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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2616-2635. doi: 10.3934/mbe.2020143
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A predator-prey model with genetic differentiation both in the predator and prey

  • School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, China
  • Received: 27 November 2019 Accepted: 26 February 2020 Published: 04 March 2020

  • In this paper, we propose a predator-prey model with genetic differentiation both in the predator and prey. First, we analyze two special cases: a model without the predators and a model with one genotype in both the predator and prey, and for each model show that the positive equilibria are always globally stable when they exist, while the boundary equilibria are always unstable. Then, for the newly proposed model, we give the results that the positive equilibrium is always local stable when it exists, the boundary equilibrium at the origin is always unstable, and the stability of another boundary equilibrium is determined by the existence of the positive equilibrium. Moreover, our discussions show the existence of local center manifolds near the equilibria. Finally, we give some examples to illustrate our results.

    Citation: Lisha Wang, Jiandong Zhao. A predator-prey model with genetic differentiation both in the predator and prey[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2616-2635. doi: 10.3934/mbe.2020143

    Related Papers:

  • In this paper, we propose a predator-prey model with genetic differentiation both in the predator and prey. First, we analyze two special cases: a model without the predators and a model with one genotype in both the predator and prey, and for each model show that the positive equilibria are always globally stable when they exist, while the boundary equilibria are always unstable. Then, for the newly proposed model, we give the results that the positive equilibrium is always local stable when it exists, the boundary equilibrium at the origin is always unstable, and the stability of another boundary equilibrium is determined by the existence of the positive equilibrium. Moreover, our discussions show the existence of local center manifolds near the equilibria. Finally, we give some examples to illustrate our results.


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    [1] A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.
    [2] V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Menor. Accad. Lincei., 6 (1926), 31-113.
    [3] C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.
    [4] M. L. Rosenzweig, R. H. MacArthur, Graphical representation and stability conditions of predatorprey interactions, Am. Nat., 97 (1963), 209-223.
    [5] 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.
    [6] J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology 56 (1975), 855-867.
    [7] R. M. Anderson, R. M. May, Regulation and stability of host-parasite population interactions I. Regulation processes, J. Anim. Ecol., 47 (1978), 219-247.
    [8] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 2001.
    [9] M. Mobilia, I. T. Georgiev, U. C. Täuber, Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka-Volterra models, J. Stat. Phys., 128 (2007), 447-483.
    [10] U. Dobramysl, M. Mobilia, M. Pleimling, U. C. Täuber, Stochastic population dynamics in spatially extended predator-prey systems, J. Phys. A Math. Theor., 51 (2018), 063001.
    [11] J. A. Langa, A. Rodríguez-Bernal, A. Suárez, On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method, J. Differ. Equations, 249 (2010), 414-445.
    [12] F. M. De Oca, M. Vivas, Extinction in a two dimensional Lotka-Volterra system with infinite delay, Nonlinear Anal. Real World Appl., 7 (2006), 1042-1047.
    [13] R. Bhattacharyya, B. Mukhopadhyay, On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives, Nonlinear Anal. Real World Appl., 11 (2010), 3824-3833.
    [14] B. W. Kooi, E. Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosci., 274 (2016), 58-72.
    [15] Y. Xiao, L. Chen, Analysis of a three species eco-epidemiological model, J. Math. Anal. Appl., 258 (2001), 733-754.
    [16] J. Zhao, J. Jiang, Permanence in nonautonomous Lotka-Volterra system with predator-prey, Appl. Math. Comput., 152 (2004), 99-109.
    [17] J. Chattopadhyay, S. Chatterjee, E. Venturino, Patchy agglomeration as a transition from monospecies to recurrent plankton blooms, J. Theor. Biol., 253 (2008), 289-295.
    [18] V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal. Real World Appl., 12 (2011), 2319-2338.
    [19] P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. Real World Appl., 13 (2012), 1837-1843.
    [20] T. Zhang, Y. Xing, H. Zang, M. Han, Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality, Nonlinear Dynam., 78 (2014), 265-277.
    [21] C. Xu, S. Yuan, T. Zhang, Global dynamics of a predator-prey model with defence mechanism for prey, Appl. Math. Lett., 62 (2016), 42-48.
    [22] E. Venturino, An ecogenetic model, Appl. Math. Lett., 25 (2012), 1230-1233.
    [23] M. D. Rahman, Modelling of an eco-genetic system: A mathematical model, Int. J. Ecol. Econ. Stat., 37 (2016), 102-119.
    [24] C. Viberti, E. Venturino, A predator-prey model with genetically distinguishable predators, in Recent Advances in Environmental Sciences, Proceedings of the 9th International Conference on Energy, Environment, Ecosystems and Sustainable Development (EEESD13), Lemesos, Cyprus, March 21st-23rd 2013 (eds. A. Kanarachos, N. E. Mastorakis), WSEAS Press, (2013), 87-92.
    [25] C. Viberti, E. Venturino, An ecosystem with HTII response and predators' genetic variability, Math. Model. Anal., 19 (2014), 371-394.
    [26] L. Castellino, S. Peretti, S. Rivoira, E. Venturino, A mathematical ecogenetic predator-prey model where both populations are genetically distinguishable, AIP Conference Proceedings, 1776 (2016), 020006.
    [27] I. Azzali, G. Marcaccio, R. Turrisi, E. Venturino, A genetically distinguishable competition model, in Modern Mathematical Methods and High Performance Computing in Science and Technology, Springer Proceedings in Mathematics and Statistics (eds. V. Singh, H. Srivastava, E. Venturino, M. Resch, V. Gupta), Springer, Singapore, 171 (2016), 129-140.
    [28] R. M. Anderson, R. May, The invasion persistence and spread of infectious diseases within animal and plant communities, Philos. T. R. Soc. B, 314 (1986), 533-570.
    [29] K. P. Hadeler, H. I. Freedman, Predator-prey populations with parasitic infection, J. Math. Biol., 27 (1989), 609-631.
    [30] E. Venturino, The influence of diseases on Lotka-Volterra systems, Rocky Mt. J. Math., 24 (1994), 381-402.
    [31] J. Chattopadhyay, O. Arino, A predator-prey model with disease in the prey, Nonlinear Anal., 36 (1999), 747-766.
    [32] Y. Xiao, L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci., 171 (2001), 59-82.
    [33] L. Han, Z. Ma, H. W. Hethcote, Four predator prey models with infectious diseases, Math. Comput. Model., 34 (2001), 849-858.
    [34] E. Venturino, Epidemics in predator-prey models: disease in the predators, IMA J. Math. Appl. Med. Biol., 19 (2002), 185-205.
    [35] P. van den Driessche, M. L. Zeeman, Disease induced oscillations between two computing species, SIAM J. Appl. Dyn. Syst., 3 (2004), 601-619.
    [36] H. W. Hethcote, W. Wang, L. Han, Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268.
    [37] M. Haque, J. Zhen, E. Venturino, An ecoepidemiological predator-prey model with standard disease incidence, Math. Method. Appl. Sci., 32 (2009), 875-898.
    [38] X. Zhou, J. Cui, X. Shi, X. Song, A modified Leslie-Grower predator-prey model with prey infection, J. Appl. Math. Comput., 33 (2010), 471-487.
    [39] S. Belvisi, E. Venturino, An ecoepidemic model with diseased predators and prey group defense, Simul. Model. Pract. Th., 34 (2013), 144-155.
    [40] S. Sharma, G. P. Samanta, A ratio-dependent predator-prey model with Allee effect and disease in prey, J. Appl. Math. Comput., 47 (2015), 345-364.
    [41] C. Chicone, Ordinary Differential Equations with Applications, 2nd edition, Texts in Applied Mathematics 34, Springer, New York, 2006.
    [42] J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, Cambridge University Press, Cambridge, 1988.
    [43] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981.
    [44] T. C. Sideris, Ordinary Differential Equations and Dynamical Systems, Atlantis Press, Paris France, 2013.
    [45] J. Zhao, X. Guo, Z. Han, Z. Chen, Average conditions for competitive exclusion in a nonautonomous two dimensional Lotka-Volterra system, Math. Comput. Model., 57 (2013), 1131-1138.
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