A discrete-time model with non-monotonic functional response and strong Allee effect in prey

In this paper, we investigate the impact of strong Allee effect on the stability of a discrete-time predator–prey model with a non-monotonic functional response. The dynamics of discrete-time predator–prey models with strong Allee effect is studied earlier. But, the mathematical investigations of predator–prey dynamics in discrete-time set up with Holling type-IV functional response and strong Allee effect in prey are lacking. The proposed model supports the coexistence of two steady states, and the mathematical features of the model are analyzed based on local stability and bifurcation theory. By considering the Allee parameter as the bifurcation parameter, we provide sufficient conditions for the flip and the Neimark–Sacker bifurcations. We observe that Allee parameter plays a significant role in the dynamics of the system.     

MODELING THE EFFECT OF RARITY VALUE ON THE EXPLOITATION OF A WILDLIFE SPECIES SUBJECTED TO THE ALLEE EFFECT

The overexploitation of wildlife species is a serious problem in the field of biodiversity conservation. The species subjected to natural Allee effects are even more threatened by exploitation. Moreover, for many wildlife species, their rarity can fuel their exploitation by making them disproportionately desirable and consequently increasing their market price. In this paper, a mathematical model is proposed and analyzed to study how the value that consumers place on rarity can threaten the survival of a species subjected to natural Allee effects. It is assumed that the value of a species increases as its density declines. The analysis of model shows that the increase in the consumers' response to rarity can drive the system to admit Hopf‐bifurcation and heteroclinic bifurcation. The occurrence of the heteroclinic cycle indicates that the increase in consumers' response to rarity can cause the extinction of the species. It is found that an increase in the Allee threshold causes a decrease in the threshold value of consumers' response below which extinction is inevitable.     

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

The chaotic behaviors of a particle in a triple well ϕ⁶ potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.     

Study of a Tritrophic Food Chain Model with Non-differentiable Functional Response

We study a model of three interacting species in a food chain composed by a prey, an specific predator and a generalist predator. The capture of the prey by the specific predator is modelled as a modified Holling-type II non-differentiable functional response. The other predatory interactions are both modelled as Holling-type I. Moreover, our model follows a Leslie-Gower approach, in which the function that models the growth of each predator is of logistic type, and the corresponding carrying capacities depend on the sizes of their associated available preys. The resulting model has the form of a set of nonlinear ordinary differential equations which includes a non-differentiable term. By means of topological equivalences and suitable changes of parameters, we find that there exists an Allee threshold for the survival of the prey population in the food chain, given, effectively, as a critical level for the generalist predator. The dynamics of the model is studied with analytical and computational tools for bifurcation theory. We present two-parameter bifurcation diagrams that contain both local phenomena (Hopf, saddle-node transcritical, cusp, Bogdanov-Takens bifurcations) and global events (homoclinic and heteroclinic connections). In particular, we find that two types of heteroclinic cycles can be formed, both of them containing connections to the origin. One of these cycles is planar involving the absence of the specific predator. In turn, the other heteroclinic cycle is formed by connections in the full three-dimensional phase space.

     

Complex dynamics of a discrete predator–prey model with the prey subject to the Allee effect

In this paper, complex dynamics of the discrete predator–prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark–Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark–Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.     

Non-autonomous periodic systems with Allee effects

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with three fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper, the properties and stability of the three fixed points are studied in the setting of non-autonomous periodic dynamical systems or difference equations. Finally, we investigate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps.     

An extended discrete Ricker population model with Allee effects

Based on the classical discrete Ricker population model, we incorporate Allee effects by assuming rectangular hyperbola, or Holling-II type functional form, for the birth or growth function and formulate an extended Ricker model. We explore the dynamics features of the extended Ricker model. We obtain domains of attraction for the trivial fixed point. We determine conditions for the existence and stability of positive fixed points and find regions where there exist no positive fixed points, two positive fixed points one of which is stable and two positive fixed points both of which are unstable. We demonstrate that the model exhibits period-doubling bifurcations and investigate the existence and stability of the cycles. We also confirm that Allee effects have stabilization effects, by different measures, through numerical simulations.     

Dynamics in a discrete-time predator-prey system with Allee effect

In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases (within a limited value). Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.     

Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a ϕ6 potential

This paper investigates the chaotic behavior of an extended Duffing Van der pol oscillator in a ϕ⁶ potential under additive harmonic and bounded noise excitations for a specific parameter choice. From Melnikov theorem, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation in the case of the ϕ⁶ potential is bounded, which are complemented by the numerical simulations from which we illustrate the bifurcation surfaces and the fractality of the basins of attraction. The results show that the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further validated by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps is also investigated.     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Dynamical analysis in a diffusive predator-prey system with a delay and strong Allee effect

In this paper, we consider the dynamics of a diffusive predator-prey system with strong Allee effect and delayed Ivlev-type functional response. At first, we apply the method of upper-lower solutions and the comparison principle in proving the nonnegativity of the solutions. Then by analyzing the distribution of the eigenvalues, we obtain the bistability of the system and the existence of Hopf bifurcation. Furthermore, by using the center manifolds theory and normal form method, we study the properties of the Hopf bifurcations. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.     

Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non‐hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω ? limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

Bifurcations and Exact Solutions of the Raman Soliton Model in Nanoscale Optical Waveguides with Metamaterials

In this paper, we study Raman soliton model in nanoscale optical waveguides with metamaterials, having polynomial law non-linearity. By using the bifurcation theory method of dynamical systems to the equations of $\phi(\xi)$, under 24 different parameter conditions, we obtain bifurcations of phase portraits and different traveling wave solutions including periodic solutions, homoclinic and heteroclinic solutions for planar dynamical system of the Raman soliton model. Under different parameter conditions, 24 exact explicit parametric representations of the traveling wave solutions are derived. The dynamic behavior of these traveling wave solutions are meaningful and helpful for us to understand the physical structures of the model.     

Rich dynamical behaviours for predator–prey model with weak Allee effect

The weak Allee effect on the predator is introduced into the classic predator–prey model of Lotka–Volterra type. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. It is shown that the weak Allee effect can bring rich and complicated dynamics to the previous simple model, such as the saddle–node bifurcation, subcritical and supercritical Hopf bifurcations, and Bogdanov–Takens bifurcations, implying that weak Allee effect can be one of the simple reasons for many complicated behaviours in the predator–prey communities.     

基于Allee效应诱导的Filippov生态系统的动力学行为研究

该文建立了一类由Allee效应诱导的非光滑Filippov切换系统.运用Filippov系统的定性分析方法,从理论上研究了系统的滑动区域、滑动模态和各类平衡点的存在性.同时用数值方法研究了系统的滑动模态分支、边界焦点分支及全局动力学行为.研究发现:Allee效应的强度可使种群的动态不稳定,不利于濒危生物种群的管理.     

Bifurcations of heterodimensional cycles with two saddle points

The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed.     

Transitions in a logistic population model incorporating an Allee effect

In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations. Copyright © 2015 John Wiley & Sons, Ltd.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.