# American Institute of Mathematical Sciences

May  2013, 18(3): 693-720. doi: 10.3934/dcdsb.2013.18.693

## Nonlocal generalized models of predator-prey systems

 1 Institute for Analysis and Scientific Computing, Vienna University of Technology, 1040 Vienna, Austria 2 Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom

Received  February 2012 Revised  April 2012 Published  December 2012

The method of generalized modeling has been used to analyze differential equations arising in applications. It makes minimal assumptions about the precise functional form of the differential equation and the quantitative values of the steady-states which it aims to analyze from a dynamical systems perspective. The method has been applied successfully in many different contexts, particularly in ecology and systems biology, where the key advantage is that one does not have to select a particular model but is able to provide directly applicable conclusions for sets of models simultaneously. Although many dynamical systems in mathematical biology exhibit steady-state behaviour one also wants to understand nonlocal dynamics beyond equilibrium points. In this paper we analyze predator-prey dynamical systems and extend the method of generalized models to periodic solutions. First, we adapt the equilibrium generalized modeling approach and compute the unique Floquet multiplier of the periodic solution which depends upon so-called generalized elasticity and scale functions. We prove that these functions also have to satisfy a flow on parameter (or moduli) space. Then we use Fourier analysis to provide computable conditions for stability and the moduli space flow. The final stability analysis reduces to two discrete convolutions which can be interpreted to understand when the predator-prey system is stable and what factors enhance or prohibit stable oscillatory behaviour. Finally, we provide a sampling algorithm for parameter space based on nonlinear optimization and the Fast Fourier Transform which enables us to gain a statistical understanding of the stability properties of periodic predator-prey dynamics.
Citation: Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693
##### References:
 [1] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fré and T. Magri, $N=2$ supergravity and $N=2$ super Yang-Mills theory on general scalar manifolds: Symplectic covariance gaugings and the momentum map, J. Geom. Phys., 23 (1997), 111-189. doi: 10.1016/S0393-0440(97)00002-8. [2] U. M. Ascher, R. M. M. Mattheij and R. D. Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations," Classics in Applied Mathematics, 13, SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611971231. [3] M. Baurmann, T. Gross and U. Feudel, Instabilities in sptially extended predator-prey systems: Spatio-temporal patterns in the neighbourhood of Turing-Hopf bifurcations, J. Theor. Bio., 245 (2007), 220-229. doi: 10.1016/j.jtbi.2006.09.036. [4] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations," Edited and with a foreword by Alexander I. Khibnik and Bernd Krauskopf, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 11, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798725. [5] N. Berglund and B. Gentz, "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach," Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2006. [6] A. A. Berryman, The origins and evolution of predator-prey theory, Ecol., 73 (1992), 1530-1535. [7] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. [8] M. Braun, "Differential Equations and their Applications," Hochschultext, Springer-Verlag, Berlin-New York, 1979. [9] C. Chicone, Inertial and slow manifolds for delay differential equations, J. Diff. Eqs., 190 (2003), 364-406. doi: 10.1016/S0022-0396(02)00148-1. [10] C. Chicone, "Ordinary Differential Equations with Applications," Second edition, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 2006. [11] E. J. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang and C. Zhang, Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont), http://cmvl.cs.concordia.ca/auto, 2007. [12] T. F. Fairgrieve and A. D. Jepson, O. K. Floquet multipliers, SIAM J. Numer. Anal., 28 (1991), 1446-1462. doi: 10.1137/0728075. [13] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0611-8. [14] C. Gardiner, "Stochastic Methods. A Handbook for the Natural and Social Sciences," Fourth edition, Springer Series in Synergetics, Springer-Verlag, Berlin, 2009. [15] E. Gehrmann and B. Drossel, Boolean versus continuous dynamics on simple two-gene modules, Phys. Rev. E (3), 82 (2010), 046120, 9 pp. doi: 10.1103/PhysRevE.82.046120. [16] B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318. [17] T. Gross, M. Baurmann, U. Feudel and B. Blasius, Generalized models - a new tool for the investigation of ecological systems, in "Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics" (eds. B. Blasius, J. Kurths, and L. Stone), World Scientific, Singapore, (2006), 21-48. [18] T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network, Phys. Rev. Lett., 96 (2006), 208701. [19] T. Gross, W. Ebenhöh and U. Feudel, Enrichment and foodchain stability: The impact of different functional forms, J. Theor. Bio., 227 (2004), 349-358. doi: 10.1016/j.jtbi.2003.09.020. [20] T. Gross, W. Ebenhöh and U. Feudel, Long food chains are in general chaotic, Oikos, 109 (2005), 133-155. [21] T. Gross and U. Feudel, Analytical search for bifurcation surfaces in parameter space, Physica D, 195 (2004), 292-302. doi: 10.1016/j.physd.2004.03.019. [22] T. Gross and U. Feudel, Generalized models as an universal approach to the analysis of nonlinear dynamical systems, Phys. Rev. E, 73 (2006), 016205-14. [23] T. Gross and U. Feudel, Local dynamical equivalence of certain food webs, Ocean Dynamics, 59 (2009), 417-427. [24] T. Gross, L. Rudolf, S. A. Levin and U. Dieckmann, Generalized models reveal stabilizing factors in food webs, Science, 325 (2009), 747-750. [25] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. [26] Joe Harris, "Algebraic Geometry. A First Course," Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1992. [27] Robin Hartshorne, "Algebraic Geometry," Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. [28] A. Hastings, Global stability of two-species systems, J. Math. Biol., 5 (1977/78), 399-403. doi: 10.1007/BF00276109. [29] Y. Katznelson, "An Introduction to Harmonic Analysis," Third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. [30] M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics, Proc. R. Soc. B, 264 (1997), 1149-1156. [31] C. A. Klausmeier, Floquet theory: A useful tool for understanding nonequilibrium dynamics, Theor. Ecol., 1 (2008), 153-161. [32] T. W. Körner, "Fourier Analysis," CUP, 1989. [33] M. Kot, "Elements of Mathematical Ecology," CUP, 2003. [34] C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications, submitted, arXiv:1101.2908, 1-46, 2011. [35] C. Kuehn, S. Siegmund and T. Gross, On the analysis of evolution equations via generalized models, accepted, IMA J. Appl. Math., arXiv:1012.4340, 2012. [36] Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004. [37] S. J. Lade and T. Gross, Early warning signals for critical transitions: A generalized modeling approach, PLoS Comp. Biol., 8 (2012), e1002360-6. [38] J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim., 9 (1998), 112-147. doi: 10.1137/S1052623496303470. [39] K. Lust, Improved numerical Floquet multipliers, Int. J. Bif. Chaos Appl. Sci. Engrg., 11 (2001), 2389-2410. doi: 10.1142/S0218127401003486. [40] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems" (eds. B. Hasselblatt and A. Katok), Vol. 1A, North Holland, Amsterdam, (2002), 1015-1090. doi: 10.1016/S1874-575X(02)80015-7. [41] The MathWorks, Matlab 2010b, 2010. (with Control and Optimization Toolboxes). [42] S. M. Moghadas and M. E. Alexander, Dynamics of a generalized {Gauss-type predator-prey model with a seasonal functional response}, Chaos, Solitons and Fractals, 23 (2005), 55-65. doi: 10.1016/j.chaos.2004.04.030. [43] J. Nocedal and S. J. Wright, "Numerical Optimization," Springer Series in Operations Research, Springer-Verlag, New York, 1999. [44] E. Reznik and D. Segré, On the stability of metabolic cycles, J. Theor. Biol., 266 (2010), 536-549. [45] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. [46] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkhin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk, and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59. [47] J. Smillie, Introduction to rational billards, MSRI Workshop on Geometric Group Theory, 2007. Available from: http://www.math.cornell.edu/~vogtmann/MSRI/index.html. [48] L. Socha, "Linearization Methods for Stochastic Dynamic Systems," Lecture Notes in Physics, 730, Springer, Berlin, 2008. [49] R. Steuer, T. Gross, J. Selbig and B. Blasius, Structural kinetic modeling of metabolic networks, Proc. Natl. Acad. Sci., 103 (2006), 11868-11873. [50] R. Steuer, A. Nunes Nesi, A. R. Fernie, T. Gross, B. Blasius and J. Selbig, From structure to dynamics of metabolic pathways, Bioinformatics, 23 (2007), 1378-1385. [51] D. Stiefs, T. Gross, R. Steuer and U. Feudel, Computation and visualization of bifurcation surfaces, Int. J. Bif. Chaos, 18 (2008), 2191-2206. doi: 10.1142/S0218127408021658. [52] D. Stiefs, G. A. K. van Voorn, B. W. Kooi, U. Feudel and T. Gross, Food quality in producer-grazer models, Am. Nat., 176 (2010), 367-380. [53] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606. doi: 10.1137/0148033. [54] J. D. Yeakel, D. Stiefs, M. Novak and T. Gross, Generalized modeling of ecological population dynamics, Theor. Ecol., 4 (2011), 179-194. [55] M. Zumsande, D. Stiefs, S. Siegmund and T. Gross, General analysis of mathematical models for bone remodeling, Bone, 48 (2011), 910-917. [56] A. Zygmund, "Trigonometric Series," Vol. 1 & 2, CUP, 1988.

show all references

##### References:
 [1] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fré and T. Magri, $N=2$ supergravity and $N=2$ super Yang-Mills theory on general scalar manifolds: Symplectic covariance gaugings and the momentum map, J. Geom. Phys., 23 (1997), 111-189. doi: 10.1016/S0393-0440(97)00002-8. [2] U. M. Ascher, R. M. M. Mattheij and R. D. Russell, "Numerical Solution of Boundary Value Problems for Ordinary Differential Equations," Classics in Applied Mathematics, 13, SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611971231. [3] M. Baurmann, T. Gross and U. Feudel, Instabilities in sptially extended predator-prey systems: Spatio-temporal patterns in the neighbourhood of Turing-Hopf bifurcations, J. Theor. Bio., 245 (2007), 220-229. doi: 10.1016/j.jtbi.2006.09.036. [4] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations," Edited and with a foreword by Alexander I. Khibnik and Bernd Krauskopf, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 11, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798725. [5] N. Berglund and B. Gentz, "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach," Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2006. [6] A. A. Berryman, The origins and evolution of predator-prey theory, Ecol., 73 (1992), 1530-1535. [7] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. [8] M. Braun, "Differential Equations and their Applications," Hochschultext, Springer-Verlag, Berlin-New York, 1979. [9] C. Chicone, Inertial and slow manifolds for delay differential equations, J. Diff. Eqs., 190 (2003), 364-406. doi: 10.1016/S0022-0396(02)00148-1. [10] C. Chicone, "Ordinary Differential Equations with Applications," Second edition, Texts in Applied Mathematics, 34, Springer-Verlag, New York, 2006. [11] E. J. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang and C. Zhang, Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont), http://cmvl.cs.concordia.ca/auto, 2007. [12] T. F. Fairgrieve and A. D. Jepson, O. K. Floquet multipliers, SIAM J. Numer. Anal., 28 (1991), 1446-1462. doi: 10.1137/0728075. [13] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0611-8. [14] C. Gardiner, "Stochastic Methods. A Handbook for the Natural and Social Sciences," Fourth edition, Springer Series in Synergetics, Springer-Verlag, Berlin, 2009. [15] E. Gehrmann and B. Drossel, Boolean versus continuous dynamics on simple two-gene modules, Phys. Rev. E (3), 82 (2010), 046120, 9 pp. doi: 10.1103/PhysRevE.82.046120. [16] B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318. [17] T. Gross, M. Baurmann, U. Feudel and B. Blasius, Generalized models - a new tool for the investigation of ecological systems, in "Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics" (eds. B. Blasius, J. Kurths, and L. Stone), World Scientific, Singapore, (2006), 21-48. [18] T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network, Phys. Rev. Lett., 96 (2006), 208701. [19] T. Gross, W. Ebenhöh and U. Feudel, Enrichment and foodchain stability: The impact of different functional forms, J. Theor. Bio., 227 (2004), 349-358. doi: 10.1016/j.jtbi.2003.09.020. [20] T. Gross, W. Ebenhöh and U. Feudel, Long food chains are in general chaotic, Oikos, 109 (2005), 133-155. [21] T. Gross and U. Feudel, Analytical search for bifurcation surfaces in parameter space, Physica D, 195 (2004), 292-302. doi: 10.1016/j.physd.2004.03.019. [22] T. Gross and U. Feudel, Generalized models as an universal approach to the analysis of nonlinear dynamical systems, Phys. Rev. E, 73 (2006), 016205-14. [23] T. Gross and U. Feudel, Local dynamical equivalence of certain food webs, Ocean Dynamics, 59 (2009), 417-427. [24] T. Gross, L. Rudolf, S. A. Levin and U. Dieckmann, Generalized models reveal stabilizing factors in food webs, Science, 325 (2009), 747-750. [25] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. [26] Joe Harris, "Algebraic Geometry. A First Course," Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1992. [27] Robin Hartshorne, "Algebraic Geometry," Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. [28] A. Hastings, Global stability of two-species systems, J. Math. Biol., 5 (1977/78), 399-403. doi: 10.1007/BF00276109. [29] Y. Katznelson, "An Introduction to Harmonic Analysis," Third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. [30] M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics, Proc. R. Soc. B, 264 (1997), 1149-1156. [31] C. A. Klausmeier, Floquet theory: A useful tool for understanding nonequilibrium dynamics, Theor. Ecol., 1 (2008), 153-161. [32] T. W. Körner, "Fourier Analysis," CUP, 1989. [33] M. Kot, "Elements of Mathematical Ecology," CUP, 2003. [34] C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications, submitted, arXiv:1101.2908, 1-46, 2011. [35] C. Kuehn, S. Siegmund and T. Gross, On the analysis of evolution equations via generalized models, accepted, IMA J. Appl. Math., arXiv:1012.4340, 2012. [36] Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004. [37] S. J. Lade and T. Gross, Early warning signals for critical transitions: A generalized modeling approach, PLoS Comp. Biol., 8 (2012), e1002360-6. [38] J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim., 9 (1998), 112-147. doi: 10.1137/S1052623496303470. [39] K. Lust, Improved numerical Floquet multipliers, Int. J. Bif. Chaos Appl. Sci. Engrg., 11 (2001), 2389-2410. doi: 10.1142/S0218127401003486. [40] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in "Handbook of Dynamical Systems" (eds. B. Hasselblatt and A. Katok), Vol. 1A, North Holland, Amsterdam, (2002), 1015-1090. doi: 10.1016/S1874-575X(02)80015-7. [41] The MathWorks, Matlab 2010b, 2010. (with Control and Optimization Toolboxes). [42] S. M. Moghadas and M. E. Alexander, Dynamics of a generalized {Gauss-type predator-prey model with a seasonal functional response}, Chaos, Solitons and Fractals, 23 (2005), 55-65. doi: 10.1016/j.chaos.2004.04.030. [43] J. Nocedal and S. J. Wright, "Numerical Optimization," Springer Series in Operations Research, Springer-Verlag, New York, 1999. [44] E. Reznik and D. Segré, On the stability of metabolic cycles, J. Theor. Biol., 266 (2010), 536-549. [45] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. [46] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkhin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk, and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59. [47] J. Smillie, Introduction to rational billards, MSRI Workshop on Geometric Group Theory, 2007. Available from: http://www.math.cornell.edu/~vogtmann/MSRI/index.html. [48] L. Socha, "Linearization Methods for Stochastic Dynamic Systems," Lecture Notes in Physics, 730, Springer, Berlin, 2008. [49] R. Steuer, T. Gross, J. Selbig and B. Blasius, Structural kinetic modeling of metabolic networks, Proc. Natl. Acad. Sci., 103 (2006), 11868-11873. [50] R. Steuer, A. Nunes Nesi, A. R. Fernie, T. Gross, B. Blasius and J. Selbig, From structure to dynamics of metabolic pathways, Bioinformatics, 23 (2007), 1378-1385. [51] D. Stiefs, T. Gross, R. Steuer and U. Feudel, Computation and visualization of bifurcation surfaces, Int. J. Bif. Chaos, 18 (2008), 2191-2206. doi: 10.1142/S0218127408021658. [52] D. Stiefs, G. A. K. van Voorn, B. W. Kooi, U. Feudel and T. Gross, Food quality in producer-grazer models, Am. Nat., 176 (2010), 367-380. [53] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606. doi: 10.1137/0148033. [54] J. D. Yeakel, D. Stiefs, M. Novak and T. Gross, Generalized modeling of ecological population dynamics, Theor. Ecol., 4 (2011), 179-194. [55] M. Zumsande, D. Stiefs, S. Siegmund and T. Gross, General analysis of mathematical models for bone remodeling, Bone, 48 (2011), 910-917. [56] A. Zygmund, "Trigonometric Series," Vol. 1 & 2, CUP, 1988.
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