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A study is made of the waves which may occur at the interface of two fluids and which arise from an interaction between the two lowest adjacent harmonics of the motion. The problem is studied by reformulating it, via a conformal mapping transformation, into a pair of coupled nonlinear functional differential equations which are invariant under the action of the group [Formula: see text] The classical method of Lyapunov–Schmidt is then employed to replace this formulation by a pair of algebraic equations, known as the bifurcation equations. Throughout our analysis we impose a physical condition which essentially means that the densities and phase speeds in each fluid are approximately equal. This means that the bifurcation equations assume a cubic form rather than the quadratic form which they generically possess in problems of this type. The bifurcation equations are then analyzed by means of singularity theory by which means we are able to draw the solution curves of the system. A large variety of solution diagrams are found to exhibit such behavior as multiple primary bifurcations, secondary bifurcations and bifurcation from infinity.

In this paper, a commensal model with additive Allee effect on the first species and nonlinear growth rate on the second species is proposed. The classification, type and stability of all possible equilibria of the system are investigated. We also prove the presence of saddle-node bifurcation and transcritical bifurcation by Sotomayor’s theorem. At the end of this study, an example and its simulations are given to illustrate our main results.

We apply the [Formula: see text]-equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of [Formula: see text]-equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described.

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

In this paper, we propose a diffusive SIR model with general incidence rate, saturated treatment rate and spatially heterogeneous diffusion coefficients. We first prove the global existence of bounded solutions for the model and compute the basic reproduction number. We study the local and global stabilities of the disease-free equilibrium and the uniform persistence. In the case when the diffusion rate of infected individuals is constant, we carry out a bifurcation analysis of equilibria by considering the maximal treatment rate as the bifurcation parameter. Finally, we perform some numerical simulations, which show that the solutions to our model present periodic oscillations for certain values of the parameters.

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

This paper aims to study the bifurcation structures of the homographic [Formula: see text]-Ricker maps in a four-dimensional parameter space. The generalized Lambert [Formula: see text] functions are used to establish upper bounds for the number of fixed points of these population growth models. The variation of the number of fixed points and the cusp points organization is stipulated. This study also observes a vital characteristic on the Allee effect phenomenon in a class of bimodal Allee’s maps. Some numerical studies are included to illustrate the Allee effect and big bang local bifurcations.

This paper is concerned with the spatiotemporal heterogeneity in a modified Leslie–Gower predator–prey system with Beddington–DeAngelis functional response and prey-taxis. Using Crandall–Rabinowitz bifurcation theory, we investigate the steady-state bifurcation of the nonlinear system by choosing the prey-tactic sensitivity coefficient as a bifurcating parameter. It is rigorously proved that a branch of nonconstant solution exists near the positive equilibrium when the prey-tactic sensitivity is repulsive. Moreover, we study the existence, direction and stability of periodic orbits around the interior constant equilibrium by selecting the intrinsic growth rate of the prey as a bifurcating parameter. A priori estimates play a critical role in the verification procedure. Some numerical simulations are carried out to support our main theoretical results.

The Leslie–Gower model, a kind of predator–prey model with weak Allee effect, is studied in this paper. The existence and stability of non-negative equilibria are first discussed. Then, we investigate several bifurcation phenomena undergoing positive equilibria, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, etc. Some possible dynamical behaviors of this model are illustrated by numerical simulation. The bifurcation diagrams for the cases of codimensions 2 and 3 are given respectively. The coexistence of a periodic cycle and a homoclinic cycle, and two limit cycles enclosing an unstable equilibrium are also proved. This appears to be the first study of the Leslie–Gower model including the influence of weak Allee effect on prey.