American Institute of Mathematical Sciences

Advanced
October  2010, 14(3): 1095-1103. doi: 10.3934/dcdsb.2010.14.1095

Global properties of a general predator-prey model with non-symmetric attack and consumption rate

Andrei Korobeinikov 1,

1. 

MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick

Received  October 2009 Revised  December 2009 Published  July 2010

Identification of conditions for stable coexistence of interacting populations is a problem of the highest priority in mathematical biology. This problem is usually considered under specific assumptions made regarding the functional forms of non-linear feedbacks. Apparently, such an approach is lacking generality. In this paper, we consider the dynamics of two species with interaction of predator--prey (consumer-supplier) type with the assumption that a part of the resource is neglected or wasted by the predator (consumer). This model describes, for instance, killing for fun; such behaviour is typical for many predators when the prey is abundant.
   We assume that the functional responses that are usually included in such models are given by unspecified functions. Using the direct Lyapunov method, we derive the conditions which ensure global asymptotic stability of this model. It is remarkable that these conditions impose much weaker constraints on the system properties than that are usually assumed.
Keywords: direct Lyapunov method, Ecosystem, mass action, compartment model, endemic equilibrium state, prey-predator model, Lyapunov function, consumer-supplier model, Lotka-Volterra model, global stability, non-linear incidence rate, coexistence, infectious disease, wastage., non-linear attack rate.
Mathematics Subject Classification: Primary: 592D30; Secondary: 34D2.
Citation: Andrei Korobeinikov. Global properties of a general predator-prey model with non-symmetric attack and consumption rate. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1095-1103. doi: 10.3934/dcdsb.2010.14.1095
[1]

Shuhua Zhang, Longzhou Cao, Zuliang Lu. An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021156
[2]

Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3111-3133. doi: 10.3934/dcdsb.2020053
[3]

Hamza Khalfi, Amal Aarab, Nour Eddine Alaa. Energetics and coarsening analysis of a simplified non-linear surface growth model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 161-177. doi: 10.3934/dcdss.2021014
[4]

Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747
[5]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217
[6]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536
[7]

Paolo Buttà, Franco Flandoli, Michela Ottobre, Boguslaw Zegarlinski. A non-linear kinetic model of self-propelled particles with multiple equilibria. Kinetic and Related Models, 2019, 12 (4) : 791-827. doi: 10.3934/krm.2019031
[8]

Francesca Biagini, Katharina Oberpriller. Reduced-form setting under model uncertainty with non-linear affine intensities. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 159-188. doi: 10.3934/puqr.2021008
[9]

Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823
[10]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[11]

Anugu Sumith Reddy, Amit Apte. Stability of non-linear filter for deterministic dynamics. Foundations of Data Science, 2021, 3 (3) : 647-675. doi: 10.3934/fods.2021025
[12]

Hiroshi Ito. Input-to-state stability and Lyapunov functions with explicit domains for SIR model of infectious diseases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5171-5196. doi: 10.3934/dcdsb.2020338
[13]

H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 221-251. doi: 10.3934/dcds.2007.18.221
[14]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
[15]

Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180
[16]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[17]

Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056
[18]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[19]

Tomás Caraballo, Mohamed El Fatini, Idriss Sekkak, Regragui Taki, Aziz Laaribi. A stochastic threshold for an epidemic model with isolation and a non linear incidence. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2513-2531. doi: 10.3934/cpaa.2020110
[20]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy