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Modelling Effects of Rapid Evolution on Persistence and Stability in Structured Predator-Prey Systems

  • J. Z. Farkas (a1) and A. Y. Morozov (a2) (a3)


In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of prey by the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published results show that population structuring and rapid evolution in such predator-prey system can stabilize an otherwise globally unstable dynamics even with an unlimited carrying capacity of prey. However, those findings were only based on direct numerical simulation of equations and obtained for particular parameterizations of model functions, which obviously calls into question the correctness and generality of the previous results. The main objective of the current study is to treat the model analytically and consider various parameterizations of predator selectivity and inheritance kernel. We investigate the existence of a coexistence stationary state in the model and carry out stability analysis of this state. We derive expressions for the Hopf bifurcation curve which can be used for constructing bifurcation diagrams in the parameter space without the need for a direct numerical simulation of the underlying integro-differential equations. We analytically show the possibility of stabilization of a globally unstable predator-prey system with prey structuring. We prove that the coexistence stationary state is stable when the saturation in the predation term is low. Finally, for a class of kernels describing genetic inheritance and mutation we show that stability of the predator-prey interaction will require a selectivity of predation according to the life trait.


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[1] Abrams, P.A., Walters, C.J.. Invulnerable prey and the paradox of enrichment. Ecology, 77 (1996), 11251133.
[2] Ackleh, A. S., Farkas, J. Z.. On the net reproduction rate of continuous structured populations with distributed states at birth. Comput. Math. Appl., 66 (2013), 16851694.
[3] L. J. S. Allen. An introduction to mathematical biology. Pearson Prentice Hall, Upper Saddle River, NJ, 2007.
[4] Barles, G., Perthame, B.. Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics. Contemp. Math., 439 (2007), 5768.
[5] Bouin, E., Calvez, V., Meunier, N., Mirrahimi, S., Perthame, B., Raoul, G., Vouituriez, R.. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. C. R. Math. Acad. Sci. Paris, 350 (2012), 761766.
[6] À. Calsina, J. Z. Farkas. Positive steady states of evolution equations with finite dimensional nonlinearities. to appear in SIAM J. Math. Anal.
[7] Calsina, À., Farkas, J. Z.. Steady states in a structured epidemic model with Wentzell boundary condition. J. Evol. Equ., 12 (2012), 495512.
[8] Calsina, À., Palmada, J. M.. Steady states of a selection-mutation model for an age structured population. J. Math. Anal. Appl., 400 (2013), 386395.
[9] J.M. Cushing. An introduction to structured population dynamics. SIAM, Philadelphia, PA, 1998.
[10] Diekmann, O., Jabin, P.-E., Mischler, S., Perthame, B.. The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. Th. Pop. Biol., 67 (2005), 257271.
[11] Dube, D., Kim, K., Alker, A. P., Harvell, C.D.. Size structure and geographic variation in chemical resistance of sea fan corals Gorgonia ventalina to a fungal pathogen. Mar. Ecol. Prog. Ser., 231 (2002), 139150.
[12] Duffy, M. A., Sivars-Becker, L.. Rapid evolution and ecological host-parasite dynamics. Ecol. Lett., 10 (2007), 4453.
[13] Ellner, S. P., Geber, M. A., Hairston, N. G.. Does rapid evolution matter? Measuring the rate of contemporary evolution and its impacts on ecological dynamics. Ecol. Lett., 14 (2011), 603614.
[14] Farkas, J. Z., Green, D. M., Hinow, P.. Semigroup analysis of structured parasite populations. Math. Model. Nat. Phenom., 5 (2010), 94114.
[15] Farkas, J. Z., Hagen, T.. Linear stability and positivity results for a generalized size-structured Daphnia model with inflow. Appl. Anal., 86 (2007), 10871103.
[16] Fussmann, G. F., Gonzalez, A.. Evolutionary rescue can maintain an oscillating community undergoing environmental change. Interface Focus, 3 (2013), 20130036.
[17] Gentleman, W., Leising, A., Frost, B., Storm, S., Murray, J.. Functional responses for zooplankton feeding on multiple resources: a review of assumptions and biological dynamics. Deep-Sea Res. II Top Stud., Oceanog., 50 (2003), 28472875.
[18] Hadeler, K. P.. Structured populations with diffusion in state space. Math. Biosci. Eng., 7 (2010), 3749.
[19] Hairston, J. N. G., De Meester, L.. Daphnia paleogenetics and environmental change: deconstructing the evolution of plasticity. Int. Rev. Hydrobiol., 93 (2008), 578592.
[20] Johnson, M. T. J., Vellend, M., Stinchcombe, J. R.. Evolution in plant populations as a driver of ecological changes in arthropod communities. Phil. Trans. R. Soc. B., 364 (2009), 15931605.
[21] Jones, L.E., Ellner, S.P.. Effects of rapid prey evolution on predator-prey cycles. J. Math. Biol., 55 (2007), 541573.
[22] Jones, L. E., Becks, L., Ellner, S. P., Hairston, N. G. Jr., Yoshida, T., Fussmann, G. F.. Rapid contemporary evolution and clonal food web dynamics. Phil. Trans. R. Soc. B, 364 (2009), 15791591
[23] D. Henry. Geometric theory of semilinear parabolic equations. Springer, Berlin-New York, 1981.
[24] Holling, C. S.. The components of predation as revealed by a study of small mammal predation on the European pine sawfly. Can. Entomol., 91 (1959), 293320.
[25] T. Kato. Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg, 1995.
[26] Lorz, A., Mirrahimi, S., Perthame, B.. Dirac mass dynamics in multidimensional nonlocal parabolic equations. Comm. Partial Differential Equations, 36 (2011), 1071-1098.
[27] M. Kot. Elements of Mathematical Ecology, Cambridge University Press, 2001.
[28] M. A. Krasnoselskii. Positive solutions of operator equations. P. Noordhoff Ltd., Groningen, 1964.
[29] Marek, I.. Frobenius theory for positive operators: : Comparison theorems and applications. SIAM J. Appl. Math., 19 (1970), 607-628.
[30] Matthews B, B., et al. Toward an integration of evolutionary biology and ecosystem science. Ecol. Lett., 14 (2011), 690701.
[31] Michel, P., Touaoula, T. M.. Asymptotic behavior for a class of the renewal nonlinear equation with diffusion. Math. Methods Appl. Sci., 36 (2013), 323335.
[32] A. Yu. Morozov. Incorporating complex foraging of zooplankton in models: role of micro and mesoscale processes in macroscale patterns. In Dispersal, individual movement and spatial ecology: a mathematical perspective (eds M Lewis, P Maini & S Petrovskii). New York, NY: Springer, (2011), 1–10.
[33] Morozov, A. Yu., Arashkevich, E.G., Nikishina, A., Solovyev, K. Nutrient-rich plankton communities stabilized via predator-prey interactions: revisiting the role of vertical heterogeneity. Math. Med. Biol., 28 (2011), 185215
[34] Morozov, A. Yu., Pasternak, A. F., Arashkevich, E. G.. Revisiting the Role of Individual Variability in Population Persistence and Stability. PLoS ONE 8 (8) (2013), e70576
[35] Oaten, A., Murdoch, W.W.. Functional response and stability in predator-prey systems. Amer. Nat., 109 (1975), 289298.
[36] L. Perko. Differential Equations and Dynamical Systems. Springer, New York, 2001
[37] Petrovskii, S. V., Morozov, A. Y.. Dispersal in a statistically structured population: Fat tails revisited. Amer. Nat., 173 (2010), 278289
[38] Q. I. Rahman, G. Schmeisser. Analytic theory of polynomials. London Mathematical Society Monographs. New Series 26. Oxford: Oxford University Press, 2002.
[39] Reznick, D. N., Ghalambor, C. K., Crooks, K.. Experimental studies of evolution in guppies: a model for understanding the evolutionary consequences of predator removal in natural communities. Mol. Ecol. 17 (2008), 97107.
[40] Rosenzweig, M. L.. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171 (1971), 385387.
[41] Rosenzweig, M. L., MacArthur, R. H.. Graphical representation and stability conditions of predator-prey interactions. Am. Nat., 97 (1963), 209223.
[42] H. H. Schäfer. Banach lattices and positive operators. Springer-Verlag, Berlin, 1974.
[43] Thompson, J. N.. Rapid evolution as an ecological process. Trends Ecol. Evol., 13 (1998), 329332.
[44] Yu. V. Tyutyunov, O. V. Kovalev, L. I. Titova. Spatial demogenetic model for studying phenomena observed upon introduction of the ragweed leaf beetle in the South of Russia. Math. Mod. Nat. Phen., (2013).
[45] Venturino, E.. An ecogenetic model. Appl. Math. Letters, 25 (2012), 1230-1233.
[46] Wolf, M., Weissing, F. J.. Animal personalities: consequences for ecology and evolution. Trends Ecol. Evolut., 8 (2012), 452461.
[47] Yoshida, T., Jones, L. E., Ellner, S. P., Fussmann, G. F., Hairston, J.. Rapid evolution drives ecological dynamics in a predator-prey system. Nature, 424 (2003), 303306
[48] K. Yosida. Functional analysis. Springer-Verlag, Berlin, 1995.


Modelling Effects of Rapid Evolution on Persistence and Stability in Structured Predator-Prey Systems

  • J. Z. Farkas (a1) and A. Y. Morozov (a2) (a3)


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