Leslie Type Research Articles

Stability and Bifurcation Analysis in a Discrete Predator–Prey System of Leslie Type with Radio-Dependent Simplified Holling Type IV Functional Response

In this paper, we use a semi-discretization method to consider the predator–prey model of Leslie type with ratio-dependent simplified Holling type IV functional response. First, we discuss the existence and stability of the positive fixed point in total parameter space. Subsequently, through using the central manifold theorem and bifurcation theory, we obtain sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation of this system to occur. Finally, the numerical simulations illustrate the existence of Neimark–Sacker bifurcation and obtain some new dynamical phenomena of the system—the existence of a limit cycle. Corresponding biological meanings are also formulated.

Bifurcations and global dynamics of a predator–prey mite model of Leslie type

AbstractIn this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

Bogdanov–Takens Bifurcation of Codimensions 3 and 4 in a Holling and Leslie type Predator–Prey System with Strong Allee Effect

Bogdanov–Takens Bifurcation of Codimensions 3 and 4 in a Holling and Leslie type Predator–Prey System with Strong Allee Effect

Singular perturbations of generalized Holling type III predator-prey models with two canard points

We study the coexistence of limit cycles in a predator-prey model of Leslie type with generalized Holling type III functional response. When the prey reproduces much faster than the predator, we prove for this model that: (i) the existence of the configuration of one large stable limit cycle enclosing two small unstable limit cycles, (ii) the cyclicity of singular double-head canard cycles is three and reached, and (iii) the coexistence of two stable limit cycles surrounding three equilibria. The last result gives a positive answer to Coleman's problem on the coexistence of two ecologically stable limit cycles in predator-prey models.

Dynamics of the Leslie type predator-prey model with effect of fear and delay in the prey population

Dynamics of the Leslie type predator-prey model with effect of fear and delay in the prey population

The Fractional Discrete Predator–Prey Model: Chaos, Control and Synchronization

This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and bifurcation diagrams. These behaviours imply that the fractional predator–prey discrete system of Leslie type has rich and complex dynamical properties that are influenced by commensurate and incommensurate orders. Moreover, the sample entropy test is carried out to measure the complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model.

Multiple bifurcations of a discrete modified Leslie-Gower predator-prey model.

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.

Multiple bifurcations in a predator–prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting

In this paper, a modified Leslie-type predator–prey model with simplified Holling type IV functional response is established, in which double Allee effect on prey and nonlinear prey harvesting are considered. The analysis of the model shows that there exists a Bogdanov–Takens singularity (focus case) of codimension 4, and also multiple other nonhyperbolic and degenerate equilibria. Bifurcations are explored and it is found that transcritical bifurcation, saddle–node bifurcation, Bogdanov–Takens bifurcation of codimension 2, degenerate cusp type Bogdanov–Takens bifurcation of codimension 3, and degenerate focus type Bogdanov–Takens bifurcation of codimension 4 occur as parameters vary. The bifurcations result in complex dynamic behaviors, such as double limit cycle, triple limit cycle, quadruple limit cycle, cuspidal loop, (multiple) homoclinic loop, saddle–node loop, and limit cycle(s) simultaneously with homoclinic loop. We run numerical simulations to verify the theoretical results, and it is found that the system admits bistability, tristability, or even tetrastability.

Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting

Abstract In this paper, a discrete Leslie-Gower predator-prey system with Michaelis-Menten type harvesting is studied. Conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are presented to illustrate the main theoretical results. Compared to the continuous analog, the discrete system here possesses much richer dynamical behaviors including orbits of period-16, 21, 35, 49, 54, invariant cycles, cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.

Complex Dynamics on a Discrete Tritrophic Model of Leslie Type with General Functional Responses

In this paper, by averaging the growth rate on each state, we analyze the dynamics of a discrete dynamical system coming from a system of ODEs. This differential system corresponds to a tritrophic Leslie type model which is formed by three populations (prey (P), mesopredator (MP) and superpredator (SP)), where the last two populations are generalist predators. We give sufficient conditions where the discrete model undergoes a Neimark–Sacker bifurcation at a coexistence point. This analysis is independent of the functional responses that govern the interactions. To illustrate our results, several applications are given, under the assumptions that the population P has logistic growth and that the relations MP–P and SP–MP are carried out through Holling type functional responses. From these applications, we conclude that there are sufficient conditions to guarantee that the three species coexist by means of a supercritical Neimark–Sacker bifurcation. Moreover, numerically we can detect that the discrete system exhibits a chaotic behavior.

DYNAMICS OF A DELAYED PREDATOR-PREY MODEL WITH CONSTANT-YIELD PREY HARVESTING

<abstract> In this paper, we study a delayed predator-prey model of Holling and Leslie type with constant-yield prey harvesting, in which two types of delays caused by maturation time of prey and the gestation time of predator are considered. We mainly investigate the local dynamics of the model with emphasis on the impact of delays. The stability of equilibrium and the existence conditions of Hopf bifurcation are discussed. First, based on the different values of delays, five cases of Hopf bifurcation are studied in detail. The critical values of Hopf bifurcation for each case are presented. In addition, we explore the properties of Hopf bifurcation. The direction of Hopf bifurcation and the stability of periodic solutions by using the normal form theory and central manifold theorem are determined. The qualitative analyses have demonstrated that the values of time delays can affect the stability of equilibrium and induce small amplitude period oscillations of population densities. Numerical simulations are carried out for illustrating the theoretical results. Meanwhile, we further investigate the effects of delay on the period of periodic solutions and the influence of the harvesting term on the stability of the equilibrium with time delays. </abstract>

The dynamics of a Leslie type predator\u2013prey model with fear and Allee effect

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

Bifurcation analysis and global dynamics in a predator–prey system of Leslie type with an increasing functional response

The dynamical behaviors of a Leslie type predator–prey system are explored when the functional response is increasing for both predator and prey. Qualitative and quantitative analysis methods based on stability theory, bifurcation theory and numerical simulation are adopted. It is showed that the system is dissipative and permanent, and its solutions are bounded. Global stability of the unique positive equilibrium is investigated by constructing Dulac function and applying Poincaré–Bendixson theorem. The bifurcation behaviors are further explored and the number of limit cycles is determined. By calculating the first Lyapunov number and the first two focus values, it is proved that the positive equilibrium is not a center but a weak focus of multiplicity at most two, so the system undergoes Hopf bifurcation and Bautin bifurcation. The normal form of Bautin bifurcation is also obtained by introducing the complex system. Moreover, numerical simulations are run to demonstrate the validity of theoretical results.

Dynamics of a Discrete-time Predator-Prey Model of Leslie type with Predator Harvesting

In the present work, we attempt to analyze the discrete form of Leslie type predator-prey system with the extension of nonlinear type harvesting of predators. The proposed model is obtained with the help of theory of piecewise constant argument for differential equations. We give the local stability properties of all possible non-negative fixed points. Also, our study reveals that the discrete system admits two bifurcations which are flip and Neimark-Sacker by making the use of center manifold argument and bifurcation theory, where the harvesting parameter is varied in order to take place of such bifurcations. Some simulations are carried out to depicts the obtained analytical results such as bifurcation plots and phase portraits. Also, it can be confirmed from numerical simulations that the considered system exhibits chaotic behavior for smaller values of harvesting parameter and becomes stable for larger values of same parameter. The largest Lyapunov exponents are plotted to show the sensitivity of chaotic regime.

Fitting stochastic predator–prey models using both population density and kill rate data

Most mechanistic predator–prey modelling has involved either parameterization from process rate data or inverse modelling. Here, we take a median road: we aim at identifying the potential benefits of combining datasets, when both population growth and predation processes are viewed as stochastic. We fit a discrete-time, stochastic predator–prey model of the Leslie type to simulated time series of densities and kill rate data. Our model has both environmental stochasticity in the growth rates and interaction stochasticity, i.e., a stochastic functional response. We examine what the kill rate data brings to the quality of the estimates, and whether estimation is possible (for various time series lengths) solely with time series of population counts or biomass data. Both Bayesian and frequentist estimation are performed, providing multiple ways to check model identifiability. The Fisher Information Matrix suggests that models with and without kill rate data are all identifiable, although correlations remain between parameters that belong to the same functional form. However, our results show that if the attractor is a fixed point in the absence of stochasticity, identifying parameters in practice requires kill rate data as a complement to the time series of population densities, due to the relatively flat likelihood. Only noisy limit cycle attractors can be identified directly from population count data (as in inverse modelling), although even in this case, adding kill rate data – including in small amounts – can make the estimates much more precise. Overall, we show that under process stochasticity in interaction rates, interaction data might be essential to obtain identifiable dynamical models for multiple species. These results may extend to other biotic interactions than predation, for which similar models combining interaction rates and population counts could be developed.

Complex Dynamics of a Discrete-Time Prey–Predator System with Leslie Type: Stability, Bifurcation Analyses and Chaos

Dynamic behavior of a discrete-time prey–predator system with Leslie type is analyzed. The discrete mathematical model was obtained by applying the forward Euler scheme to its continuous-time counterpart. First, the local stability conditions of equilibrium point of this system are determined. Then, the conditions of existence for flip bifurcation and Neimark–Sacker bifurcation arising from this positive equilibrium point are investigated. More specifically, by choosing integral step size as a bifurcation parameter, these bifurcations are driven via center manifold theorem and normal form theory. Finally, numerical simulations are performed to support and extend the theoretical results. Analytical results show that an integral step size has a significant role on the dynamics of a discrete system. Numerical simulations support that enlarging the integral step size causes chaotic behavior.

Probabilistic mechanisms of the noise-induced oscillatory transitions in a Leslie type predator-prey model

Abstract A phenomenon of the noise-induced oscillatory transitions in a predator-prey model of Leslie type with generalized Holling type III functional response is studied. The original deterministic model can exhibit different kinds of phase portraits (one, two or three stable states) for various parameter values. When the predator-prey model is subjected to environmental noise, we find that the stochastic trajectory started near one of the deterministic attractors may experience the oscillatory transitions between different zones. To reveal the probabilistic mechanisms of the noise-induced transitions, we construct the confidence domains of stochastic attractors by applying the technique of stochastic sensitivity functions. It is showed that increasing the noise intensity results in an intersection between different confidence domains, and then the phenomenon of oscillatory transitions can occur.

On a novel approach for modeling liquid crystalline flows

In this paper, we derive a new model for the description of liquid crystalline flows. While microscopic Doi type models suffer from the high dimensionality of the underlying product space, the more macroscopic Ericksen--Leslie type models describe only the long time behavior of the flow and are valid only close to equilibrium. By applying an energetic variational approach, we derive a new macroscopic model which shall provide an improved description far from equilibrium. The novelty of our approach lies in the way the energy is minimized. Distinguishing between the velocities of particles and fluid allows us to define the energy dissipation not in terms of chemical potentials but in terms of friction induced by the discrepancies in the considered velocities. We conclude this publication by establishing the existence of weak solutions to the newly derived model.

Bogdanov-Takens Bifurcation in a Leslie Type Tritrophic Model with General Functional Responses

The dynamics of a differential system modeling a tritrophic food chain of Leslie type is analyzed. It is assumed that the prey has general growth rate and the predator (superpredator) population eats the prey (predator) through a general functional response. The parameter conditions that ensure a coexistence equilibrium point, where the differential system exhibits a codimension 2 Bogdanov-Takens bifurcation (BTb), is given. Some numerical examples where the functional responses are Holling type and the prey population has logistic growth rate are shown. The techniques that have been used to obtain these results may be applied to ecological models with other functional responses types.

Four limit cycles in a predator–prey system of Leslie type with generalized Holling type III functional response

This paper, as a complement to the works by Hsu et al [SIAM. J. Appl. Math. 55 (1995)] and Huang et al [J. Differential Equations 257 (2014)], aims to examine the Hopf bifurcation and global dynamics of a predator–prey system of Leslie type with generalized Holling type III functional response for the two cases: (A) system has a unique anti-saddle positive equilibrium, which is not semi-hyperbolic or nilpotent; (B) system has three distinct positive equilibria. For each case, the type and stability of each equilibrium, Hopf bifurcation at each weak focus, and the number and distribution of limit cycles in the first quadrant are studied. It is shown that every equilibrium is not a center for all parameter values. For the case (A), i limit cycle(s) can appear near the unique positive equilibrium, i=1,…,4. For i=3 or 4, system has two stable limit cycles, which gives a positive answer to the open problem proposed by Coleman [Differential equations model,1983]: finding at least two ecologically stable cycles in any natural (or laboratory) predator–prey or other interacting system. For the case (B), one positive equilibrium is a saddle and the others are both anti-saddle. If one of the two anti-saddles is a weak focus and the other is not, then the order of the weak focus is at most 3. If both anti-saddles are weak foci, then they are unstable weak foci of order one. Moreover, one limit cycle can bifurcate from each of them simultaneously. Numerical simulations demonstrate that there is also a big stable limit cycle enclosing these two limit cycles. Our results indicate that the maximum number of limit cycles in the model of this kind is at least 4, which improves the preceding results that this number is at least 2.