# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022127
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## Studying the fear effect in a predator-prey system with apparent competition

 Department of Mathematics, Trent University, Peterborough, ON K9L 0G2, Canada

* Corresponding author: Xiaoying Wang

Received  December 2021 Revised  May 2022 Early access July 2022

Fund Project: The first author is supported by the NSERC of Canada (RGPIN-2020-06825 and DGECR-2020-00369)

Recent experimental evidence shows that the mere presence of predators may largely reduce the reproduction success of prey. The loss of prey's reproduction rate is attributed to the cost of anti-predator defense of prey when the prey perceives predation risks. We propose a predator-prey model where the prey shares a common enemy that leads to apparent competition between the prey and also the cost of anti-predator defense. Analytical results give the persistence conditions for the population densities of the prey and the predator. Numerical simulations demonstrate rich dynamics, such as the bi-stability of an equilibrium and a limit cycle. Results also reveal how the prey and the predator may coexist when the anti-predator defense level varies in prey. A relatively strong anti-predator defense in the prey may drive the population density of the prey to extinction and change the original coexistence of all the prey and the predator where the population densities oscillate periodically. Alternatively, strong anti-predator defense in the prey may facilitate the coexistence of the prey and the predator at a steady state.

Citation: Xiaoying Wang, Alexander Smit. Studying the fear effect in a predator-prey system with apparent competition. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022127
##### References:
 [1] P. Abrams and H. Matsuda, Effects of adaptive predatory and anti-predator behaviour in a two-prey-one-predator system, Evolutionary Ecology, 7 (1993), 312-326.  doi: 10.1007/BF01237749. [2] C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics, 1993, 33–50. [3] P. Cong, M. Fan and X. Zou, Dynamics of a three-species food chain model with fear effect, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105809, 19 pp. doi: 10.1016/j.cnsns.2021.105809. [4] S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.  doi: 10.1016/j.tree.2007.12.004. [5] S. Creel, D. Christianson, S. Liley and J. A. Winnie, Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960-960.  doi: 10.1126/science.1135918. [6] W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.  doi: 10.1007/s10336-010-0638-1. [7] S. Eggers, M. Griesser, M. Nystrand and J. Ekman, Predation risk induces changes in nest-site selection and clutch size in the Siberian jay, Proceedings of the Royal Society B: Biological Sciences, 273 (2006), 701-706.  doi: 10.1098/rspb.2005.3373. [8] J. Grasman, F. van den Bosch and O. A. van Herwaarden, Mathematical conservation ecology: A one-predator-two-prey system as case study, Bulletin of Mathematical Biology, 63 (2001), 259-269.  doi: 10.1006/bulm.2000.0218. [9] C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7. [10] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv. [11] S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math., 9 (2005), 151-173.  doi: 10.11650/twjm/1500407791. [12] J. P. LaSalle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, 1976. [13] A. Li and X. Zou, Evolution and adaptation of anti-predation response of prey in a two-patchy environment, Bull. Math. Biol., 83 (2021), Paper No. 59, 27 pp. doi: 10.1007/s11538-021-00893-5. [14] S. L. Lima, Nonlethal effects in the ecology of predator-prey interactions, BioScience, 48 (1998), 25-34.  doi: 10.2307/1313225. [15] S. L. Lima, Predators and the breeding bird: Behavioural and reproductive flexibility under the risk of predation, Biological Reviews, 84 (2009), 485-513.  doi: 10.1111/j.1469-185X.2009.00085.x. [16] D. Mukherjee, Study of fear mechanism in predator-prey system in the presence of competitor for the prey, Ecological Genetics and Genomics, 15 (2020), 100052.  doi: 10.1016/j.egg.2020.100052. [17] D. Mukherjee, Effect of fear on two-predator-one prey model in deterministic and fluctuating environment, Mathematics in Applied Sciences and Engineering, 2 (2021), 1-71.  doi: 10.5206/mase/13541. [18] S. Samaddar, M. Dhar and P. Bhattacharya, Effect of fear on prey-predator dynamics: Exploring the role of prey refuge and additional food, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30 (2020), 063129, 17 pp. doi: 10.1063/5.0006968. [19] M. J. Sheriff, C. J. Krebs and R. Boonstra, The sensitive hare: Sublethal effects of predator stress on reproduction in snowshoe hares, Journal of Animal Ecology, 78 (2009), 1249-1258.  doi: 10.1111/j.1365-2656.2009.01552.x. [20] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [21] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026. [22] X. Wang, L. Y. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179-1204.  doi: 10.1007/s00285-016-0989-1. [23] X. Wang and X. Zou, Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators, Bulletin of Mathematical Biology, 79 (2017), 1325-1359.  doi: 10.1007/s11538-017-0287-0. [24] X. Wang and X. Zou, Pattern formation of a predator-prey model with the cost of anti-predator behaviors, Math. Biosci. Eng., 15 (2018), 775-805.  doi: 10.3934/mbe.2018035. [25] Y. Wang and X. Zou, On mechanisms of trophic cascade caused by anti-predation response in food chain systems, Mathematics in Applied Sciences and Engineering, 1 (2020), 91-206.  doi: 10.5206/mase/10739. [26] Y. Wang and X. Zou, On a predator-prey system with digestion delay and anti-predation strategy, J. Nonlinear Sci., 30 (2020), 1579-1605.  doi: 10.1007/s00332-020-09618-9. [27] M. J. Weterings, S. P. Ewert, J. N. Peereboom, H. J. Kuipers, D. P. J. Kuijper, H. H. Prins, P. A. Jansen, F. van Langevelde and S. E. van Wieren, Implications of shared predation for space use in two sympatric leporids, Ecology and Evolution, 9 (2019), 3457-3469.  doi: 10.1002/ece3.4980. [28] A. J. Wirsing and W. J. Ripple, A comparison of shark and wolf research reveals similar behavioural responses by prey, Frontiers in Ecology and the Environment, 9 (2011), 335-341. [29] L. Y. Zanette, A. F. White, M. C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401.  doi: 10.1126/science.1210908. [30] X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics, Springer, 2017. doi: 10.1007/978-3-319-56433-3.

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##### References:
 [1] P. Abrams and H. Matsuda, Effects of adaptive predatory and anti-predator behaviour in a two-prey-one-predator system, Evolutionary Ecology, 7 (1993), 312-326.  doi: 10.1007/BF01237749. [2] C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics, 1993, 33–50. [3] P. Cong, M. Fan and X. Zou, Dynamics of a three-species food chain model with fear effect, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105809, 19 pp. doi: 10.1016/j.cnsns.2021.105809. [4] S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends in Ecology & Evolution, 23 (2008), 194-201.  doi: 10.1016/j.tree.2007.12.004. [5] S. Creel, D. Christianson, S. Liley and J. A. Winnie, Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960-960.  doi: 10.1126/science.1135918. [6] W. Cresswell, Predation in bird populations, Journal of Ornithology, 152 (2011), 251-263.  doi: 10.1007/s10336-010-0638-1. [7] S. Eggers, M. Griesser, M. Nystrand and J. Ekman, Predation risk induces changes in nest-site selection and clutch size in the Siberian jay, Proceedings of the Royal Society B: Biological Sciences, 273 (2006), 701-706.  doi: 10.1098/rspb.2005.3373. [8] J. Grasman, F. van den Bosch and O. A. van Herwaarden, Mathematical conservation ecology: A one-predator-two-prey system as case study, Bulletin of Mathematical Biology, 63 (2001), 259-269.  doi: 10.1006/bulm.2000.0218. [9] C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7. [10] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv. [11] S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math., 9 (2005), 151-173.  doi: 10.11650/twjm/1500407791. [12] J. P. LaSalle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, 1976. [13] A. Li and X. Zou, Evolution and adaptation of anti-predation response of prey in a two-patchy environment, Bull. Math. Biol., 83 (2021), Paper No. 59, 27 pp. doi: 10.1007/s11538-021-00893-5. [14] S. L. Lima, Nonlethal effects in the ecology of predator-prey interactions, BioScience, 48 (1998), 25-34.  doi: 10.2307/1313225. [15] S. L. Lima, Predators and the breeding bird: Behavioural and reproductive flexibility under the risk of predation, Biological Reviews, 84 (2009), 485-513.  doi: 10.1111/j.1469-185X.2009.00085.x. [16] D. Mukherjee, Study of fear mechanism in predator-prey system in the presence of competitor for the prey, Ecological Genetics and Genomics, 15 (2020), 100052.  doi: 10.1016/j.egg.2020.100052. [17] D. Mukherjee, Effect of fear on two-predator-one prey model in deterministic and fluctuating environment, Mathematics in Applied Sciences and Engineering, 2 (2021), 1-71.  doi: 10.5206/mase/13541. [18] S. Samaddar, M. Dhar and P. Bhattacharya, Effect of fear on prey-predator dynamics: Exploring the role of prey refuge and additional food, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30 (2020), 063129, 17 pp. doi: 10.1063/5.0006968. [19] M. J. Sheriff, C. J. Krebs and R. Boonstra, The sensitive hare: Sublethal effects of predator stress on reproduction in snowshoe hares, Journal of Animal Ecology, 78 (2009), 1249-1258.  doi: 10.1111/j.1365-2656.2009.01552.x. [20] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [21] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.  doi: 10.1137/0524026. [22] X. Wang, L. Y. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179-1204.  doi: 10.1007/s00285-016-0989-1. [23] X. Wang and X. Zou, Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators, Bulletin of Mathematical Biology, 79 (2017), 1325-1359.  doi: 10.1007/s11538-017-0287-0. [24] X. Wang and X. Zou, Pattern formation of a predator-prey model with the cost of anti-predator behaviors, Math. Biosci. Eng., 15 (2018), 775-805.  doi: 10.3934/mbe.2018035. [25] Y. Wang and X. Zou, On mechanisms of trophic cascade caused by anti-predation response in food chain systems, Mathematics in Applied Sciences and Engineering, 1 (2020), 91-206.  doi: 10.5206/mase/10739. [26] Y. Wang and X. Zou, On a predator-prey system with digestion delay and anti-predation strategy, J. Nonlinear Sci., 30 (2020), 1579-1605.  doi: 10.1007/s00332-020-09618-9. [27] M. J. Weterings, S. P. Ewert, J. N. Peereboom, H. J. Kuipers, D. P. J. Kuijper, H. H. Prins, P. A. Jansen, F. van Langevelde and S. E. van Wieren, Implications of shared predation for space use in two sympatric leporids, Ecology and Evolution, 9 (2019), 3457-3469.  doi: 10.1002/ece3.4980. [28] A. J. Wirsing and W. J. Ripple, A comparison of shark and wolf research reveals similar behavioural responses by prey, Frontiers in Ecology and the Environment, 9 (2011), 335-341. [29] L. Y. Zanette, A. F. White, M. C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401.  doi: 10.1126/science.1210908. [30] X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics, Springer, 2017. doi: 10.1007/978-3-319-56433-3.
Steady-state solutions for (1) with the linear functional response. Parameters are: 1(a) $r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, $$k_2 = 2, d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 6.5; 1(b) r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, k_2 = 10,$$ d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 1;$ 1(c) $r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, k_2 = 2, $$d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 0.2; 1(d) r_1 = 2, k_1 = 1, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, k_2 = 2, d_2 = 0.3,$$a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 2.$ Initial condition for 1(a)–1(d) is $(u_1(0),u_2(0),v(0)) = (5,1,3).$
Existence of the positive equilibrium $E(\hat{u}_1,\hat{u}_2,\hat{v})$ of (1) with the linear functional response under varying $k_1$ and $k_2.$ Parameters are: $r_1 = 8, d_1 = 0.2, a_1 = 1, p_1 = 0.6, r_2 = 6, d_2 = 1, a_2 = 0.2, p_2 = 0.2, c_1 = 0.6, c_2 = 0.1, m = 0.6.$
Steady-state solutions for (1) with the linear functional response, when $k_1$ or $k_2$ varies. Parameters are: 3(a) $r_1 = 2, d_1 = 0.2, a_1 = 0.1, p_1 = 0.4, r_2 = 5, d_2 = 0.3,$$a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 2, k_2 = 1; 3(b) r_1 = 2, d_1 = 0.2,$$ a_1 = 0.1, p_1 = 0.4, r_2 = 5, d_2 = 0.3, a_2 = 0.2, p_2 = 0.3, c_1 = 0.6, c_2 = 0.3, m = 2, k_1 = 1.$
Bi-stability of the boundary equilibrium $E_4$ and a limit cycle in the interior of the $u_1-u_2-v$ space. Parameters are: $r_1 = 6.92, k_1 = 5.5, d_1 = 0.24, a_1 = 0.22, p_1 = 0.38, h_1 = 3.8, r_2 = 5.15, k_2 = 9.11, d_2 = 0.09,$$a_2 = 0.06, p_2 = 0.36, h_2 = 2.45, c_1 = 0.44, c_2 = 0.4, m = 0.04. Initial condition for 4(a) is IC1 = (2.6, 0.5, 1.3). Initial condition for 4(b) is IC2 = (0, 0.8, 1.6). Initial condition for 4(c) is IC3 = (5, 5, 5). Basin of attraction for the boundary equilibrium E_4 and the positive periodic solution. Parameters are the same as Figure 4 except the initial conditions IC = (u_1(0),u_2(0),v(0)). In 5(a), u_2(0) = 0.5; in 5(b), v(0) = 2. Oscillatory solutions of (1) with the Holling type Ⅱ functional response when k_1 varies. Parameters are identical to Figure 4 except k_1. For 6(a), k_1 = 7.5 and for 6(b), k_1 = 10. The initial condition is IC = (5,5,5). Bi-stability of the boundary equilibrium E_5 and a periodic solution on the boundary. Parameters are: r_1 = 6.2, d_1 = 0.63, a_1 = 0.11, p_1 = 0.35, h_1 = 1.48, r_2 = 5.85,$$d_2 = 0.25, a_2 = 0.23, p_2 = 0.66, h_2 = 1.34, c_1 = 0.98, c_2 = 0.29,$$m = 0.12, k_1 = 1, k_2 = 0.8. Initial condition of 7(a) is IC1 = (0.2,3.9,3.5). Initial condition of 7(b) is IC2 = (0.5,3.9,3.5). Periodic solutions/steady-state solutions of (1) with the Holling type Ⅱ functional response when k_2 varies. Parameters are: r_1 = 3.25, d_1 = 0.09, a_1 = 0.2, p_1 = 0.18, h_1 = 0.01,$$r_2 = 7, d_2 = 0.8, a_2 = 0.22, p_2 = 0.5, h_2 = 0.25, c_1 = 0.19, c_2 = 0.44, m = 0.3, k_1 = 1.$ For 8(a), $k_2 = 0.3$; for 8(b), $k_2 = 0.52$; for 8(c), $k_2 = 1.$ Initial condition is $IC = (5,5,5).$
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