Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.     

Relaxation oscillations in a slow–fast modified Leslie–Gower model

In this paper, we prove the existence and uniqueness of relaxation oscillation cycle of a slow–fast modified Leslie–Gower model via the entry–exit function and geometric singular perturbation theory. Numerical simulations are also carried out to illustrate our theoretical result.     

Canard cycle transition at a slow–fast passage through a jump point

We introduce transitory canard cycles for slow–fast vector fields in the plane. Such cycles separate “canards without head” and “canards with head”, like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle under whatever condition on the related slow divergence integral I  , including the challenging case I=0$I=0$.     

Global asymptotic stability of a general stochastic Lotka–Volterra system with delays

This paper discusses a general stochastic Lotka–Volterra system with delays. Some conditions for the global asymptotic stability are established.     

Discretized fast–slow systems near pitchfork singularities

ABSTRACT

Motivated by the normal form of a fast–slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of the explicit Euler method. Tracking trajectories in the vicinity of the singularity we show, how the slow manifold extends beyond the singular point and give an estimate on the contraction rate of a transition mapping. The proof relies on the blow-up method suitably adapted to the discrete setting where precise estimates for a cubic map in the central rescaling chart make a key technical contribution.     

Slow–fast dynamics of tri-neuron Hopfield neural network with two timescales

The slow-fast dynamics of a tri-neuron Hopfield neural network with two timescales is stated in present paper. On the basis of geometric singular perturbation theory, the transition of the solution trajectory is illuminated, and the existence of the relaxation oscillation with rapid movement process alternating with slow movement process is proved. It is indicated the characteristic of the relaxation oscillation is dependent on the structure of the slow manifold. Moreover, the approximate expression of the relaxation oscillation and its period are obtained analytically. Case studies are given to demonstrate the validity of theoretical results.     

Global stability of a nonlinear stochastic predator–prey system with Beddington–DeAngelis functional response

Stochastically asymptotic stability in the large of a predator–prey system with Beddington–DeAngelis functional response with stochastic perturbation is considered. The result shows that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided the noise is sufficiently small. Some simulation figures are introduced to support the analytical findings.     

Asymptotic properties of a stochastic predator–prey system with Holling II functional response

A stochastic predator–prey system with Holling II functional response is proposed and investigated. We show that there is a unique positive solution to the model for any positive initial value. And we show that the positive solution to the stochastic system is stochastically bounded. Moreover, under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean.     

λ-Automorphisms of a Riemannian foliation

We study geometric properties of -automorphisms of a Riemannian foliationF which is not harmonic. This notion was first introduced in [KTT] for the case whereF is harmonic. Transversal Killing, affine, conformal, projective fields are all examples of -automorphisms. We derive several general identities for a -automorphism. In particular, we extend the results on the transversal conformal and Killing fields obtained in [PrY], [NY1,2]. Furthermore, we analyse the geometric meaning of the condition appearing in our results.The present studies were supported (in part) by the Basic Science Research Institute Program, Ministry of Education, 1994, Project No. BSRI-94-1404     

Local solutions of the fast–slow model of an optoelectronic oscillator with delay

Theoretical and Mathematical Physics - We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the...     

Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system

The existence of suitable weak solutions of 3D Navier–Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Existence of statistically stationary solutions is also proved.     

A remark on a stochastic predator–prey system with time delays

An autonomous stochastic predator–prey model with time delays is investigated. Almost sufficient and necessary conditions for stability in the mean and extinction of each population are established. Numerical simulations are introduced to support the results.     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.     

Canard-cycle transition at a fast–fast passage through a jump point

We consider transitory canard cycles that consist of a generic breaking mechanism, i.e. a Hopf or a jump breaking mechanism, in combination with a fast–fast passage through a jump point. Such cycle separates two types of canard cycles with a different shape. We obtain upper bounds on the number of periodic orbits that can appear near the canard cycle, and this under very general conditions.     

Persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise

In this paper, we consider the persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the system are established. The threshold between weak persistence and extinction is obtained. From the results we can see that both persistence and extinction have close relationships with Lévy noise. Some simulation figures are introduced to demonstrate the analytical findings.