Holling Tanner Research Articles

Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out.

A spatially heterogeneous predator-prey model

This paper introduces a spatially heterogeneous diffusive predator–prey model unifying the classical Lotka–Volterra and Holling–Tanner ones through a prey saturation coefficient, \begin{document}$ m(x) $\end{document} , which is spatially heterogenous and it is allowed to 'degenerate'. Thus, in some patches of the territory the species can interact according to a Lotka–Volterra kinetics, while in others the prey saturation effects play a significant role on the dynamics of the species. As we are working under general mixed boundary conditions of non-classical type, we must invoke to some very recent technical devices to get some of the main results of this paper.

A Holling–Tanner Predator–Prey Model with Strong Allee Effect

We analyze a modified Holling–Tanner predator–prey model where the predation functional response is of Holling type II and we incorporate a strong Allee effect associated with the prey species production. The analysis complements the results of previous articles by Saez and González-Olivares [1999] and Arancibia-Ibarra and González-Olivares [2015] discussing Holling–Tanner models which incorporate a weak Allee effect. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to coexistence and extinction of the species. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogdanov–Takens bifurcations.

The effect of additional food in Holling Tanner type models

Biological control, the use of predators and pathogens to control target pests, is a promising alternative to chemical control. It is hypothesized that the introduced predators efficacy can be boosted by providing them with an additional food source. The current literature primarily considers models of additional food in predator–pest systems for symmetric predator functional responses—that is when the functional and numerical responses are of the same form. The purpose of the current manuscript is to show that providing predators with additional food in models where their functional response is not symmetric, such as Holling–Tanner models, is also effective in pest eradication. Results on stability, cyclicity and Turing instability in such a class of models is investigated. Our results have large scale implications for the effective design of biological control methods involving additional food.

Dynamics of a stochastic Holling–Tanner predator–prey model

In the paper, we study a stochastic Holling–Tanner predator–prey system. First, by constructing a suitable stochastic Lyapunov function we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions to the system. After that, sufficient causes for extinction of the predator population is obtained. Meanwhile, we also obtain the threshold between persistence and extinction of the predator population. At last, some examples and numerical simulations are provided to illustrate our theoretical results.

Spatial Nonhomogeneous Periodic Solutions Induced by Nonlocal Prey Competition in a Diffusive Predator–Prey Model

The diffusive Holling–Tanner predator–prey model with no-flux boundary conditions and nonlocal prey competition is considered in this paper. We show the existence of spatially nonhomogeneous periodic solutions, which is induced by nonlocal prey competition. In particular, the constant positive steady state may lose the stability through Hopf bifurcation when the given parameter passes through some critical values, and the bifurcating periodic solutions near such values could be spatially nonhomogeneous and orbitally asymptotically stable.

The basins of attraction in a modified May–Holling–Tanner predator–prey model with Allee affect

I analyse a modified May–Holling–Tanner predator–prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator–prey population. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle–node, Hopf, and Bogdanov–Takens bifurcations. We use simulations to illustrate the behaviour of the model.

Fisher-KPP dynamics in diffusive Rosenzweig–MacArthur and Holling–Tanner models

We prove the existence of traveling fronts in diffusive Rosenzweig–MacArthur and Holling–Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig–MacArthur predator-prey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling–Tanner population model. In the second situation for the Rosenzweig–MacArthur model we prove the existence of the fronts but without observing a direct relation with Fisher-KPP equation. The analysis suggests that, in a variety of reaction–diffusion systems that rise in population modeling, parameter regimes may be found when the dynamics of the system is inherited from the scalar Fisher-KPP equation.

Spatio-Temporal Pattern Formation in Holling–Tanner Type Model with Nonlocal Consumption of Resources

A wide variety of spatio-temporal models are available in literature which are unable to generate stationary patterns through Turing bifurcation. Introduction of nonlocal terms to the same model can produce Turing patterns and this is true even for a single species population model. In this paper, we consider a prey–predator model of Holling–Tanner type with a generalist predator and a nonlocal interaction in the intra-specific competition term of the prey population. Nonmonotonic functional response is assumed to describe consumption rate of the prey by the predator. The Turing instability condition has been studied for the model without the nonlocal term around coexisting steady states. We also determine the Turing domain in the presence of nonlocal interaction term. The spatial-Hopf bifurcation has been studied and it plays an important role to find the pure Turing domain for the nonlocal model. Furthermore, in the presence of nonlocal interaction, the nonlocal model produces traveling wave solution. Using linear stability analysis, we have obtained the wave speed for the traveling wave front analytically. With the help of numerical simulation, we have verified that the speed of the traveling wave front for the complete nonlinear nonlocal model matches with the analytical approximation. The emergence of wave trains has also been established for higher range of nonlocal interaction.

Global bifurcation for a Holling–Tanner predator–prey model with prey-taxis

The local existence and stability of non-constant positive steady states for the Holling–Tanner predator–prey model with prey-taxis was discussed by Wang et al. (2017) through local bifurcation analysis. It is our purpose in this paper to make a detailed description for the structure of the set of the non-constant steady states. We will prove a global bifurcation theorem by treating prey-taxis as a bifurcation parameter, which gives the existence of non-constant steady states under a rather natural condition. A priori estimates for steady state solutions will play a key role in the proof. In addition, the stability results for the homogeneous equilibrium will extend to the case that the derivative of prey’s functional response with prey is positive, and it is found that attractive prey-taxis can stabilize homogeneous equilibrium even diffusion-driven instability has occurred.

The Holling–Tanner predator–prey model with strong Allee effect on prey

The Holling–Tanner predator–prey model with strong Allee effect on prey

Positive solutions of a nonlocal singular elliptic equation by means of a non-standard bifurcation theory

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tanner.

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.

Diffusive Holling–Tanner predator–prey models in periodic environments

In this paper, by using the Lyapunov method, we establish sufficient conditions for the global asymptotic stability of the positive periodic solution to diffusive Holling–Tanner predator–prey models with periodic coefficients and no-flux conditions.

Predator-prey model of exploited fish population Arowana (Scleropages spp.)

Interactions in the ecosystem are inseparable from predator-prey interactions. Population model of two species that the predator interacts with its prey for its survival has been developed. The growth rate of a predator depends on predation on its prey. The model used to describe this predation is the Holling Tanner II function. This model also includes a catch variables of prey and predator populations. This formulation provides more realistic picture of exploited fisheries, such as Scleropages spp. (arowana) that prey on small fish and shrimp. The purpose of this study is to analyze the model of predator-prey and to interpret on the model based on simulation. High of exploitation rate will lead to rapid population decline. Exploitation rate ranges from 0 ≤ h ≤ h max, where h max is the maximum exploitation rate. Results of these simulation show that population of the predator fish will become extinct if exploitation rate exceeds the maximum exploitation rate.

A diffusive Holling–Tanner prey–predator model with free boundary

This paper is concerned with a diffusive Holling–Tanner prey–predator model in a bounded domain with Dirichlet boundary condition and a free boundary. The global existence of the unique solution is proved. Moreover, the criteria governing spreading–vanishing are derived by mainly using the comparison principle. The results show that if the length of the occupying line is bigger than a threshold value (spreading barrier), then the spreading of predators will make an achievement, and, if the length of the occupying line is smaller than this spreading barrier and the spreading coefficient is relatively small depending on initial size of predators, then the predators will fail in establishing themselves and eventually die out.

Qualitative analysis of a strongly coupled predator\u2013prey system with modified Holling\u2013Tanner functional response

In this paper, we deal with a strong coupled predator–prey model with modified Holling–Tanner functional response under homogeneous Neumann boundary conditions. We mainly discuss the following two problems: (1) stability of the positive constant solution; (2) existence and non-existence results as regards the non-constant positive solutions.

Traveling waves of some Holling–Tanner predator–prey system with nonlocal diffusion

This paper is devoted to establish the existence and non-existence of the traveling waves for the nonlocal Holling–Tanner predator–prey model. By applying the Schauder’s fixed point theorem, we can obtain the existence of the traveling waves. Moreover, in order to prove the limit behavior of the traveling waves at infinity, we construct a sequence that converges to the coexistence state. For the proof of the nonexistence of the traveling waves, we use the property of the two-sided Laplace transform. Finally, we give the effect of the nonlocal diffusion term for the traveling waves.

Spatial patterns of a fractional type cross-diffusion Holling–Tanner model

In this paper, a fractional type cross-diffusion Holling–Tanner model under homogeneous Neumann boundary condition is considered. Using the maximum principle and the Harnack inequality, we obtain a priori estimates of the positive stationary solution. Nonexistence and existence of nonconstant positive stationary solution are obtained, which reveal that fractional type cross-diffusion can create spatial patterns but the random diffusion fails.

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.