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

Published online by Cambridge University Press:  28 May 2014

J. Z. Farkas  and
A. Y. Morozov
J. Z. Farkas*
Affiliation:
Division of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, UK
A. Y. Morozov
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK Shirshov Institute of Oceanology, Moscow, 117997, Russia
*
⋆ Corresponding author. E-mail: jozsef.farkas@stir.ac.uk
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Abstract

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.

Keywords

Integro-differential equationsstructured populationspopulation persistencestability analysisspectral theory of operators.
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 3: Biological evolution , 2014 , pp. 26 - 46
Copyright
© EDP Sciences, 2014

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References

Abrams, P.A., Walters, C.J.. Invulnerable prey and the paradox of enrichment. Ecology, 77 (1996), 1125–1133. CrossRefGoogle Scholar
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), 1685–1694. CrossRefGoogle Scholar
L. J. S. Allen. An introduction to mathematical biology. Pearson Prentice Hall, Upper Saddle River, NJ, 2007.
Barles, G., Perthame, B.. Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics. Contemp. Math., 439 (2007), 57–68. CrossRefGoogle Scholar
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), 761–766. CrossRefGoogle Scholar
À. Calsina, J. Z. Farkas. Positive steady states of evolution equations with finite dimensional nonlinearities. to appear in SIAM J. Math. Anal.
Calsina, À., Farkas, J. Z.. Steady states in a structured epidemic model with Wentzell boundary condition. J. Evol. Equ., 12 (2012), 495–512. CrossRefGoogle Scholar
Calsina, À., Palmada, J. M.. Steady states of a selection-mutation model for an age structured population. J. Math. Anal. Appl., 400 (2013), 386–395. CrossRefGoogle Scholar
J.M. Cushing. An introduction to structured population dynamics. SIAM, Philadelphia, PA, 1998.
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), 257–271. CrossRefGoogle Scholar
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), 139–150. CrossRefGoogle Scholar
Duffy, M. A., Sivars-Becker, L.. Rapid evolution and ecological host-parasite dynamics. Ecol. Lett., 10 (2007), 44–53. CrossRefGoogle Scholar
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), 603–614. CrossRefGoogle ScholarPubMed
Farkas, J. Z., Green, D. M., Hinow, P.. Semigroup analysis of structured parasite populations. Math. Model. Nat. Phenom., 5 (2010), 94–114. CrossRefGoogle Scholar
Farkas, J. Z., Hagen, T.. Linear stability and positivity results for a generalized size-structured Daphnia model with inflow. Appl. Anal., 86 (2007), 1087–1103. CrossRefGoogle Scholar
Fussmann, G. F., Gonzalez, A.. Evolutionary rescue can maintain an oscillating community undergoing environmental change. Interface Focus, 3 (2013), 20130036. CrossRefGoogle ScholarPubMed
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), 2847–2875. CrossRefGoogle Scholar
Hadeler, K. P.. Structured populations with diffusion in state space. Math. Biosci. Eng., 7 (2010), 37–49. CrossRefGoogle Scholar
Hairston, J. N. G., De Meester, L.. Daphnia paleogenetics and environmental change: deconstructing the evolution of plasticity. Int. Rev. Hydrobiol., 93 (2008), 578–592. CrossRefGoogle Scholar
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), 1593–1605. CrossRefGoogle Scholar
Jones, L.E., Ellner, S.P.. Effects of rapid prey evolution on predator-prey cycles. J. Math. Biol., 55 (2007), 541–573. CrossRefGoogle Scholar
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), 1579–1591 CrossRefGoogle Scholar
D. Henry. Geometric theory of semilinear parabolic equations. Springer, Berlin-New York, 1981.
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), 293–320. CrossRefGoogle Scholar
T. Kato. Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg, 1995.
Lorz, A., Mirrahimi, S., Perthame, B.. Dirac mass dynamics in multidimensional nonlocal parabolic equations. Comm. Partial Differential Equations, 36 (2011), 1071-1098. CrossRefGoogle Scholar
M. Kot. Elements of Mathematical Ecology, Cambridge University Press, 2001.
M. A. Krasnoselskii. Positive solutions of operator equations. P. Noordhoff Ltd., Groningen, 1964.
Marek, I.. Frobenius theory for positive operators: : Comparison theorems and applications. SIAM J. Appl. Math., 19 (1970), 607-628. CrossRefGoogle Scholar
Matthews B, B., et al. Toward an integration of evolutionary biology and ecosystem science. Ecol. Lett., 14 (2011), 690–701. CrossRefGoogle Scholar
Michel, P., Touaoula, T. M.. Asymptotic behavior for a class of the renewal nonlinear equation with diffusion. Math. Methods Appl. Sci., 36 (2013), 323–335. CrossRefGoogle Scholar
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.
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), 185–215 CrossRefGoogle ScholarPubMed
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 CrossRefGoogle ScholarPubMed
Oaten, A., Murdoch, W.W.. Functional response and stability in predator-prey systems. Amer. Nat., 109 (1975), 289–298. CrossRefGoogle Scholar
L. Perko. Differential Equations and Dynamical Systems. Springer, New York, 2001
Petrovskii, S. V., Morozov, A. Y.. Dispersal in a statistically structured population: Fat tails revisited. Amer. Nat., 173 (2010), 278–289 CrossRefGoogle Scholar
Q. I. Rahman, G. Schmeisser. Analytic theory of polynomials. London Mathematical Society Monographs. New Series 26. Oxford: Oxford University Press, 2002.
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), 97–107. CrossRefGoogle ScholarPubMed
Rosenzweig, M. L.. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171 (1971), 385–387. CrossRefGoogle ScholarPubMed
Rosenzweig, M. L., MacArthur, R. H.. Graphical representation and stability conditions of predator-prey interactions. Am. Nat., 97 (1963), 209–223. CrossRefGoogle Scholar
H. H. Schäfer. Banach lattices and positive operators. Springer-Verlag, Berlin, 1974.
Thompson, J. N.. Rapid evolution as an ecological process. Trends Ecol. Evol., 13 (1998), 329–332. CrossRefGoogle Scholar
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).
Venturino, E.. An ecogenetic model. Appl. Math. Letters, 25 (2012), 1230-1233. CrossRefGoogle Scholar
Wolf, M., Weissing, F. J.. Animal personalities: consequences for ecology and evolution. Trends Ecol. Evolut., 8 (2012), 452–461. CrossRefGoogle Scholar
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), 303–306 CrossRefGoogle Scholar
K. Yosida. Functional analysis. Springer-Verlag, Berlin, 1995.