Dynamics of predator-prey community with age structures and its changing due to harvesting

G. P. Neverova ; O. L. Zhdanova ; E. Ya. Frisman

Matematičeskaâ biologiâ i bioinformatika, Tome 15 (2020) no. 1, pp. 73-92.

Résumé

The paper studies dynamic modes of discrete-time model of structured predator-prey community like “arctic fox – rodent” and changing its dynamic modes due to interspecific interaction. We paid special attention to the analysis of situations in which changes in the dynamic modes are possible. In particularly, 3-cycle emerging in prey population can result in predator extinction. Moreover, this solution corresponding to an incomplete community simultaneously coexists with the solution describing dynamics of complete community, which can be both stable and unstable. The anthropogenic impact on the community dynamics is studied, that is realized as harvest of some part of predator or prey population. It is shown that prey harvesting leads to expansion of parameter space domain with non-trivial stable numbers of community populations. In this case, the prey harvest has little effect on the predator dynamics; changes are mainly associated with multistability areas. In particular, the multistability domain narrows, in which changing initial conditions leads to different dynamic regimes, such as the transition to a stable state or periodic oscillations. As a result, community dynamics becomes more predictable. It is shown that the dynamics of prey population is sensitive to its harvesting. Even a small harvest rate results in disappearance of population size fluctuations: the stable state captures the entire phase space in multistability areas. In the case of the predator population harvest, stability domain of the nontrivial fixed point expands along the parameter of the predator birth rate. Accordingly, a case where predator determines the prey population dynamics is possible only at high values of predator reproductive potential. It is shown that in the case of predator harvest, a change in the community dynamic mode is possible because of a shifting dynamic regime in the prey population initiating the same nature fluctuations in the predator population. The dynamic regimes emerging in the community models with and without harvesting were compared.

Bibliographie

[1] A. J. Lotka, Analytical theory of biological populations, Springer Science Business Media, 1998, 220 pp. | MR

[2] V. Volterra, Matematicheskaya teoriya borby za suschestvovanie, Nauka, M., 2004, 288 pp.

[3] Yu. V. Tyutyunov, L. I. Titova, “Ot Lotki-Volterra k Arditi-Ginzburgu: 90 let evolyutsii troficheskikh funktsii”, Zhurnal obschei biologii, 79:6 (2018), 428–448

[4] C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation”, Mem. Ent. Soc. Can., 45 (1965), 1–60

[5] H. Caswell, Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer Associates. Inc., Sunderland, MA, 2001

[6] A. D. Bazykin, Matematicheskaya biofizika vzaimodeistvuyuschikh populyatsii, Nauka, M., 1985, 181 pp.

[7] J. L. Sabo, “Stochasticity, predator-prey dynamics, and trigger harvest of nonnative predators”, Ecology, 86:9 (2005), 2329–2343 | DOI

[8] E. V. Pakht, A. I. Abakumov, “Neopredelennost pri modelirovanii ekosistemy ozera”, Matematicheskaya biologiya i bioinformatika, 6:1 (2011), 102–114 | DOI

[9] I. A. Bashkirtseva, P. V. Boyarshinova, T. V. Ryazanova, L. B. Ryashko, “Analiz indutsirovannogo shumom razrusheniya rezhimov sosuschestvovaniya v populyatsionnoi sisteme «khischnik–zhertva»”, Kompyuternye issledovaniya i modelirovanie, 8:4 (2016), 647–660

[10] E. P. Abramova, T. V. Ryazanova, “Dinamicheskie rezhimy stokhasticheskoi modeli «khischnik–zhertva» s uchetom konkurentsii i nasyscheniya”, Kompyuternye issledovaniya i modelirovanie, 11:3 (2019), 515–531

[11] Yu. M. Aponin, E. A. Aponina, “Matematicheskaya model soobschestva khischnik-zhertva s nizhnim porogom chislennosti zhertvy”, Kompyuternye issledovaniya i modelirovanie, 1:1 (2009), 51–56

[12] C. Xu, Y. Wu, L. Lu, “Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays”, Advances in Difference Equations, 1 (2014), 1–5 | MR

[13] E. Ya. Frisman, M. P. Kulakov, O. L. Revutskaya, O. L. Zhdanova, G. P. Neverova, “Osnovnye napravleniya i obzor sovremennogo sostoyaniya issledovanii dinamiki strukturirovannykh i vzaimodeistvuyuschikh populyatsii”, Kompyuternye issledovaniya i modelirovanie, 11:1 (2019), 119–151 | DOI | MR

[14] Y. Saito, Y. Takeuchi, “A time-delay model for prey-predator growth with stage structure”, Canadian Applied Mathematics Quarterly, 11:3 (2003), 293–302 | MR | Zbl

[15] S. A. Gourley, Y. Kuang, “A stage structured predator-prey model and its dependence on maturation delay and death rate”, Journal of mathematical Biology, 49:2 (2004), 188–200 | DOI | MR | Zbl

[16] X. K. Sun, H. F. Huo, H. Xiang, “Bifurcation and stability analysis in predator-prey model with a stage-structure for predator”, Nonlinear Dynamics, 58:3 (2009), 497–513 | DOI | MR | Zbl

[17] R. Xu, “Global dynamics of a predator-prey model with time delay and stage structure for the prey”, Nonlinear Analysis: Real World Applications, 12:4 (2011), 2151–2162 | DOI | MR | Zbl

[18] K. Chakraborty, S. Jana, T. K. Kar, “Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting”, Applied Mathematics and Computation, 218:18 (2012), 9271–9290 | DOI | MR | Zbl

[19] S. Kundu, S. Maitra, “Dynamics of a delayed predator-prey system with stage structure and cooperation for preys”, Chaos, Solitons Fractals, 114 (2018), 453–460 | DOI | MR | Zbl

[20] P. A. Abrams, C. Quince, “The impact of mortality on predator population size and stability in systems with stage-structured prey”, Theoretical Population Biology, 68:4 (2005), 253–266 | DOI | Zbl

[21] S. Khajanchi, S. Banerjee, “Role of constant prey refuge on stage structure predator-prey model with ratio dependent functional response”, Applied Mathematics and Computation, 314 (2017), 193–198 | DOI | MR | Zbl

[22] J. Bhattacharyya, S. Pal, “Stage-structured cannibalism in a ratio-dependent system with constant prey refuge and harvesting of matured predator”, Differential Equations and Dynamical Systems, 24:3 (2016), 345–366 | DOI | MR | Zbl

[23] A. I. Abakumov, O. I. Ilin, N. S. Ivanko, “Igrovye zadachi sbora urozhaya v biologicheskom soobschestve”, Matematicheskaya teoriya igr i ee prilozheniya, 3:2 (2011), 3–17 | MR | Zbl

[24] A. I. Abakumov, Yu. G. Izrailskii, “Effekty promyslovogo vozdeistviya na rybnuyu populyatsiyu”, Matematicheskaya biologiya i bioinformatika, 11:2 (2016), 191–204 | DOI

[25] P. Walters, V. Christensen, B. Fulton, A. D. Smith, R. Hilborn, “Predictions from simple predator-prey theory about impacts of harvesting forage fishes”, Ecological modelling, 337 (2016), 272–280 | DOI

[26] A. I. Abakumov, O. I. Il'in, N. S. Ivanko, “Game problems of harvesting in a biological community”, Automation and Remote Control, 77:4 (2016), 697–707 | DOI | MR

[27] C. Liu, Q. Zhang, X. Duan, “Dynamical behavior in a harvested differential-algebraic prey-predator model with discrete time delay and stage structure”, Journal of the Franklin Institute, 346:10 (2009), 1038–1059 | DOI | MR | Zbl

[28] C. Liu, Q. Zhang, X. Zhang, X. Duan, “Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting”, Journal of Computational and Applied Mathematics, 231:2 (2009), 612–625 | DOI | MR | Zbl

[29] G. Kolli, Analiz populyatsii pozvonochnykh, Mir, M., 1979, 362 pp.

[30] G. P. Neverova, A. I. Abakumov, E. Ya. Frisman, “Rezhimy dinamiki limitirovannoi strukturirovannoi populyatsii pri izbiratelnom promysle”, Matematicheskaya biologiya i bioinformatika, 12:2 (2017), 327–342 | DOI

[31] G. P. Neverova, Abakumov A.I, I. P. Yarovenko, E. Ya. Frisman, “Mode change in the dynamics of exploited limited population with age structure”, Nonlinear Dynamics, 94 (2018), 827–844 | DOI

[32] O. L. Revutskaya, G. P. Neverova, E. Ya. Frisman, “Vliyanie promyslovogo iz'yatiya na dinamiku populyatsii s vozrastnoi i polovoi strukturoi”, Matematicheskaya biologiya i bioinformatika, 13:1 (2018), 270–289 | DOI

[33] H. N. Agiza, E. M. Elabbasy, H. El-Metwally, A. A. Elsadany, “Chaotic dynamics of a discrete prey-predator model with Holling type II”, Nonlinear Analysis: Real World Applications, 10:1 (2009), 116–129 | DOI | MR | Zbl

[34] Z. Hu, Z. Teng, L. Zhang, “Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response”, Nonlinear Analysis: Real World Applications, 12:4 (2011), 2356–2377 | DOI | MR | Zbl

[35] J. Zhao, Y. Yan, “Stability and bifurcation analysis of a discrete predator-prey system with modified Holling-Tanner functional response”, Advances in Difference Equations, 2018, 402 | DOI | MR

[36] D. P. Mistro, L. A.D. Rodrigues, S. Petrovskii, “Spatiotemporal complexity of biological invasion in a space- and time-discrete predator-prey system with the strong Allee effect”, Ecological Complexity, 9 (2012), 16–32 | DOI | MR

[37] T. Huang, H. Zhang, “Bifurcation, chaos and pattern formation in a space-and time-discrete predator-prey system”, Chaos, Solitons Fractals, 91 (2016), 92–107 | DOI | MR | Zbl

[38] T. Huang, H. Zhang, H. Yang, N. Wang, F. Zhang, “Complex patterns in a space-and time-discrete predator-prey model with Beddington-DeAngelis functional response”, Communications in Nonlinear Science and Numerical Simulation, 43 (2017), 182–199 | DOI | MR | Zbl

[39] J. Zhong, Z. Yu, “Qualitative properties and bifurcations of Mistro-Rodrigues-Petrovskii model”, Nonlinear Dynamics, 91:4 (2018), 2063–2075 | DOI | Zbl

[40] J. R. Reimer, H. Brown, E. Beltaos-Kerr, G. de Vries, “Evidence of intraspecific prey switching: stage-structured predation of polar bears on ringed seals”, Oecologia, 189:1 (2018), 133–148 | DOI

[41] A. Wikan, Ø. Kristensen, “Prey-Predator Interactions in Two and Three Species Population Models”, Discrete Dynamics in Nature and Society, 2019 (2019), 1–14 | DOI | MR

[42] C. C. Wilmers, E. Post, A. Hastings, “The anatomy of predator-prey dynamics in a changing climate”, Journal of Animal Ecology, 76:6 (2007), 1037–1044 | DOI

[43] R. Kon, “Multiple attractors in host-parasitoid interactions: Coexistence and extinction”, Mathematical Biosciences, 201:1-2 (2006), 172–183 | DOI | MR | Zbl

[44] O. L. Revutskaya, M. P. Kulakov, E. Ya. Frisman, “Bistabilnost i bifurkatsii v modifitsirovannoi modeli Nikolsona–Beili pri uchete vozrastnoi struktury zhertvy”, Matematicheskaya biologiya i bioinformatika, 14:1 (2019), 257–278 | DOI

[45] Y. Kang, D. Armbruster, Y. Kuang, “Dynamics of a plant-herbivore model”, Journal of Biological Dynamics, 2:2 (2008), 89–101 | DOI | MR | Zbl

[46] Y. Kang, D. Armbruster, “Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions”, International Journal of Biomathematics, 4:3 (2011), 255–274 | DOI | MR | Zbl

[47] A. Wikan, “An analysis of discrete stage-structured prey and prey-predator population models”, Discrete Dynamics in Nature and Society, 2017 (2017) | DOI | MR

[48] M. Basson, M. J. Fogarty, “Harvesting in discrete-time predator-prey systems”, Mathematical biosciences, 141:1 (1997), 41–74 | DOI | Zbl

[49] B. Chen, J. Chen, “Complex dynamic behaviors of a discrete predator-prey model with stage structure and harvesting”, International Journal of Biomathematics, 10:1 (2017), 1750013 | DOI | MR | Zbl

[50] O. L. Zhdanova, G. P. Neverova, E. Ya. Frisman, “Modelirovanie dinamiki soobschestva khischnik zhertva s uchetom vozrastnoi struktury vzaimodeistvuyuschikh vidov”, Informatika i sistemy upravleniya, 2018, no. 4 (58), 34–45

[51] G. P. Neverova, O. L. Zhdanova, E. Ya. Frisman, “Modelirovanie dinamiki soobschestva «khischnik–zhertva» pri nalichii vozrastnykh struktur”, Matematicheskaya biologiya i bioinformatika, 14:1 (2019), 77–93 | DOI

[52] G. P. Neverova, O. L. Zhdanova, Bapan Ghosh, E. Ya. Frisman, “Dynamics of a discrete-time stage-structured predator-prey system with Holling type II response function”, Nonlinear dynamics, 98:1 (2019), 427–446 | DOI | Zbl

[53] A. Angerbjorn, M. Tannerfeldt, S. Erlinge, “Predator-prey relationships: arctic foxes and lemmings”, Journal of Animal Ecology, 68:1 (1999), 34–49 | DOI

[54] A. P. Kuznetsov, A. V. Savin, Yu. V. Sedova, L. V. Tyuryukina, Bifurkatsii otobrazhenii, OOO Izdatelskii tsentr «Nauka», Saratov, 2012, 196 pp.

[55] P. Hersteinsson, D. W. Macdonald, “Diet of Arctic foxes (Alopex lagopus) in Iceland”, J. Zool., 240 (1996), 457–474 | DOI