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Positive solutions of a diffusive two competitive species model with saturation
Effects of fear and anti-predator response in a discrete system with delay
1. | Indian Institute of Engineering Science and Technology, Shibpur, Howrah -711103, India |
2. | Vivekananda College, Thakurpukur, Kolkata - 700063, India |
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.
References:
[1] |
S. Aoki, U. Kurosu and S. Usuba,
First instar larvae of the sugar-cane wooly aphid, ceratovacuna lanigera (homotera, pemphigidae), attack its predators, Kontyu, 52 (1984), 458-460.
|
[2] |
R. Banerjee, P. Das and D. Mukherjee,
Stability and permanence of a discrete-time two-prey one-predator system with Holling Type-Ⅲ functional response, Chaos, Solitons & Fractals, 117 (2018), 240-248.
doi: 10.1016/j.chaos.2018.10.032. |
[3] |
R. Banerjee, P. Das and D. Mukherjee,
Global dynamics of a Holling Type-Ⅲ two prey–one predator discrete model with optimal harvest strategy, Nonlinear Dynamics, 99 (2020), 3285-3300.
doi: 10.1007/s11071-020-05490-0. |
[4] |
M. C. and A. Barkai, Predator-prey role reversal in a marine benthic ecosystem, Science, (1988), 62–64. |
[5] |
M. Clinchy, M. J. Sheriff and L. Y. Zanette,
Predator-induced stress and the ecology of fear, Functional Ecology, 27 (2013), 56-65.
doi: 10.1111/1365-2435.12007. |
[6] |
R. Kaushik and S. Banerjee, Predator-prey system: Prey's counter-attack on juvenile predators shows opposite side of the same ecological coin, Applied Mathematics and Computation, 388 (2021), 125530.
doi: 10.1016/j.amc.2020.125530. |
[7] |
R. H. MacArthur and E. R. Pianka,
On optimal use of a patchy environment, The American Naturalist, 100 (1966), 603-609.
doi: 10.1086/282454. |
[8] |
R. J. Mrowicki and N. E. O'Connor,
Wave action modifies the effects of consumer diversity and warming on algal assemblages, Ecology, 96 (2015), 1020-1029.
doi: 10.1890/14-0577.1. |
[9] |
P. Panday, N. Pal, S. Samanta and J. Chattopadhyay, A three species food chain model with fear induced trophic cascade, Int. J. Appl. Comput. Math., 5 (2019), 26 pp.
doi: 10.1007/s40819-019-0688-x. |
[10] |
P. Panja, S. Jana and S. k. Mondal,
Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey, Numerical Algebra, Control & Optimization, 11 (2021), 391-405.
doi: 10.3934/naco.2020033. |
[11] |
W. Ripple, L. Painter, R. Beschta and C. Gates, Wolves, elk, bison, and secondary trophic cascades in Yellowstone National Park, The Open Ecology Journal, 3 (2010). |
[12] |
W. J. Ripple and R. L. Beschta,
Wolves and the ecology of fear: Can predation risk structure ecosystems?, BioScience, 54 (2004), 755-766.
|
[13] |
W. J. Ripple and R. L. Beschta,
Trophic cascades in Yellowstone: The first 15 years after wolf reintroduction, Biological Conservation, 145 (2012), 205-213.
|
[14] |
Y. Saito,
Prey kills predator: Counter-attack success of a spider mite against its specific phytoseiid predator, Experimental & Applied Acarology, 2 (1986), 47-62.
doi: 10.1007/BF01193354. |
[15] |
O. J. Schmitz, A. P. Beckerman and K. M. O'Brien,
Behaviorally mediated trophic cascades: Effects of predation risk on food web interactions, Ecology, 78 (1997), 1388-1399.
|
[16] |
Y. N. P. Service, \em 2019 Late Winter Survey of Northern Yellowstone Elk, 2019. Available from: https://www.nps.gov/yell/learn/news/2019-late-winter-survey-of-northern-yellowstone-elk.htm. |
[17] |
Y. N. P. Service, \em Questions & Answers About Bison Management, 2021. Available from: https://www.nps.gov/yell/learn/news/2019-late-winter-survey-of-northern-yellowstone-elk.htm. |
[18] |
D. W. Smith, L. D. Mech, M. Meagher, W. E. Clark, R. Jaffe, M. K. Phillips and J. A. Mack,
Wolf–bison interactions in Yellowstone National Park, Journal of Mammalogy, 81 (2000), 1128-1135.
|
[19] |
J. P. Suraci, M. Clinchy, L. M. Dill, D. Roberts and L. Y. Zanette, Fear of large carnivores causes a trophic cascade, Nat Commun., 7 (2016), 10698. |
[20] |
V. Tiwari, J. P. Tripathi, S. Mishra and R. K. Upadhyay, Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems, Applied Mathematics and Computation, 371 (2020), 124948.
doi: 10.1016/j.amc.2019.124948. |
[21] |
H. Zhang, Y. Cai, S. Fu and W. Wang,
Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied Mathematics and Computation, 356 (2019), 328-337.
doi: 10.1016/j.amc.2019.03.034. |
show all references
References:
[1] |
S. Aoki, U. Kurosu and S. Usuba,
First instar larvae of the sugar-cane wooly aphid, ceratovacuna lanigera (homotera, pemphigidae), attack its predators, Kontyu, 52 (1984), 458-460.
|
[2] |
R. Banerjee, P. Das and D. Mukherjee,
Stability and permanence of a discrete-time two-prey one-predator system with Holling Type-Ⅲ functional response, Chaos, Solitons & Fractals, 117 (2018), 240-248.
doi: 10.1016/j.chaos.2018.10.032. |
[3] |
R. Banerjee, P. Das and D. Mukherjee,
Global dynamics of a Holling Type-Ⅲ two prey–one predator discrete model with optimal harvest strategy, Nonlinear Dynamics, 99 (2020), 3285-3300.
doi: 10.1007/s11071-020-05490-0. |
[4] |
M. C. and A. Barkai, Predator-prey role reversal in a marine benthic ecosystem, Science, (1988), 62–64. |
[5] |
M. Clinchy, M. J. Sheriff and L. Y. Zanette,
Predator-induced stress and the ecology of fear, Functional Ecology, 27 (2013), 56-65.
doi: 10.1111/1365-2435.12007. |
[6] |
R. Kaushik and S. Banerjee, Predator-prey system: Prey's counter-attack on juvenile predators shows opposite side of the same ecological coin, Applied Mathematics and Computation, 388 (2021), 125530.
doi: 10.1016/j.amc.2020.125530. |
[7] |
R. H. MacArthur and E. R. Pianka,
On optimal use of a patchy environment, The American Naturalist, 100 (1966), 603-609.
doi: 10.1086/282454. |
[8] |
R. J. Mrowicki and N. E. O'Connor,
Wave action modifies the effects of consumer diversity and warming on algal assemblages, Ecology, 96 (2015), 1020-1029.
doi: 10.1890/14-0577.1. |
[9] |
P. Panday, N. Pal, S. Samanta and J. Chattopadhyay, A three species food chain model with fear induced trophic cascade, Int. J. Appl. Comput. Math., 5 (2019), 26 pp.
doi: 10.1007/s40819-019-0688-x. |
[10] |
P. Panja, S. Jana and S. k. Mondal,
Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey, Numerical Algebra, Control & Optimization, 11 (2021), 391-405.
doi: 10.3934/naco.2020033. |
[11] |
W. Ripple, L. Painter, R. Beschta and C. Gates, Wolves, elk, bison, and secondary trophic cascades in Yellowstone National Park, The Open Ecology Journal, 3 (2010). |
[12] |
W. J. Ripple and R. L. Beschta,
Wolves and the ecology of fear: Can predation risk structure ecosystems?, BioScience, 54 (2004), 755-766.
|
[13] |
W. J. Ripple and R. L. Beschta,
Trophic cascades in Yellowstone: The first 15 years after wolf reintroduction, Biological Conservation, 145 (2012), 205-213.
|
[14] |
Y. Saito,
Prey kills predator: Counter-attack success of a spider mite against its specific phytoseiid predator, Experimental & Applied Acarology, 2 (1986), 47-62.
doi: 10.1007/BF01193354. |
[15] |
O. J. Schmitz, A. P. Beckerman and K. M. O'Brien,
Behaviorally mediated trophic cascades: Effects of predation risk on food web interactions, Ecology, 78 (1997), 1388-1399.
|
[16] |
Y. N. P. Service, \em 2019 Late Winter Survey of Northern Yellowstone Elk, 2019. Available from: https://www.nps.gov/yell/learn/news/2019-late-winter-survey-of-northern-yellowstone-elk.htm. |
[17] |
Y. N. P. Service, \em Questions & Answers About Bison Management, 2021. Available from: https://www.nps.gov/yell/learn/news/2019-late-winter-survey-of-northern-yellowstone-elk.htm. |
[18] |
D. W. Smith, L. D. Mech, M. Meagher, W. E. Clark, R. Jaffe, M. K. Phillips and J. A. Mack,
Wolf–bison interactions in Yellowstone National Park, Journal of Mammalogy, 81 (2000), 1128-1135.
|
[19] |
J. P. Suraci, M. Clinchy, L. M. Dill, D. Roberts and L. Y. Zanette, Fear of large carnivores causes a trophic cascade, Nat Commun., 7 (2016), 10698. |
[20] |
V. Tiwari, J. P. Tripathi, S. Mishra and R. K. Upadhyay, Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems, Applied Mathematics and Computation, 371 (2020), 124948.
doi: 10.1016/j.amc.2019.124948. |
[21] |
H. Zhang, Y. Cai, S. Fu and W. Wang,
Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied Mathematics and Computation, 356 (2019), 328-337.
doi: 10.1016/j.amc.2019.03.034. |








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