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In this paper, we work on the discrete modified Leslie type predator-prey model with Holling type II functional response. The existence and local stability of the fixed points of this system are studied. According to bifurcation theory and normal forms, we investigate the codimension 1 and 2 bifurcations of positive fixed points, including the fold, 1:1 strong resonance, fold-flip and 1:2 strong resonance bifurcations. In particular, the discussion of discrete codimension 2 bifurcation is rare and difficult. Our work can be seen as an attempt to complement existing research on this topic. In addition, numerical analysis is used to demonstrate the correctness of the theoretical results. Our analysis of this discrete system revealed quite different dynamical behaviors than the continuous one.

The dynamic behavior of a discrete-time two-patch model with the Allee effect and nonlinear dispersal is studied in this paper. The model consists of two patches connected by the dispersal of individuals. Each patch has its own carrying capacity and intraspecific competition, and the growth rate of one patch exhibits the Allee effect. The existence and stability of the fixed points for the model are explored. Then, utilizing the central manifold theorem and bifurcation theory, fold and flip bifurcations are investigated. Finally, numerical simulations are conducted to explore how the Allee effect and nonlinear dispersal affect the dynamics of the system.

This paper proposes a novel method for predicting the presence of saddle-node bifurcations in dynamical systems. The method exploits the effect that saddle-node bifurcations have on transient dynamics in the surrounding phase space and parameter space, and does not require any information about the steady-state solutions associated with the bifurcation. Specifically, trajectories of a system obtained for parameters close to the saddle-node bifurcation present local minima of the logarithmic decrement trend in the vicinity of the bifurcation. By tracking the logarithmic decrement for these trajectories, the saddle-node bifurcation can be accurately predicted. The method does not strictly require any mathematical model of the system, but only a few time series, making it directly implementable for gray- and black-box models and experimental apparatus. The proposed algorithm is tested on various systems of different natures, including a single-degree-of-freedom system with nonlinear damping, the mass-on-moving-belt, a time-delayed inverted pendulum, and a pitch-and-plunge wing profile. Benefits, limitations, and future perspectives of the method are also discussed. The proposed method has potential applications in various fields, such as engineering, physics, and biology, where the identification of saddle-node bifurcations is crucial for understanding and controlling complex systems.

This paper investigates bifurcations analysis and resonances in a discrete-time prey-predator model analytically and numerically as well. The local stability conditions of all the fixed points in the system are determined. Here, codim-1 and codim-2 bifurcation including multiple and generic bifurcations in the discrete model are explored. The model undergoes fold bifurcation, flip bifurcation, Neimark–Sacker bifurcation and resonances 1:2, 1:3, 1:4 at different fixed points. Using the critical normal form theorem and bifurcation theory, normal form coefficients are calculated for each bifurcation. The different bifurcation curves of fixed points are drawn which validate the analytical findings. The numerical simulation gives a wide range of periodic cycles including codim-1 bifurcation and resonance curves in the system. The results in this manuscript reveal that the dynamics of the discrete-time model in both single-parameter and two-parameter spaces are inherently rich and complex. The resonance bifurcation in the discrete-time map indicates that both species coincide till order 4 in stable periodic cycles near some critical parametric values.

A combined analysis of smooth and non-smooth bifurcations captures the interplay of different qualitative transitions in a canonical model of an impact pair, a forced capsule in which a ball moves freely between impacts on either end of the capsule. The analysis, generic for the impact pair context, is also relevant for applications. It is applied to a model of an inclined vibro-impact energy harvester device, where the energy is generated via impacts of the ball with a dielectric polymer on the capsule ends. While sequences of bifurcations have been studied extensively in single- degree-of-freedom impacting models, there are limited results for two-degree-of-freedom impacting systems such as the impact pair. Using an analytical characterization of impacting solutions and their stability based on the maps between impacts, we obtain sequences of period doubling and fold bifurcations together with grazing bifurcations, a particular focus here. Grazing occurs when a sequence of impacts on either end of the capsule are augmented by a zero-velocity impact, a transition that is fundamentally different from the smooth bifurcations that are instead characterized by eigenvalues of the local behavior. The combined analyses allow identification of bifurcations also on unstable or unphysical solutions branches, which we term ghost bifurcations. While these ghost bifurcations are not observed experimentally or via simple numerical integration of the model, nevertheless they can influence the birth or death of complex behaviors and additional grazing transitions, as confirmed by comparisons with the numerical results. The competition between the different bifurcations and their ghosts influences the parameter ranges for favorable energy output; thus, the analyses of bifurcation sequences yield important design information.

The phenomenon of relaxation oscillations is a typical fast–slow dynamical behavior. In this paper, we take a type of nonlinear oscillator involving multiple coexisting attractors as an example and aim to reveal interesting patterns of relaxation oscillations, namely the so-called compound relaxation oscillations. To begin with, a relaxation oscillation pattern with asymmetrical transitions is obtained. Then, a two-parameter bifurcation diagram is plotted to explore the transitions of relaxation oscillations. It is found that the number and stability of the attractors near fold bifurcation points may change when the system parameters vary. This is manifested in two ways: First, when a fold bifurcation occurs in the upper equilibrium branch, the system is bi-stable; while a fold bifurcation occurs in the lower equilibrium branch, the system is stable. Second, the system is bi-stable when fold bifurcations occur in the upper and lower equilibrium branches. For the above two cases, we use fast–slow analysis and attraction domain analysis to explore the dynamical mechanisms of the relaxation oscillation behaviors. As a result, several different relaxation oscillation patterns are obtained. In particular, relaxation oscillations with asymmetric structure, i.e., compound relaxation oscillations, characterized by the system trajectory transitions passing through the middle branch at one end and crossing it at the other, are discovered and researched.

Matsuda and Abrams (Theor Popul Biol 45(1):76–91, 1994) initiated the exploration of self-extinction in species through evolution, focusing on the advantageous position of mutants near the extinction boundary in a prey-predator system with evolving foraging traits. Previous models lacked theoretical investigation into the long-term effects of harvesting. In our model, we introduce constant-effort prey and predator harvesting, along with individual logistic growth of predators. The model reveals two distinct evolutionary outcomes: (i) Evolutionary suicide, marked by a saddle-node bifurcation, where prey extinction results from the invasion of a lower forager mutant; and (ii) Evolutionary reversal, characterized by a subcritical Hopf bifurcation, leading to cyclic prey evolution. Employing an innovative approach based on Gröbner basis computation, we identify various bifurcation manifolds, including fold, transcritical, cusp, Hopf, and Bogdanov-Takens bifurcations. These contrasting scenarios emerge from variations in harvesting parameters while keeping other factors constant, rendering the model an intriguing subject of study.

Ecosystems close to a critical threshold lose resilience, in the sense that perturbations can more easily push them into an alternative state. Recently, it has been proposed that such loss of resilience may be detected from elevated autocorrelation and variance in the fluctuations of the state of an ecosystem due to critical slowing down; the underlying generic phenomenon that occurs at critical thresholds. Here we explore the robustness of autocorrelation and variance as indicators of imminent critical transitions. We show both analytically and in simulations that variance may sometimes decrease close to a transition. This can happen when environmental factors fluctuate stochastically and the ecosystem becomes less sensitive to these factors near the threshold, or when critical slowing down reduces the ecosystem's capacity to follow high-frequency fluctuations in the environment. In addition, when available data is limited, variance can be systematically underestimated due to the prevalence of low frequencies close to a transition. By contrast, autocorrelation always increases toward critical transitions in our analyses. To exemplify this point, we provide cases of rising autocorrelation and increasing or decreasing variance in time series prior to past climate transitions.

Concerned in this paper is a discrete predator-prey system with Allee effect and other food resources for the predators. The conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation and flip bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are provided to illustrate the feasibility of the main results and the influence of Allee effect on the stability of the system. Our study indicates that other food resources for the predator can enrich the dynamical behaviours of the system, including cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.

In this work, we give some results obtained on the dynamics of a symmetrically decoupled three-dimensional point transformation. We are interested, in particular, in the study of its parametric plane and in its phase space by highlighting the existence of chaotic attractors.