Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

AbstractA new way to study the harvested predator–prey system is presented by analyzing the dynamics of two-prey and one-predator model, in which two teams of prey are interacting with one team of predators and the harvesting functions for two prey species takes different forms. Firstly, we make a brief analysis of the dynamics of the two subsystems which include one predator and one prey, respectively. The positivity and boundedness of the solutions are verified. The existence and stability of seven equilibrium points of the three-species model are further studied. Specifically, the global stability analysis of the coexistence equilibrium point is investigated. The problem of maximum sustainable yield and dynamic optimal yield in finite time is studied. Numerical simulations are performed using MATLAB from four aspects: the role of the carrying capacity of prey, the simulation about the model equations around three biologically significant steady states, simulation for the yield problem of model system, and the comparison between the two forms of harvesting functions. We obtain that the new form of harvesting function is more realistic than the traditional form in the given model, which may be a better reflection of the role of human-made disturbance in the development of the biological system.

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Feedback control and its impact on generalist predator–prey system with prey harvesting

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2019.5.3 ◽

2019 ◽

Vol 24 (5) ◽

Author(s):

Debabrata Das ◽

Tapan Kumar Kar

Keyword(s):

Feedback Control ◽

Sustainable Management ◽

Prey Species ◽

Fishing Effort ◽

Maximum Sustainable Yield ◽

Management Policy ◽

Generalist Predator ◽

Predator Density ◽

Predator Prey ◽

Prey System

This article examines the effectiveness of feedback control as a management policy on a generalist predator–prey system with prey harvesting. We discuss the result of implementing feedback control with respect to prey and predator separately. This paper also depicts the effect of exploitations up to maximum sustainable yield (MSY). We observe that with a constant fishing effort MSY policy is a sustainable management policy to protect both the species. However, further increase of fishing effort may cause the extinction of prey species. But considering feedback control of fishing effort may restrict the extinction of prey species. When fishing effort is controlled in terms of prey density, the extinction of prey population can be avoided. In this case, there may be coexistences of prey, predator and fishery or extinction of fishery. But when fishing effort is controlled by predator density, it is difficult to manage the coexistences of prey, predator and fishery.

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Effects of fear and anti-predator response in a discrete system with delay

Discrete & Continuous Dynamical Systems - B ◽

10.3934/dcdsb.2021200 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Ritwick Banerjee ◽

Pritha Das ◽

Debasis Mukherjee

Keyword(s):

Discrete System ◽

Prey Species ◽

Equilibrium Points ◽

Sufficient Conditions ◽

The Other ◽

Predator Prey ◽

Predator Response ◽

Predator Model ◽

The Cost ◽

Predator Behavior

<p style='text-indent:20px;'>In this paper a discrete-time two prey one predator model is considered with delay and Holling Type-Ⅲ functional response. The cost of fear of predation and the effect of anti-predator behavior of the prey is incorporated in the model, coupled with inter-specific competition among the prey species and intra-specific competition within the predator. The conditions for existence of the equilibrium points are obtained. We further derive the sufficient conditions for permanence and global stability of the co-existence equilibrium point. It is observed that the effect of fear induces stability in the system by eliminating the periodic solutions. On the other hand the effect of anti-predator behavior plays a major role in de-stabilizing the system by giving rise to predator-prey oscillations. Finally, several numerical simulations are performed which support our analytical findings.</p>

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The uneasy coexistence of predators and pathogens

The European Physical Journal E ◽

10.1140/epje/i2020-11966-7 ◽

2020 ◽

Vol 43 (7) ◽

Author(s):

Andreas Eilersen ◽

Kim Sneppen

Keyword(s):

Host Species ◽

Prey Species ◽

Scaling Relations ◽

Predator Species ◽

Predator Prey ◽

Mass Scaling ◽

Prey System ◽

Model Equations ◽

Extinction Events ◽

Parameter Values

Abstract. Disease and predation are both highly important in ecology, and pathogens with multiple host species have turned out to be common. Nonetheless, the interplay between multi-host epidemics and predation has received relatively little attention. Here, we analyse a model of a predator-prey system with disease in both prey and predator populations and determine reasonable parameter values using allometric mass scaling relations. Our analysis focuses on the possibility of extinction events rather than the linear stability of the model equations, and we derive approximate relations for the parameter values at which we expect these events to occur. We find that if the predator is a specialist, epidemics frequently drive the predator species to extinction. If the predator has an additional, immune prey species, predators will usually survive. Coexistence of predator and disease is impossible in the single-prey model. We conclude that for the prey species, carrying a pathogen can be an effective weapon against predators, and that being a generalist is a major advantage for a predator in the event of an epidemic affecting the prey or both species. Graphical abstract

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The uneasy coexistence of predators and pathogens

10.1101/721506 ◽

2019 ◽

Cited By ~ 1

Author(s):

Andreas Eilersen ◽

Kim Sneppen

Keyword(s):

Host Species ◽

Prey Species ◽

Scaling Relations ◽

Predator Species ◽

Predator Prey ◽

Mass Scaling ◽

Prey System ◽

Model Equations ◽

Extinction Events ◽

Parameter Values

Disease and predation are both highly important in ecology, and pathogens with multiple host species have turned out to be common. Nonetheless, the interplay between multi-host epidemics and predation has received relatively little attention. Here, we analyse a model of a predator-prey system with disease in both prey and predator populations and determine reasonable parameter values using allometric mass scaling relations. Our analysis focuses on the possibility of extinction events rather than the linear stability of the model equations. We find that if the predator is a specialist, epidemics frequently drive the predator species to extinction. If the predator has an additional, immune prey species, predators will usually survive. Coexistence of predator and disease is impossible in the single-prey model. We conclude that for the prey species, carrying a pathogen can be an effective weapon against predators, and that being a generalist is a major advantage for a predator.

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Stability and Bifurcation Analysis in a Predator–Prey System with Constant Harvesting and Prey Group Defense

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417501796 ◽

2017 ◽

Vol 27 (11) ◽

pp. 1750179 ◽

Cited By ~ 9

Author(s):

Jianfeng Luo ◽

Yi Zhao

Keyword(s):

Hopf Bifurcation ◽

Maximum Sustainable Yield ◽

Sustainable Yield ◽

Defensive Strategy ◽

Predator Prey ◽

Interior Equilibrium ◽

Prey System ◽

Constant Rate ◽

The Given ◽

Existence Of Equilibria

In this paper, we study a predator–prey system that the prey population gathers in herds to defend its predator and both are harvested by constant rate. The defensive strategy of the gathered prey makes the individuals at the border of the herd mostly suffer from the attacks of the predators. This behavior can be described by a modified Holling-type II functional response in mathematics. Notably, we consider harvesting under two cases: prey harvesting only and predator harvesting only. We investigate the existence of equilibria for both cases, and then find there exists the maximum sustainable yield for two cases to guarantee predator and prey to coexist. Moreover, both species can coexist under some conditions and initial values through investigation of stability of the interior equilibrium in the given system. These results demonstrate that, when hunting the prey or predator for economic interest, harvesting rate must be chosen at a suitable value (not merely less than the maximum sustainable yield) to maintain the coexistence of the predator and prey as well as ecological balance. Finally, we analyze the saddle-node bifurcation and Hopf bifurcation, and determine the direction of Hopf bifurcation by calculating the first Lyapunov number for both cases. In particular, Bogdanov–Takens bifurcation occurs only in the given system with predator harvesting.

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Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

Advances in Difference Equations ◽

10.1186/s13662-021-03324-w ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sekson Sirisubtawee ◽

Nattawut Khansai ◽

Akapak Charoenloedmongkhon

Keyword(s):

Periodic Solution ◽

Functional Response ◽

Impulsive Control ◽

Positive Periodic Solution ◽

Existence Condition ◽

Predator Prey ◽

Type Iv ◽

Prey System ◽

Existence And Stability ◽

Predator Behavior

AbstractIn the present article, we propose and analyze a new mathematical model for a predator–prey system including the following terms: a Monod–Haldane functional response (a generalized Holling type IV), a term describing the anti-predator behavior of prey populations and one for an impulsive control strategy. In particular, we establish the existence condition under which the system has a locally asymptotically stable prey-eradication periodic solution. Violating such a condition, the system turns out to be permanent. Employing bifurcation theory, some conditions, under which the existence and stability of a positive periodic solution of the system occur but its prey-eradication periodic solution becomes unstable, are provided. Furthermore, numerical simulations for the proposed model are given to confirm the obtained theoretical results.

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On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

Abstract and Applied Analysis ◽

10.1155/2010/161978 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12

Author(s):

Xiangzeng Kong ◽

Zhiqin Chen ◽

Li Xu ◽

Wensheng Yang

Keyword(s):

Functional Response ◽

Prey Species ◽

Necessary Conditions ◽

Stage Structure ◽

Predator Prey ◽

Prey System ◽

Sufficient And Necessary Conditions ◽

Holling Iii Functional Response

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

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Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

Abstract and Applied Analysis ◽

10.1155/2008/371632 ◽

2008 ◽

Vol 2008 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

Can-Yun Huang ◽

Min Zhao ◽

Hai-Feng Huo

Keyword(s):

Functional Response ◽

Prey Species ◽

Necessary Conditions ◽

Stage Structure ◽

Functional Responses ◽

Predator Prey ◽

Predator Prey Model ◽

Prey System ◽

Sufficient And Necessary Conditions ◽

Holling Ii Functional Response

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

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