Ratio-dependent Holling-Tanner Research Articles

Local and Global Dynamics of a Ratio-Dependent Holling–Tanner Predator–Prey Model with Strong Allee Effect

In this paper, the impact of the strong Allee effect and ratio-dependent Holling–Tanner functional response on the dynamical behaviors of a predator–prey system is investigated. First, the positivity and boundedness of solutions of the system are proved. Then, stability and bifurcation analysis on equilibria is provided, with explicit conditions obtained for Hopf bifurcation. Moreover, global dynamics of the system is discussed. In particular, the degenerate singular point at the origin is proved to be globally asymptotically stable under various conditions. Further, a detailed bifurcation analysis is presented to show that the system undergoes a codimension-[Formula: see text] Hopf bifurcation and a codimension-[Formula: see text] cusp Bogdanov–Takens bifurcation. Simulations are given to illustrate the theoretical predictions. The results obtained in this paper indicate that the strong Allee effect and proportional dependence coefficient have significant impact on the fundamental change of predator–prey dynamics and the species persistence.

BIFURCATIONS EMERGING FROM DIFFERENT DELAYS IN A FRACTIONAL-ORDER PREDATOR–PREY MODEL

This paper characterizes the stability and bifurcation of fractional-order ratio-dependent Holling–Tanner type model with fractional domain [Formula: see text]. The stability intervals and bifurcation conditions of the developed model are attained by viewing different delays as bifurcation parameters. Then, two numerical examples are employed to corroborate the correctness of the theoretical analysis consisting of figuring out the bifurcation point and checking the veracity of the acquired bifurcation results via the plotted bifurcation diagrams. It revamps the deficiency of fractional-order model with unique delay. The procured results are instrumental in exploring the intrinsic convolution of predator–prey models.

Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

A discrete-time Holling-Tanner model with ratio-dependent functional response is examined. We show that the system experiences a flip bifurcation and Neimark-Sacker bifurcation or both together at positive fixed point in the interior of mathbb {R}^{2}_{+} when one of the model parameter crosses its threshold value. We concentrate our attention to determine the existence conditions and direction of bifurcations via center manifold theory. To validate analytical results, numerical simulations are employed which include bifurcations, phase portraits, stable orbits, invariant closed circle, and attracting chaotic sets. In addition, the existence of chaos in the system is justified numerically by the sign of maximum Lyapunov exponents and fractal dimension. Finally, we control chaotic trajectories exists in the system by feedback control strategy.

Note on the stability property of a ratio-dependent Holling-Tanner model

A Holling-Tanner system with ratio-dependence functional response is revisited in this paper. By de9 veloping the new analysis technique, two set of new conditions which ensure the global attractivity of the positive 10 equilibrium of the system are obtained. Our results essential improving the main results of Liang and Pan [Quali11 tative analysis of a ratio-dependent Holling-Tanner model, J. Math. Anal. Appl. 334 (2007) 954-964].

Turing–Hopf Bifurcation and Spatio-Temporal Patterns of a Ratio-Dependent Holling–Tanner Model with Diffusion

In this paper, the dynamics of a diffusive ratio-dependent Holling–Tanner model subject to Neumann boundary conditions is considered. We derive the conditions for the existence of Hopf, Turing, Turing–Hopf, Turing–Turing, Hopf-double-Turing and triple-Turing bifurcations at the unique positive equilibrium. Furthermore, we study the detailed dynamics in the neighborhood of the Turing–Hopf bifurcation by using the normal form method. Our results show that the Turing–Hopf bifurcation can give rise to the formation of the temporal and spatio-temporal patterns. In particular, we theoretically prove the existence of the spatially inhomogeneous periodic and quasi-periodic solutions, which can be used to explain the phenomenon of spatio-temporal resonance of the populations. Finally, the numerical simulations are given to illustrate the analytical results.

Qualitative analysis of a stochastic ratio-dependent Holling-Tanner system

This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stochastic differential equations. Secondly, in the case of persistence, we prove that there exists a ergodic stationary distribution. Finally, numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.

Exploring the dynamics of a Holling–Tanner model with cannibalism in both predator and prey population

Cannibalism is an intriguing life history trait, that has been considered primarily in the predator, in predator–prey population models. Recent experimental evidence shows that prey cannibalism can have a significant impact on predator–prey population dynamics in natural communities. Motivated by these experimental results, we investigate a ratio-dependent Holling–Tanner model, where cannibalism occurs simultaneously in both the predator and prey species. We show that depending on parameters, whilst prey or predator cannibalism acting alone leads to instability, their joint effect can actually stabilize the unstable interior equilibrium. Furthermore, in the spatially explicit model, we find that depending on parameters, prey and predator cannibalism acting jointly can cause spatial patterns to form, while not so acting individually. We discuss ecological consequences of these findings in light of food chain dynamics, invasive species control and climate change.

Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay

The existence of traveling wave solutions and wave train solutions of a diffusive ratio-dependent predator-prey system with distributed delay is proved. For the case without distributed delay, we first establish the existence of traveling wave solution by using the upper and lower solutions method. Second, we prove the existence of periodic traveling wave train by using the Hopf bifurcation theorem. For the case with distributed delay, we obtain the existence of traveling wave and traveling wave train solutions when the mean delay is sufficiently small via the geometric singular perturbation theory. Our results provide theoretical basis for biological invasion of predator species.

Global dynamics of a ratio-dependent Holling–Tanner predator–prey system

A ratio-dependent Holling–Tanner predator–prey model is studied. By investigating the local stability properties of equilibrium points and periodic solutions of a transformed system, and by applying the Poincaré–Bendixson theorem, we are able to give a complete classification of the global dynamics of the ratio-dependent Holling–Tanner predator–prey model.

Phase Portraits, Hopf Bifurcations and Limit Cycles of the Ratio Dependent Holling-Tanner Models for Predator-prey Interactions

In this paper we consider a ratio dependent Holling Tanner predator prey model with type II functional responses. We analyzed the local stability, phase portraits, existence and uniqueness of stable limit cycles and Hopf bifurcation. The ranges of the parameter involved are provided under which the unique interior equilibrium can be determined for a stable (or an unstable) node or focus without diffusion. Furthermore the Turing instability analysis of the system with diffusion are studied. Numerical simulations using MATLAB are carried out to demonstrate the theoretical results obtained.

Stability and Hopf Bifurcation in a delayed ratio dependent Holling–Tanner type model

In this study, a delayed ratio dependent Holling–Tanner type predator–prey model is investigated. First, the local stability of a positive equilibrium is studied and then the existence of Hopf bifurcations is established. By using the normal form theory and center manifold theorem, the explicit algorithm determining the stability, direction of the bifurcating periodic solutions are derived. Finally, we perform the numerical simulations for justifying the theoretical results.

Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting

The bifurcation properties of a predator prey system with refuge and constant harvesting are investigated. The number of the equilibria and the properties of the system will change due to refuge and harvesting, which leads to the occurrence of several kinds bifurcation phenomena, for example, the saddle-node bifurcation, Bogdanov-Takens bifurcation, Hopf bifurcation, backward bifurcation, separatrix connecting a saddle-node and a saddle bifurcation and heteroclinic bifurcation, and so forth. Our main results reveal much richer dynamics of the system compared to the system with no refuge and harvesting.

Turing instabilities and spatio-temporal chaos in ratio-dependent Holling–Tanner model

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling–Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing–Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing–Hopf bifurcation leads to onset of spatio-temporal chaos when predator’s diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.

Dynamical analysis of a delayed ratio-dependent Holling–Tanner predator–prey model

In this paper, a delayed Holling–Tanner predator–prey model with ratio-dependent functional response is considered. It is proved that the model system is permanent under certain conditions. The local asymptotic stability and the Hopf-bifurcation results are discussed. Qualitative behaviour of the singularity ( 0 , 0 ) is explored by using a blow up transformation. Global asymptotic stability analysis of the positive equilibrium is carried out. Numerical simulations are presented for the support of our analytical findings.

Qualitative analysis of a ratio-dependent Holling–Tanner model

In this paper, we consider a Holling–Tanner system with ratio-dependence. First, we establish the sufficient conditions for the global stability of positive equilibrium by constructing Lyapunov function. Second, through a simple change of variables, we transform the ratio-dependent Holling–Tanner model into a better studied Liénard equation. As a result, the uniqueness of limit cycle can be solved.