Temporal Response of Predator-Prey System with the Allee Effect

Year 2020, Volume: 7 Issue: 1, 54 - 65, 28.06.2020

Yadigar Sekerci

https://doi.org/10.35193/bseufbd.648992

Abstract

Temporal dynamics of a predator-prey model, in which predator mortality rate varies linearly in time, is considered with the assumption that prey growth is stressed by the strong Allee effect. In this work, temporal structure of prey-predator system consisting of two coupled ordinary differential equations is examined. By means of extensive numerical simulations it is shown that the system has rich temporal structure and the system can be stabilizes with the change in predator mortality rate. Obtained results show that with an increasing stress on prey due to Allee affect and the increase in predator population size, the system push the prey to extinct and then the system is extinct due to absence of prey for predator.

Keywords

Predator-Prey system, Allee Effect, Mathematical Modelling

Supporting Institution

Amasya Universitesi

Project Number

FBM-BAP 17-0258

Thanks

This research has been supported by Amasya University Scientific Research Projects Coordination Unit. Project Number: FBM-BAP 17-0258.

Allee Etkisi Altındaki Av-Avcı Sisteminin Zamana Bağlı Değişimi

Abstract

Allee etkisi altındaki avın büyüme katkısının baskılandığı ve avcı ölüm oranının zaman içinde doğrusal olarak değiştiği av-avcı modelinin zamana bağlı dinamikleri ele alınmıştır. Bu çalışmada, av-avcı sisteminin zamana bağlı dinamik yapısı ikili adi diferansiyel denklem ile incelenmiştir. Çok sayıdaki nümerik simülasyonlar sayesinde sistemin çeşitli zamansal yapıya sahip olduğu ve sistemin avcının ölüm katsayısındaki değişim ile dengelenebileceği gösterilmiştir. Elde edilen sonuçlar göstermektedir ki Allee etkisi altında ve avcı popülasyonunun artmasıyla birlikte av üzerindeki baskının artması ile system, avı neslinin tükenmesine zorlamıştır ve dolayısıyla avın olmaması avcının neslini de tüketmiştir.

Keywords

Av-Avcı Sistemi, Allee Etkisi, Matematiksel Modelleme

Project Number

FBM-BAP 17-0258

References

• Allee, W. C. (1931). Animal Aggregations, a Study in General Sociology. University of Chicago Press, Chicago, IL
• Allee, W. C., & Bowen, E. S. (1932). Studies in animal aggregations: mass protection against colloidal silver among goldfishes. Journal of Experimental Zoology, 61(2), 185-207.
• Stephens, P. A., Sutherland, W. J., & Freckleton, R. P. (1999). What is the Allee effect? Oikos, 185-190.
• Tsoularis, A., & Wallace, J. (2002). Analysis of logistic growth models. Mathematical Biosciences, 179(1), 21-55.
• Blumberg, A. A. (1968). Logistic growth rate functions. Journal of Theoretical Biology, 21(1), 42-44.
• Amarasekare, P. (1998). Allee effects in metapopulation dynamics. The American Naturalist, 152(2), 298-302.
• Sekerci, Y. (2020). Climate change effects on fractional order prey-predator model. Chaos, Solitons & Fractals, 134, 109690.
• Lewis, M. A., & Kareiva, P. (1993). Allee dynamics and the spread of invading organisms. Theoretical Population Biology, 43(2), 141-158.
• Courchamp, F., Clutton-Brock, T., & Grenfell, B. (1999). Inverse density dependence and the Allee effect. Trends in Ecology & Evolution, 14(10), 405-410.
• Odum, E. P. (1953). Fundamentals of ecology. xii, 387 pp. W. B. Saunders Co., Philadelphia, Pennsylvania, and London, England.
• Courchamp, F., Berec, L., & Gascoigne, J. (2008). Allee effects in ecology and conservation. Oxford University Press.
• Allee, W.C., Emerson, O., Park, T., & Schmidt, K. (1949). Principles of Animal Ecology. Saunders, Philadelphia.
• Allee, W.C. (1951). Cooperation Among Animals. Henry Shuman, New York.
• Allee, W.C. (1958). The Social Life of Animals. Beacon Press, Boston.
• Ye, Y., Liu, H., Wei, Y. M., Ma, M., & Zhang, K. (2019). Dynamic study of a predator-prey model with weak Allee effect and delay. Advances in Mathematical Physics, 27(4), 943-953.
• Wang, J., Shi, J., & Wei, J. (2011). Predator–prey system with strong Allee effect in prey. Journal of Mathematical Biology, 62(3), 291-331.
• Banerjee, M., Mukherjee, N., & Volpert, V. (2018). Prey-predator model with a nonlocal bistable dynamics of prey. Mathematics, 6(3), 41.
• Han, R., & Dai, B. (2019). Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton–zooplankton model with Allee effect. Nonlinear Analysis: Real World Applications, 45, 822-853.
• Morozov, A., S. Petrovskii, & Li, B.L. (2006). Spatiotemporal complexity of patchy invasion in a predator–prey system with the Allee effect, Journal of Theoretical Biology, 238(1), 18–35.
• Petrovskii, S., Morozov, A. & Venturino, E. (2002) Allee effect makes possible patchyinvasion in a predator–prey system, Ecological. Letters, 5(3), 345–352.
• Sun, G. Q. (2016). Mathematical modeling of population dynamics with Allee effect. Nonlinear Dynamics, 85(1), 1-12.
• Yao, S. W., Ma, Z. P., & Cheng, Z. B. (2019). Pattern formation of a diffusive predator–prey model with strong Allee effect and nonconstant death rate. Physica A: Statistical Mechanics and its Applications, 527, 121350.
• Petrovskii, S., Morozov, A., & Li, B.L. (2005). Regimes of biological invasion in a predator-prey system with the Allee effect. Bulletin of Mathematical Biology. 67(3), 637-661.
• Murray, J.D. (1989). Mathematical Biology. Springer, Berlin.
• Nisbet, R.M., & Gurney, W.S.C. (1982). Modelling Fluctuating Populations. Wiley, Chichester.
• Sherratt, J.A. (2001). Periodic travelling waves in cyclic predator–prey systems. Ecological Letters, 4, 30–37.