# 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 & 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. doi: 10.1016/S0393-0440(97)00002-8. Google Scholar [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 (1995). doi: 10.1137/1.9781611971231. Google Scholar [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. doi: 10.1016/j.jtbi.2006.09.036. Google Scholar [4] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", Edited and with a foreword by Alexander I. Khibnik and Bernd Krauskopf, 11 (1998). doi: 10.1142/9789812798725. Google Scholar [5] N. Berglund and B. Gentz, "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach,", Probability and its Applications (New York), (2006). Google Scholar [6] A. A. Berryman, The origins and evolution of predator-prey theory,, Ecol., 73 (1992), 1530. Google Scholar [7] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001). Google Scholar [8] M. Braun, "Differential Equations and their Applications,", Hochschultext, (1979). Google Scholar [9] C. Chicone, Inertial and slow manifolds for delay differential equations,, J. Diff. Eqs., 190 (2003), 364. doi: 10.1016/S0022-0396(02)00148-1. Google Scholar [10] C. Chicone, "Ordinary Differential Equations with Applications,", Second edition, 34 (2006). Google Scholar [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),, , (2007). Google Scholar [12] T. F. Fairgrieve and A. D. Jepson, O. K. Floquet multipliers,, SIAM J. Numer. Anal., 28 (1991), 1446. doi: 10.1137/0728075. Google Scholar [13] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,", Second edition, (1998). doi: 10.1007/978-1-4612-0611-8. Google Scholar [14] C. Gardiner, "Stochastic Methods. A Handbook for the Natural and Social Sciences,", Fourth edition, (2009). Google Scholar [15] E. Gehrmann and B. Drossel, Boolean versus continuous dynamics on simple two-gene modules,, Phys. Rev. E (3), 82 (2010). doi: 10.1103/PhysRevE.82.046120. Google Scholar [16] B. S. Goh, Global stability in two species interactions,, J. Math. Biol., 3 (1976), 313. Google Scholar [17] T. Gross, M. Baurmann, U. Feudel and B. Blasius, Generalized models - a new tool for the investigation of ecological systems,, in, (2006), 21. Google Scholar [18] T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network,, Phys. Rev. Lett., 96 (2006). Google Scholar [19] T. Gross, W. Ebenhöh and U. Feudel, Enrichment and foodchain stability: The impact of different functional forms,, J. Theor. Bio., 227 (2004), 349. doi: 10.1016/j.jtbi.2003.09.020. Google Scholar [20] T. Gross, W. Ebenhöh and U. Feudel, Long food chains are in general chaotic,, Oikos, 109 (2005), 133. Google Scholar [21] T. Gross and U. Feudel, Analytical search for bifurcation surfaces in parameter space,, Physica D, 195 (2004), 292. doi: 10.1016/j.physd.2004.03.019. Google Scholar [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. Google Scholar [23] T. Gross and U. Feudel, Local dynamical equivalence of certain food webs,, Ocean Dynamics, 59 (2009), 417. Google Scholar [24] T. Gross, L. Rudolf, S. A. Levin and U. Dieckmann, Generalized models reveal stabilizing factors in food webs,, Science, 325 (2009), 747. Google Scholar [25] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, 42 (1983). Google Scholar [26] Joe Harris, "Algebraic Geometry. A First Course,", Graduate Texts in Mathematics, 133 (1992). Google Scholar [27] Robin Hartshorne, "Algebraic Geometry,", Graduate Texts in Mathematics, (1977). Google Scholar [28] A. Hastings, Global stability of two-species systems,, J. Math. Biol., 5 (): 399. doi: 10.1007/BF00276109. Google Scholar [29] Y. Katznelson, "An Introduction to Harmonic Analysis,", Third edition, (2004). Google Scholar [30] M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics,, Proc. R. Soc. B, 264 (1997), 1149. Google Scholar [31] C. A. Klausmeier, Floquet theory: A useful tool for understanding nonequilibrium dynamics,, Theor. Ecol., 1 (2008), 153. Google Scholar [32] T. W. Körner, "Fourier Analysis,", CUP, (1989). Google Scholar [33] M. Kot, "Elements of Mathematical Ecology,", CUP, (2003). Google Scholar [34] C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications,, submitted, (2011), 1. Google Scholar [35] C. Kuehn, S. Siegmund and T. Gross, On the analysis of evolution equations via generalized models,, accepted, (2012). Google Scholar [36] Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Third edition, 112 (2004). Google Scholar [37] S. J. Lade and T. Gross, Early warning signals for critical transitions: A generalized modeling approach,, PLoS Comp. Biol., 8 (2012), 1002360. Google Scholar [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. doi: 10.1137/S1052623496303470. Google Scholar [39] K. Lust, Improved numerical Floquet multipliers,, Int. J. Bif. Chaos Appl. Sci. Engrg., 11 (2001), 2389. doi: 10.1142/S0218127401003486. Google Scholar [40] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar [41] The MathWorks, Matlab 2010b, 2010., (with Control and Optimization Toolboxes)., (). Google Scholar [42] S. M. Moghadas and M. E. Alexander, Dynamics of a generalized {Gauss-type predator-prey model with a seasonal functional response},, Chaos, 23 (2005), 55. doi: 10.1016/j.chaos.2004.04.030. Google Scholar [43] J. Nocedal and S. J. Wright, "Numerical Optimization,", Springer Series in Operations Research, (1999). Google Scholar [44] E. Reznik and D. Segré, On the stability of metabolic cycles,, J. Theor. Biol., 266 (2010), 536. Google Scholar [45] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385. Google Scholar [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. Google Scholar [47] J. Smillie, Introduction to rational billards,, MSRI Workshop on Geometric Group Theory, (2007). Google Scholar [48] L. Socha, "Linearization Methods for Stochastic Dynamic Systems,", Lecture Notes in Physics, 730 (2008). Google Scholar [49] R. Steuer, T. Gross, J. Selbig and B. Blasius, Structural kinetic modeling of metabolic networks,, Proc. Natl. Acad. Sci., 103 (2006), 11868. Google Scholar [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. Google Scholar [51] D. Stiefs, T. Gross, R. Steuer and U. Feudel, Computation and visualization of bifurcation surfaces,, Int. J. Bif. Chaos, 18 (2008), 2191. doi: 10.1142/S0218127408021658. Google Scholar [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. Google Scholar [53] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence,, SIAM J. Appl. Math., 48 (1988), 592. doi: 10.1137/0148033. Google Scholar [54] J. D. Yeakel, D. Stiefs, M. Novak and T. Gross, Generalized modeling of ecological population dynamics,, Theor. Ecol., 4 (2011), 179. Google Scholar [55] M. Zumsande, D. Stiefs, S. Siegmund and T. Gross, General analysis of mathematical models for bone remodeling,, Bone, 48 (2011), 910. Google Scholar [56] A. Zygmund, "Trigonometric Series,", Vol. 1 & 2, (1988). Google Scholar

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. doi: 10.1016/S0393-0440(97)00002-8. Google Scholar [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 (1995). doi: 10.1137/1.9781611971231. Google Scholar [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. doi: 10.1016/j.jtbi.2006.09.036. Google Scholar [4] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", Edited and with a foreword by Alexander I. Khibnik and Bernd Krauskopf, 11 (1998). doi: 10.1142/9789812798725. Google Scholar [5] N. Berglund and B. Gentz, "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach,", Probability and its Applications (New York), (2006). Google Scholar [6] A. A. Berryman, The origins and evolution of predator-prey theory,, Ecol., 73 (1992), 1530. Google Scholar [7] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001). Google Scholar [8] M. Braun, "Differential Equations and their Applications,", Hochschultext, (1979). Google Scholar [9] C. Chicone, Inertial and slow manifolds for delay differential equations,, J. Diff. Eqs., 190 (2003), 364. doi: 10.1016/S0022-0396(02)00148-1. Google Scholar [10] C. Chicone, "Ordinary Differential Equations with Applications,", Second edition, 34 (2006). Google Scholar [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),, , (2007). Google Scholar [12] T. F. Fairgrieve and A. D. Jepson, O. K. Floquet multipliers,, SIAM J. Numer. Anal., 28 (1991), 1446. doi: 10.1137/0728075. Google Scholar [13] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems,", Second edition, (1998). doi: 10.1007/978-1-4612-0611-8. Google Scholar [14] C. Gardiner, "Stochastic Methods. A Handbook for the Natural and Social Sciences,", Fourth edition, (2009). Google Scholar [15] E. Gehrmann and B. Drossel, Boolean versus continuous dynamics on simple two-gene modules,, Phys. Rev. E (3), 82 (2010). doi: 10.1103/PhysRevE.82.046120. Google Scholar [16] B. S. Goh, Global stability in two species interactions,, J. Math. Biol., 3 (1976), 313. Google Scholar [17] T. Gross, M. Baurmann, U. Feudel and B. Blasius, Generalized models - a new tool for the investigation of ecological systems,, in, (2006), 21. Google Scholar [18] T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network,, Phys. Rev. Lett., 96 (2006). Google Scholar [19] T. Gross, W. Ebenhöh and U. Feudel, Enrichment and foodchain stability: The impact of different functional forms,, J. Theor. Bio., 227 (2004), 349. doi: 10.1016/j.jtbi.2003.09.020. Google Scholar [20] T. Gross, W. Ebenhöh and U. Feudel, Long food chains are in general chaotic,, Oikos, 109 (2005), 133. Google Scholar [21] T. Gross and U. Feudel, Analytical search for bifurcation surfaces in parameter space,, Physica D, 195 (2004), 292. doi: 10.1016/j.physd.2004.03.019. Google Scholar [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. Google Scholar [23] T. Gross and U. Feudel, Local dynamical equivalence of certain food webs,, Ocean Dynamics, 59 (2009), 417. Google Scholar [24] T. Gross, L. Rudolf, S. A. Levin and U. Dieckmann, Generalized models reveal stabilizing factors in food webs,, Science, 325 (2009), 747. Google Scholar [25] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, 42 (1983). Google Scholar [26] Joe Harris, "Algebraic Geometry. A First Course,", Graduate Texts in Mathematics, 133 (1992). Google Scholar [27] Robin Hartshorne, "Algebraic Geometry,", Graduate Texts in Mathematics, (1977). Google Scholar [28] A. Hastings, Global stability of two-species systems,, J. Math. Biol., 5 (): 399. doi: 10.1007/BF00276109. Google Scholar [29] Y. Katznelson, "An Introduction to Harmonic Analysis,", Third edition, (2004). Google Scholar [30] M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics,, Proc. R. Soc. B, 264 (1997), 1149. Google Scholar [31] C. A. Klausmeier, Floquet theory: A useful tool for understanding nonequilibrium dynamics,, Theor. Ecol., 1 (2008), 153. Google Scholar [32] T. W. Körner, "Fourier Analysis,", CUP, (1989). Google Scholar [33] M. Kot, "Elements of Mathematical Ecology,", CUP, (2003). Google Scholar [34] C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications,, submitted, (2011), 1. Google Scholar [35] C. Kuehn, S. Siegmund and T. Gross, On the analysis of evolution equations via generalized models,, accepted, (2012). Google Scholar [36] Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Third edition, 112 (2004). Google Scholar [37] S. J. Lade and T. Gross, Early warning signals for critical transitions: A generalized modeling approach,, PLoS Comp. Biol., 8 (2012), 1002360. Google Scholar [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. doi: 10.1137/S1052623496303470. Google Scholar [39] K. Lust, Improved numerical Floquet multipliers,, Int. J. Bif. Chaos Appl. Sci. Engrg., 11 (2001), 2389. doi: 10.1142/S0218127401003486. Google Scholar [40] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7. Google Scholar [41] The MathWorks, Matlab 2010b, 2010., (with Control and Optimization Toolboxes)., (). Google Scholar [42] S. M. Moghadas and M. E. Alexander, Dynamics of a generalized {Gauss-type predator-prey model with a seasonal functional response},, Chaos, 23 (2005), 55. doi: 10.1016/j.chaos.2004.04.030. Google Scholar [43] J. Nocedal and S. J. Wright, "Numerical Optimization,", Springer Series in Operations Research, (1999). Google Scholar [44] E. Reznik and D. Segré, On the stability of metabolic cycles,, J. Theor. Biol., 266 (2010), 536. Google Scholar [45] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time,, Science, 171 (1971), 385. Google Scholar [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. Google Scholar [47] J. Smillie, Introduction to rational billards,, MSRI Workshop on Geometric Group Theory, (2007). Google Scholar [48] L. Socha, "Linearization Methods for Stochastic Dynamic Systems,", Lecture Notes in Physics, 730 (2008). Google Scholar [49] R. Steuer, T. Gross, J. Selbig and B. Blasius, Structural kinetic modeling of metabolic networks,, Proc. Natl. Acad. Sci., 103 (2006), 11868. Google Scholar [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. Google Scholar [51] D. Stiefs, T. Gross, R. Steuer and U. Feudel, Computation and visualization of bifurcation surfaces,, Int. J. Bif. Chaos, 18 (2008), 2191. doi: 10.1142/S0218127408021658. Google Scholar [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. Google Scholar [53] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence,, SIAM J. Appl. Math., 48 (1988), 592. doi: 10.1137/0148033. Google Scholar [54] J. D. Yeakel, D. Stiefs, M. Novak and T. Gross, Generalized modeling of ecological population dynamics,, Theor. Ecol., 4 (2011), 179. Google Scholar [55] M. Zumsande, D. Stiefs, S. Siegmund and T. Gross, General analysis of mathematical models for bone remodeling,, Bone, 48 (2011), 910. Google Scholar [56] A. Zygmund, "Trigonometric Series,", Vol. 1 & 2, (1988). Google Scholar
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