Holling Tanner Predator Prey Model 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.

Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and anti-predator behavior

In this paper, a Holling–Tanner predator–prey model with constant-yield prey harvesting and anti-predator behavior is investigated. It is shown that the codimension of a nilpotent cusp is 3 and a degenerate Bogdanov–Takens bifurcation of codimension 3 acts as an organizing center for rich dynamical behaviors including saddle–node bifurcation, Hopf bifurcation, saddle–node bifurcation of limit cycles, and homoclinic bifurcation. Moreover, a Hopf bifurcation of codimension 2 is found, a codimension-2 cusp of limit cycles is firstly observed, and the corresponding normal form coefficients are also obtained, which indicate that there exists an acute angle region of three coexistent limit cycles. In particular, a reversed S-shaped saddle–node bifurcation curve of limit cycles is found. In addition, we observe that the anti-predator behavior may cause nearly extinction of the predator population while the prey population reaches its maximum which is less than the carrying capacity due to the effect of harvesting; however, the constant-yield prey harvesting may drive both species to extinction suddenly. Finally, numerical simulations including bifurcation diagrams and phase portraits are given to illustrate the theoretical results.

Bifurcation analysis in a Holling-Tanner predator-prey model with strong Allee effect.

In this paper, we analyze the bifurcation of a Holling-Tanner predator-prey model with strong Allee effect. We confirm that the degenerate equilibrium of system can be a cusp of codimension 2 or 3. As the values of parameters vary, we show that some bifurcations will appear in system. By calculating the Lyapunov number, the system undergoes a subcritical Hopf bifurcation, supercritical Hopf bifurcation or degenerate Hopf bifurcation. We show that there exists bistable phenomena and two limit cycles. By verifying the transversality condition, we also prove that the system undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. The main conclusions of this paper complement and improve the previous paper [30]. Moreover, numerical simulations are given to verify the theoretical results.

Qualitative analysis on the diffusive Holling–Tanner predator–prey model

We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steady-state. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel'kov model to obtain the non-existence of positive non-constant steady-states if 0 < p ≤ 1 .

Bifurcations and pattern formation in a predator-prey model with memory-based diffusion

New spatial temporal patterns for Holling-Tanner predator-prey model with memory-based diffusion are investigated. Firstly, we show the existence of Turing, Hopf, Turing-Turing, and Turing-Hopf bifurcations, and obtain sufficient and necessary conditions for Turing instability. Secondly, we extend the formulae for coefficients in the normal form of Turing-Hopf bifurcation in Jiang et al. (2020) [1], and provide new formulae applicable to the general reaction-diffusion system with memory-based self-diffusion and cross-diffusion. These formulae are also applicable to the case that the memory delay is absent. From the bifurcation analysis, we finally find that the model can exhibit various complex spatiotemporal patterns, including a pair of stable spatially inhomogeneous periodic solutions with two spatial frequencies and heteroclinic orbits connecting non-constant steady states to a spatially inhomogeneous periodic solution. These results show that large memory-based diffusion coefficient can eliminate spatial patterns driven by Fickian diffusion, and memory delay mainly affects spatiotemporal patterns arising from Hopf and Turing-Hopf bifurcations.

Hopf bifurcation analysis of a fractional order Holling-Tanner predator-prey model with time delay

We study a fractional-order delayed predator-prey model with Holling–Tanner-type functional response. Mainly, by choosing the delay time \(\tau \) as the bifurcation parameter, we show that Hopf bifurcation can occur as the delay time \(\tau \) passes some critical values. The local stability of a positive equilibrium and the existence of the Hopf bifurcations are established, and numerical simulations for justifying the theoretical analysis are also presented. doi:https://doi.org/10.1017/S1446181122000025

Bifurcation Analysis of a Modified May–Holling–Tanner Predator–Prey Model with Allee Effect

Bifurcation Analysis of a Modified May–Holling–Tanner Predator–Prey Model with Allee Effect

Turing Instability of a Modified Reaction–Diffusion Holling–Tanner Model Over a Random Network

Turing instability is a prominent feature of reaction–diffusion systems, which is widely investigated in many fields, such as ecology, neurobiology, chemistry. However, although the inhomogeneous diffusion between prey and predators exist in their network space, there are few considerations on how network diffusion affects the stability of prey–predator models. Therefore, in this paper we study the pattern dynamics of a modified reaction–diffusion Holling–Tanner prey–predator model over a random network. Specifically, we study the relationship between the node degrees of the random network and the eigenvalues of the network Laplacian matrix. Then, we obtain conditions under which the network system instability, Hopf bifurcation as well as Turing bifurcation occur. Also, we find an approximate Turing instability region of the diffusion coefficient and the connection probability of the network. Finally, we apply the mean-field approximation theory with numerical simulation to confirm the correctness of our results. The instability region indicates the random migration of the prey and predators among different communities.

Nonlocal Competition and Spatial Multi-peak Periodic Pattern Formation in Diffusive Holling-Tanner Predator-prey Model

Nonlocal Competition and Spatial Multi-peak Periodic Pattern Formation in Diffusive Holling-Tanner Predator-prey Model

Pattern formation in the Holling–Tanner predator–prey model with predator-taxis. A nonstandard finite difference approach

A characteristic feature of living organisms is their response to the environment in search for food or reproduction opportunities. This paper is devoted to the investigation of the pattern formation of the Holling-Tanner predator–prey model with predator-taxis. We first summarise the qualitative properties of the model where a threshold for the appearance of pattern formation is specified. Then we design and analyse a coupled nonstandard finite difference and finite volume scheme for the proposed model. Numerical simulations are provided to support theoretical findings.

Complex Dynamical Behavior of Holling–Tanner Predator-Prey Model with Cross-Diffusion

Spatial predator-prey models have been studied by researchers for many years, because the exact distributions of the population can be well illustrated via pattern formation. In this paper, amplitude equations of a spatial Holling–Tanner predator-prey model are studied via multiple scale analysis. First, by amplitude equations, we obtain the corresponding intervals in which different kinds of patterns will be onset. Additionally, we get the conclusion that pattern transitions of the predator are induced by the increasing rate of conversion into predator biomass. Specifically, pattern transitions of the predator between distinct Turing pattern structures vary in an orderly manner: from spotted patterns to stripe patterns, and finally to black-eye patterns. Moreover, it is discovered that pattern transitions of prey can be induced by cross-diffusion; that is, patterns of prey transmit from spotted patterns to stripe patterns and finally to a mixture of spot and stripe patterns. Meanwhile, it is found that both effects of cross-diffusion and interaction between the prey and predator can lead to the complicated phenomenon of dynamics in the system of biology.

Normal form formulations of double-Hopf bifurcation for partial functional differential equations with nonlocal effect

This paper is concerned with the calculation of the normal form on the center manifold up to the third-order term at the double-Hopf singularity for general partial functional differential equations with the nonlocal effect. We derive explicit formulas of the normal form, that can be applied for both functional differential equations and partial differential equations with or without nonlocal effects in a bounded spatial domain. This provides an effective tool to establish existences of multi-periodic and quasi-periodic oscillations of a double-Hopf singularity for such equations. As an example, the Holling-Tanner predator-prey model with the nonlocal intraspecific competition of prey is considered. It turns out that double-Hopf bifurcation will occur because of the nonlocal interaction. Many spatio-temporal dynamics are found, including stable spatially homogeneous or nonhomogeneous periodic solutions, and stable spatially nonhomogeneous quasi-periodic solutions.

Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

AbstractWe investigate the effect of nonlocal intraspecific prey competition on the spatiotemporal dynamics of a Holling–Tanner predator–prey model with diffusion. We first establish the criteria for Hopf, Turing, double‐Hopf, and Turing–Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. For Turing–Hopf bifurcation, by analyzing the normal form truncated to the third order, we derive that, with strong nonlocal interaction, the system exhibits the tristable phenomena, that is, the coexistence of a stable spatially nonhomogeneous periodic orbit and two nonconstant stable steady states, as well as the existence of periodic orbits with two spatial wave frequencies induced by the nonlocal interaction. The main analytical difficulty arises from the nonlocal interaction that prevents the direct application of formulas for the coefficients of the normal form. Biologically, the emerging spatiotemporal patterns suggest that the global intraspecific competition can promote the coexistence of the prey and predator by allowing the prey maintain a critical total population size, which may provide an alternative approach in explaining the group formation of some prey species under the risk of predation. Some coexistence patterns are destabilized by introducing another prey species with local intraspecific competition, leading to the coexistence of two preys and one predator, namely, the prey with nonlocal interaction is concentrated at a single spatial location, and the other prey is distributed uniformly in the rest of the habitat. Accordingly, the predator is forced to change its behavior as well.

Global stability of spatially nonhomogeneous steady state solution in a diffusive Holling-Tanner predator-prey model

The global stability of the nonhomogeneous positive steady state solution to a diffusive Holling-Tanner predator-prey model in a heterogeneous environment is proved by using a newly constructed Lyapunov function and estimates of nonconstant steady state solutions. The techniques developed here can be adapted for other spatially heterogeneous consumer-resource models.

Holling-Tanner Predator-Prey Model with Type-IV Functional Response and Harvesting

In this paper we modify Holling-Tanner predator-prey model by using type-IV functional response in prey species in lieu of type-II functional response. Harvesting is used in predator as well as prey species. This model is compared with a special type of Kolmogorov model. In the case of quadratic harvesting, the fixed points are computed after nondimensionalization. For the non-existence of periodic orbits in the first quadrant we apply a condition of the general Kolmogorov model to exist a Dulac function. We show that this system does not have periodic orbits with the help of numerical simulation and graphical representation.

Traveling wave solutions to diffusive Holling-Tanner predator-prey models

In this paper, we first establish the existence of semi-traveling wave solutions to a diffusive generalized Holling-Tanner predator-prey model in which the functional response may depend on both the predator and prey populations. Then, by constructing the Lyapunov function, we apply the obtained result to show the existence of traveling wave solutions to the diffusive Holling-Tanner predator-prey models with various functional responses, including the Lotka-Volterra type functional response, the Holling type Ⅱ functional response and the Beddington-DeAngelis functional response.

Holling-Tanner prey-predator model with Beddington-DeAngelis functional response including delay

ABSTRACT This paper is designed based on the combined bioeconomic harvesting of Holling-Tanner prey-predator competition model with Beddington-DeAngelis functional response with two different delays. The situation and existence of the steady state points for proposed model are discussed. The conditions for local stability of the interior steady state points together with the existence of Hopf bifurcation at the interior steady state point are addressed. The situations for stability and the direction of bifurcating periodic solutions are obtained by using center manifold theory. Also total revenue is calculated using bionomic equilibrium, and it is maximized by using Pontryagin’s maximum principle. Some numerical simulations together with pictorial representations are placed in the paper for showing the effects of various parameters in the considered model. Conclusions with the future study are described at last.

On Wave Patterns in a Spatially Extended Holling–Tanner Model

In this paper we consider the extended spatially Holling–Tanner predator–prey model. We study traveling wave solutions for the model, where the predator equation has a singularity at zero prey population. Using a dynamical systems approach we show analytically and numerically the existence of different types of traveling waves for this model. Further, numerical simulations of the partial differential equations system problems are presented to illustrate and confirm these results.

Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis

A reaction-diffusion predator-prey system with prey-taxis and predator-taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing prey and prey evading predators. The spatial pattern formation induced by the prey-taxis and predator-taxis is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. It is shown that both attractive prey-taxis and repulsive predator-taxis compress the spatial patterns, while repulsive prey-taxis and attractive predator-taxis help to generate spatial patterns. Our results are applied to the Holling-Tanner predator-prey model to demonstrate the pattern formation mechanism.

A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.