Front dynamics in a singularly perturbed predator-prey model with spatial diffusion

<p>In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.</p>

We consider planar systems of predator-prey models with small predator death\nrate $\\epsilon>0$. Using geometric singular perturbation theory and Floquet\ntheory, we derive characteristic functions that determines the location and the\nstability of relaxation oscillations as $\\epsilon\\to 0$. When the prey-isocline\nhas a single interior local extremum, we prove that the system has a unique\nnontrivial periodic orbit, which forms a relaxation oscillation. For some\nsystems with prey-isocline possessing two interior local extrema, we show that\neither the positive equilibrium is globally stable, or the system has exact two\nperiodic orbits. In particular, for a predator-prey model with the Holling type\nIV functional response we derive a threshold value of the carrying capacity\nthat separates these two outcomes. This result supports the so-called paradox\nof enrichment.\n

This paper is concerned with the traveling wave solutions of a singularly perturbed system, which arises from the coupled arrays of Chua's circuit. By the geometric singular perturbation theory and invariant manifold theory, we prove that there exists a heteroclinic cycle consisting of the traveling front and back waves with the same wave speed. In particular, the expression of corresponding wave speed is also obtained. Furthermore, we show that the chaotic behavior induced by this heteroclinic cycle is hyperchaos.

This article aims to establish the existence of traveling waves for a predator-prey system with Beddington-DeAngelis functional response, reproductive Allee effect, and time delay. We investigate the existence of solutions for a system with two special delay kernels by geometric singular perturbation theory, invariant manifold theory, and Fredholm orthogonality theory. In addition, we discuss the asymptotic behaviors of traveling waves with the aid of the asymptotic theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/33/abstr.html

In this paper, we consider the global dynamics of a predator–prey system with simplified Holling type IV functional response and Allee effect in prey. Based on geometric singular perturbation theory, we focus on the nonstandard slow–fast dynamics of the system with a small death rate of the predator. We provide the existence and stability of the positive equilibrium and boundary equilibria under the fact that prey-isocline has a single interior local extremum in the first quadrant, whether the prey suffers from weak or strong Allee effect. In the case of weak Allee effect, we prove the existence and location of the unique relaxation oscillation type nontrivial stable periodic orbit by entry–exit function. Meanwhile, the global attractor is observed in the system with strong Allee effect. In addition, combined with the analysis of two equilibria at infinity by means of Poincaré transformation, the global phase portraits of predator–prey system are characterized.

In classes of N-particle systems and lattice models, the speed of front propagation is approximated by that of the corresponding continuum model, and for many such systems, the rate of convergence to the continuum speed is known to be slow as N → ∞. This slow convergence has been captured by including a cutoff function on the reaction terms in the continuum models. For example, the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with a cutoff has fronts that travel at the speed $${c \sim c_{\rm FKPP} - \frac{\pi^2}{(\ln(N))^2}}$$ , which agrees well with data from numerical simulations of the corresponding N-particle systems, where c FKPP is the linear spreading speed. In Panja and van Saarloos (Phys Rev E 66:015206, 2002), an example is presented in which a small enhancement of the reaction function causes the propagation speeds of fronts to be larger than c FKPP. Such front speeds are also observed in stochastic lattice models where the growth rates in the regime of few particles are modified. In this article, we analyze the dynamics of traveling fronts in the FKPP equation with the constant enhancement function employed by Panja and van Saarloos. We present formulas for the wave speeds, develop the criteria on the parameters for which the front speeds are larger than the linear spreading speed even in the limit in which the size of the cutoff domain vanishes, study the rate of approach as N → ∞, and identify the mechanisms in phase space by which the constant enhancement of the reaction function makes possible the larger than linear wave speeds. In addition, we extend these results to the FKPP equation with two other enhancement functions, which are also of interest for continuum level modeling of lattice models and many-particle systems in the regimes of small numbers of particles, namely a linear enhancement function and an enhancement that is uniform above the linearized reaction function. We also derive explicit formulas for the parameters in these problems. The mathematical techniques used herein are geometric singular perturbation theory, geometric desingularization, invariant manifold theory, and normal form theory, all from dynamical systems.

In this paper, we investigate the dynamics of a modified Leslie–Gower predator–prey model with Allee effect on prey. And the Holling type II functional response is considered in this model. When prey reproduces much faster than predator, by combining the normal form theory of slow–fast systems and the geometric singular perturbation theory, we observe much richer new dynamical phenomena than the existing ones. In the case of strong Allee effect, we prove the existence of canard cycles, homoclinic orbits and heteroclinic orbits. Furthermore, we focus on the case of weak Allee effect. In addition to the similar dynamics of that exhibited by the system with strong Allee effect, we also demonstrate the occurrence of relaxation oscillations created by entry-exit function. Moreover, the existence of canard explosion is further explained analytically and numerically with the help of sophisticated slow–fast techniques.

In this paper, the Korteweg–de Vries (KdV) equation is considered, which is a shallow water wave model in fluid mechanic fields. First the existence of solitary wave solutions for the original KdV equation and geometric singular perturbation theory are recalled. Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Melnikov method. Finally, the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory. Moreover, an interesting result is found for the equation without backward diffusion effect, there is no solitary wave solution in the case of local delay, but there is a solitary wave solution in the case of nonlocal delay.

<p style='text-indent:20px;'>In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation.

In this paper, we investigate a predator–prey model with Holling-II functional response, Allee effect and constant-yield predator harvesting, by comparing the differences between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the harvesting rate of predators. For system without harvesting, the Allee effect leads to population extinction. The system has at most one positive equilibrium and has a supercritical Hopf bifurcation which depends on the natural mortality rate of predators. Besides, by using normal form theory, we show that the system with [Formula: see text] reveals rich dynamic properties, including saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, where numerical simulations are presented to demonstrate the Bogdanov–Takens bifurcation of codimension 2 with a limit cycle and a homoclinic cycle. The system can generate up to two positive equilibria with the changes of [Formula: see text], which indicates that appropriate predator harvesting can assist in regulating the ecosystem. We then give the optimal harvesting strategy by using Pontryagin’s maximum principle. Finally, numerical simulations are performed to validate the functions of Allee effect and harvesting. Theoretical studies and numerical simulations demonstrate that the Allee effect can lead to species extinction and highlight the role of appropriate harvesting in controlling the stability of the system.

This article concerns the dynamics of mixed-mode oscillations (MMOs) emerging from the calcium-based inner hair cells (IHCs) model in the auditory cortex. The paper captures the MMOs generation mechanism based on the geometric singular perturbation theory (GSPT) after exploiting the average analysis for reducing the full model. Our analysis also finds that the critical manifold and folded surface are central to the mechanism of the existence of MMOs at the folded saddle for the perturbed system. The system parameters, such like the maximal calcium channels conductance, controls the firing patterns, and many new oscillations occur for the IHCs model. Tentatively, we conduct dynamic analysis combined with dynamic method based on GSPT by giving slow-fast analysis for the singular perturbed models and bifurcation analysis. In particular, we explore the two-slow-two-fast and three-slow-one-fast IHCs perturbed systems with layer and reduced problems so that differential-algebraic equations are obtained. This paper reveals the underlying dynamic properties of perturbed systems under singular perturbation theory.

Local analysis near a folded saddle-node singularity

Dynamical complexity induced by Allee effect in a predator–prey model

This study entails investigation of mixed-mode oscillations (MMOs) with a conductance-based pyramidal cell (PC) model located in the entorhinal cortex layer V. This six-dimensional neuron model was reduced to three dimensions by analysis of the voltage-dependent timescales to illustrate a regime in which the MMOs are generated. Additionally, the mechanism of generation of the MMOs under antiepileptic drug conditions was illustrated in the 3D PC model. Combined with geometric singular perturbation theory (GSPT), this work shows that there is a range of parameters under which the reduced model explains the emergence of MMOs caused by an underlying canard phenomenon. In particular, we theoretically calculate the number of subthreshold oscillations using the relationship with the eigenvalue ratio of the singular perturbation system at the folded singular node, which is consistent with numerical simulations. Furthermore, a slow–fast dynamics analysis of the 3D PC model is performed, where two slow/one fast and one slow/two fast systems with the layer problem and the reduced problem are considered to explain the trajectory on the critical manifold. General one- and two-parameter bifurcation types are also discussed in this work. The first Lyapunov coefficient of the Hopf bifurcation can decide whether the bifurcation is supercritical or subcritical. Bogdanov–Takens (BT) bifurcation was also analyzed in this study and associated with three bifurcation curves near the BT point. Finally, studies on the GSPT and bifurcation analysis are of great importance for further understanding the complex dynamic behaviors and crucial roles of the signal transmission and information processing pathways of the biological nervous system.

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.