Effect of Fear on a Modified Lesli-Gower Predator-Prey Eco-epidemiological model with disease in Predator

다국어 초록 (Multilingual Abstract)

The anti-predator factor due to fear of predator in eco- epidemiological models has a great importance and cannot be evaded. The present paper consists of a modified Lesli-Gower predator-prey model with contagious disease in the predator population only and also consider the fear effect in the prey population. Boundedness and positivity have been studied to ensure the eco-epidemiological model is well-behaved. The existence and stability conditions of all possible equilibria of the model have been studied thoroughly. Considering the fear constant as bifurcating parameter, the conditions for the existence of limit cycle under which the system admits a Hopf bifurcation are investigated. The detailed study for direction of Hopf bifurcation have been derived with the use of both the normal form and the central manifold theory. We observe that the increasing fear constant, not only reduce the prey density, but also stabilize the system from unstable to stable focus by excluding the existence of periodic solutions.

참고문헌 (Reference)

1 H. I. Freedman, "Uniform Persistence in Functional Differential Equations" 115 : 173-192, 1995

2 B. K. Singh, "The role of virus infection in a simple phytoplankton zooplankton system" 231 : 545-559, 2004

3 J. Wang, "The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge" 29 : 2019

4 P. Panday, "Stability and bifurcation analysis of a three-species food chain model with fear" 28 : 1850009-, 2018

5 A. K. Pal, "Stability analysis of an eco-epidemiological model incorpo-rating a prey refuge" 15 : 473-491, 2010

6 P. H. Lesli, "Some further notes on the use of matrices in population mathematics" 35 : 213-245, 1948

7 S. Pal, "Role of fear in a predator-prey model with Beddington-DeAngelis functional response" 74 : 301-327, 2019

8 R. M. Anderson, "Regulation and stability of host-parasite population inter-actions: I. Regulatory processes" 47 : 219-247, 1978

9 K. P. Hadeler, "Predator-prey populations with parasitic infection" 27 : 609-631, 1989

10 S. Creel, "Predation risk affects reproductive physiology and demography of elk" 315 : 960-, 2007

11 T.C. Gard, "Persistence in food web-1, Lotka- Volterra food chains" 41 : 302-315, 1979

12 L. Y. Zanette, "Perceived predation risk reduces the number of offspring songbirds produce per year" 334 : 1398-1401, 2011

13 G. Birkhoff, "Ordinary Differential Equations" Ginn 1982

14 A. Mondal, "On the dynamics of evolutionary Lesli-Gower predator-prey eco-epidemiological model with disease in predator" 10 : 100034-, 2019

15 J. Guckenheimer, "Nonlinear Oscillations, Dynamical Systems, and Bifur-cations of Vector Fields" Springer-Verlag 1983

16 Y. Xiao, "Modelling and analysis of a predator-prey model with disease in the prey" 171 : 59-82, 2001

17 X. Wang, "Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators" 79 : 1325-1359, 2017

18 X. Wang, "Modeling the fear effect in predator-prey interactions" 73 : 1179-1204, 2016

19 E. Venturino, "Mathematical population dynamics: analysis of heterogeneity, Theory of epidemics 1" 381-393, 1995

20 S. Wiggins, "Introduction to applied nonlinear dynamical systems and chaos" Springer 2003

21 F. Hua, "Increased perception of predation risk to adults and offspring alters avian reproductive strategy and performance" 25 : 509-519, 2014

22 S. Ruan, "Global analysis in a predator-prey system with nonmonotonic func-tional response" 61 : 1445-1472, 2001

23 L. Han, "Four predator prey models with infectious disease" 34 : 849-858, 2001

24 Y. Zhu, "Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Lesli-Gower Holling-type II schemes" 384 : 445-454, 2011

25 E. Venturino, "Epidemics in predator-prey models: disease in the predators" 19 : 185-205, 2002

26 M. Haque, "Effect of delay in a Lotka-Volterra type predator-prey model with a transmission disease in the predator species" 234 : 47-57, 2011

27 T. K. Kar, "Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide" 310 : 187-198, 2012

28 S.K. Sasmal, "Dynamics of a predator-prey system with fear and group defense" 481 : 2020

29 Das, K.P., "Disease-induced chaotic oscillations and its possible control in a predator-prey system with disease in predator" 2015

30 L. Perko, "Differential Equations and Dynamical Systems" Springer-Verlag 2001

31 A. Maiti, "Deterministic and stochastic analysis of a prey-dependent predator-prey system" 36 : 65-83, 2005

32 H.I. Freedman, "Deterministic Mathematical Models in Population Ecology" Marcel Dekker 1980

33 A. Wolf, "Determining Lyapunov exponents from a time series" 16 : 285-317, 1985

34 M. A. Aziz-Alaoui, "Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling type II schemes" 16 : 1069-1075, 2003

35 R. Gupta, "Bifurcation analysis of modified Lesli-Gower predator-prey model with Michaelis-Menten type harvesting" 398 : 278-295, 2013

36 A. Sha, "Backward bifurcation, oscillations and chaos in an eco-epidemiological model with fear effect" 13 : 301-327, 2019

37 J. Carr, "Applications of Center Manifold Theory" Springer 1982

38 A. Nindjin, "Analysis of a predator-prey model with mod-ified Lesli-Gower and Holling-type II schemes with time delay" 7 : 1104-1118, 2006

39 J. L. Orrock, "An island-wide predator manipulation reveals immediate and long-lasting matching of risk by prey" 281 (281): 2014039-, 2014

40 J.P. Tripathi, "Almost Periodicity of a Modified Lesli-Gower Predator-Prey System with Crowley-Martin functional Response" 143 : 2015

41 H. Hethcote, "A predator-prey model with infected prey" 66 : 259-268, 2004

42 D. Greenhalgh, "A predator-prey model with disease in the prey species only" 30 : 911-929, 2006

43 J. Chattopadhyay, "A predator-prey model with disease in the prey" 36 : 747-766, 1999

44 P. H. Lesli, "A Stochastic model for studying the properties of certain biological systems by numerical methods" 45 : 16-31, 1958

45 A. K. Pal, "A Ratio-dependent Eco-epidemiological Model Incorporating a Prey Refuge" 1 : 86-100, 2013

46 S. Sarwardi, "A Leslie-Gower Holling-type II ecoepidemic model" 35 : 263-280, 2011