycle zone on one side of the bifurcation curve of a 2D non-Lienard biochemical oscillator has been introduced. It is discussed how such an introduction facilitates predicting the boundaries of limit cycles at various points of parameter space, nearly accurately, by the use of perturbative Renormalization Group. Sel’kov model of Glycolytic oscillator has been chosen as the base model to introduce the method.

The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

Fast-slow Variable Dissection with Two Slow Variables Related to Calcium Concentrations: A Case Study to Bursting in a Neural Pacemaker Model

Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast-slow variable dissection method with two slow variables is proposed to analyze the complex bursting in a 4-dimensional neuronal model to describe bursting associated with pathological pain. The lumenal (Clum) and intracellular (Cin) calcium concentrations are the slowest variables respectively in the quiescent state and burst duration. Questions encountered when the traditional method with one low variable is used. When Clum is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With Cin chosen as slow variable, Clum value of initiation point is far from the saddle-node bifurcation point, due to Clum not contained in the equation of membrane potential. To overcome this problem, both Cin and Clum are regarded as slow variables, the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing Clum dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast-slow variable dissection and deep understanding to the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

Nonlinear and Stability Analysis of a Ship with General Roll-Damping Using an Asymptotic Perturbation Method

In this work, the response of a ship rolling in regular beam waves is studied. The model is one degree of freedom model for nonlinear ship dynamics. The model consists of the terms containing inertia, damping, restoring forces, and external forces. The asymptotic perturbation method is used to study the primary resonance phenomena. The effects of various parameters are studied on the stability of steady states. It is shown that the variation of bifurcation parameters affects the bending of the bifurcation curve. The slope stability theorems are also presented.

Bogdanov–Takens bifurcation of a Holling IV prey–predator model with constant-effort harvesting

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

KCC Analysis of a One-Dimensional System During Catastrophic Shift of the Hill Function: Douglas Tensor in the Nonequilibrium Region

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of the invariants in the KCC theory, affects the robustness of the trajectory during a catastrophic shift. In this analysis, the forward and backward shifts can have different Jacobi stability structures in the nonequilibrium region. Moreover, the bifurcation curve of the catastrophic shift can be interpreted geometrically, as the solution curve where the nonlinear connection and the deviation curvature become zero. KCC analysis also shows that even if the catastrophic pattern itself is similar, the stability structure in the nonequilibrium region is different in some cases, from the viewpoint of the Douglas tensor.

Precise Asymptotics for Bifurcation Curve of Nonlinear Ordinary Differential Equation

We study the following nonlinear eigenvalue problem −u″(t)=λf(u(t)),u(t)>0,t∈I:=(−1,1),u(±1)=0, where f(u)=log(1+u) and λ>0 is a parameter. Then λ is a continuous function of α>0, where α is the maximum norm α=∥uλ∥∞ of the solution uλ associated with λ. We establish the precise asymptotic formula for λ=λ(α) as α→∞ up to the third term of λ(α).