HOW DO PARTIALLY HIDING BEHAVIOR AND COMPLETELY HIDING BEHAVIOR IMPACT INTERACTING POPULATIONS?

2020 ◽  
Vol 28 (03) ◽  
pp. 641-679
Author(s):  
ZHIHUI MA ◽  
SHUFAN WANG ◽  
HAOPENG TANG
Keyword(s):  
Sufficient Conditions ◽  
Point Of View ◽  
Predator Prey ◽  
Resource Level ◽  
Prey Behavior ◽  
Special Cases ◽  
Interacting Populations ◽  
Predator Prey Models ◽  
Negative Impacts ◽  
Hiding Behavior

As the two main behaviors of prey populations in ecological systems, the partially hiding behavior (PHB) and the completely hiding behavior (CHB) play a significant role in determining the dynamics of predator–prey models. This work examines to the dynamical consequences of predator–prey systems with the PHB and the CHB. Previous research has independently studied the two behaviors, and the general conclusions are that the two behaviors can have positive and/or negative impacts on the considered population models. However, to our knowledge, no study has combined and compared the two behaviors in studying the dynamical consequences of predation interactions. Motivated by this, we investigated the dynamical consequences induced by the PHB and the CHB. From a mathematical point of view, the dynamical behaviors are studied and the corresponding sufficient conditions are given. Our findings are general and some published models are special cases of ours. From an ecological point of view, we find that the size of the ecological regions is mainly determined by the two behaviors, and which one is ecologically beneficial for the health coexistence of the interacting populations are primarily determined by the functional response and the attack coefficient of predators. Moreover, we conclude that the evolutionary and optimal choices of prey behavior (PHB or CHB) depend on the predators attack coefficient (large or small attack coefficient) and the resource level (abundant or pool resource level).

The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5811-5825
Author(s):  
Xinhong Zhang
Keyword(s):  
Mortality Rate ◽  
Stochastic Model ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Global Dynamics ◽  
Type Ii ◽  
Predator Prey ◽  
Persistence In The Mean ◽  
The Mean ◽  
Predator Prey Models

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

scholarly journals Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete Predator-Prey Models in Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Chunqing Wu ◽  
Shengming Fan ◽  
Patricia J. Y. Wong
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Studies ◽  
Patchy Environment ◽  
Predator Prey ◽  
Numerical Examples ◽  
Predator Prey Model ◽  
Prey Model ◽  
Dispersal Corridors ◽  
Predator Prey Models ◽  
Theoretical Results

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

THE PERIODIC PREDATOR-PREY LOTKA–VOLTERRA MODEL WITH IMPULSIVE EFFECT

2002 ◽  
Vol 02 (03n04) ◽  
pp. 267-296 ◽  
Author(s):  
SANYI TANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Volterra Model ◽  
Impulsive Effect ◽  
Biological Pest Control ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Special Cases

In this paper, a classical periodic Lotka–Volterra predator-prey system with impulsive effect is investigated. We analyze the dynamics of positive solutions of such models. Among other results we show that if some trivial or semi-trivial positive solution is linearly stable, then it is globally asymptotically stable with respect to the positive solutions. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution. We use bifurcation theorem to show the existence of coexistence states which arise near the sem-trivial periodic solution. As an application, we also examine some special cases of the system which can be used in the biological pest control.

scholarly journals Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Shuang Guo ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Normal Form Method ◽  
Predator Prey Models ◽  
The Stability

A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.

scholarly journals Conditions for permanence and ergodicity of certain stochastic predator–prey models

2016 ◽  
Vol 53 (1) ◽  
pp. 187-202 ◽  
Author(s):  
Nguyen Huu Du ◽  
Dang Hai Nguyen ◽  
G. George Yin
Keyword(s):  
Transition Probability ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Total Variation Norm ◽  
Distinctive Features ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Degenerate Diffusions ◽  
Predator Prey Models

Abstract In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

A MODEL FOR TWO SPECIES WITH STAGE STRUCTURE AND FEEDBACK CONTROL

2008 ◽  
Vol 01 (03) ◽  
pp. 267-286 ◽  
Author(s):  
HONG ZHANG ◽  
LANSUN CHEN
Keyword(s):  
Feedback Control ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Delay System ◽  
Point Of View ◽  
Predator Population ◽  
Integral Term ◽  
Predator Prey ◽  
Two Stages

This paper studies a periodic coefficients predator-prey delay system with mixed functional response, in which the prey has a history that takes them through two stages, immature and mature. Also, the total toxic action on the predator population expressed by an integral term is considered in our system. Furthermore, the feedback control is considered in our system. Sufficient conditions which guarantee the permanence and extinction of the system are obtained. Finally, we give a brief discussion of our results. From a biological point of view, our results can be used to help protect beneficial animals.

RATIO-DEPENDENT PREDATOR–PREY MODEL WITH STAGE STRUCTURE AND TIME DELAY

2012 ◽  
Vol 05 (04) ◽  
pp. 1250014 ◽  
Author(s):  
LIJUAN ZHA ◽  
JING-AN CUI ◽  
XUEYONG ZHOU
Keyword(s):  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent ◽  
Predator Prey Models

Ratio-dependent predator–prey models are favored by many animal ecologists recently as more suitable ones for predator–prey interactions where predation involves searching process. In this paper, a ratio-dependent predator–prey model with stage structure and time delay for prey is proposed and analyzed. In this model, we only consider the stage structure of immature and mature prey species and not consider the stage structure of predator species. We assume that the predator only feed on the mature prey and the time for prey from birth to maturity represented by a constant time delay. At first, we investigate the permanence and existence of the proposed model and sufficient conditions are derived. Then the global stability of the nonnegative equilibria are derived. We also get the sufficient criteria for stability switch of the positive equilibrium. Finally, some numerical simulations are carried out for supporting the analytic results.

scholarly journals Two Generalized Predator-Prey Models for Integrated Pest Management with Stage Structure and Disease in the Prey Population

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ruiqing Shi ◽  
Sanyi Tang ◽  
Wenli Feng
Keyword(s):  
Integrated Pest Management ◽  
Pest Management ◽  
Natural Enemies ◽  
Impulsive Control ◽  
Control Strategies ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Predator Prey Models ◽  
Impulsive Release

Stage-structured predator-prey models with disease in the prey are constructed. For the purpose of integrated pest management, two types of impulsive control strategies (impulsive release of infective prey and impulsive release of predator) are used. For Case  1, infective prey applications are more frequent than releases of predator (natural enemies). For Case  2, predator (natural enemies) releases are more frequent than infective prey applications. In both cases, we get the sufficient conditions for the global attractivity of the susceptible prey-eradication periodic solution. In addition, the persistence of the systems is also discussed. At last, the results are discussed and some possible future work is put forward.

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