# American Institute of Mathematical Sciences

January  2021, 26(1): 1-60. doi: 10.3934/dcdsb.2020231

## Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results and applications

 Institut für Mathematik, Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt, Austria

Received  January 2020 Revised  June 2020 Published  July 2020

We study local bifurcations of periodic solutions to time-periodic (systems of) integrodifference equations over compact habitats. Such infinite-dimensional discrete dynamical systems arise in theoretical ecology as models to describe the spatial dispersal of species having nonoverlapping generations. Our explicit criteria allow us to identify branchings of fold- and crossing curve-type, which include the classical transcritical-, pitchfork- and flip-scenario as special cases. Indeed, not only tools to detect qualitative changes in models from e.g. spatial ecology and related simulations are provided, but these critical transitions are also classified. In addition, the bifurcation behavior of various time-periodic integrodifference equations is investigated and illustrated. This requires a combination of analytical methods and numerical tools based on Nyström discretization of the integral operators involved.

Citation: Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $C(\Omega)$ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231
##### References:
 [1] C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ II: Discrete torus bifurcations, Commun. Pure Appl. Anal., 19 (2020), 1847-1874.   Google Scholar [2] M. Y. M. Alzoubi, The net reproductive number and bifurcation in an integro-difference system of equations, Appl. Math. Sci., 4 (2010), 191-200.   Google Scholar [3] H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, Studies in Mathematics 13, Walter de Gruyter, Berlin-New York, 1990. doi: 10.1515/9783110853698.  Google Scholar [4] M. Andersen, Properties of some density-dependent integrodifference equation population models, Math. Biosci., 104 (1991), 135-157.  doi: 10.1016/0025-5564(91)90034-G.  Google Scholar [5] T. Ando, Totally positive matrices, Linear Algebra Appl., 90 (1987), 165-219.  doi: 10.1016/0024-3795(87)90313-2.  Google Scholar [6] P. Anselone and J. Lee, Spectral properties of integral operators with nonnegative kernels, Linear Algebra Appl., 9 (1974), 67-87.  doi: 10.1016/0024-3795(74)90027-5.  Google Scholar [7] K. Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Numer. Anal., 12 (1975), 213-222.  doi: 10.1137/0712020.  Google Scholar [8] K. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. Integr. Equat. Appl., 4 (1992), 15-46.  doi: 10.1216/jiea/1181075664.  Google Scholar [9] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Monographs on Applied and Comp. Mathematics 4, University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511626340.  Google Scholar [10] N. Bacaër, Periodic matrix population models: Growth rate, basic reproduction number, and entropy, Bull. Math. Biol., 71 (2009), 1781-1792.  doi: 10.1007/s11538-009-9426-6.  Google Scholar [11] N. Bacaër and E. H. Ait Dads, On the biological interpretation of a definition for the parameter $R_0$ in periodic population models, J. Math. Biol., 65 (2012), 601-621.  doi: 10.1007/s00285-011-0479-4.  Google Scholar [12] W.-J. Beyn, T. Hüls and M.-C. Samtenschnieder., On $r$-periodic orbits of $k$-periodic maps, J. Difference Equ. Appl., 14 (2008), 865-887.  doi: 10.1080/10236190801940010.  Google Scholar [13] J. Bramburger and F. Lutscher, Analysis of integrodifference equations with a separable dispersal kernel, Acta Applicandae Mathematicae, 161 (2019), 127-151.  doi: 10.1007/s10440-018-0207-9.  Google Scholar [14] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second edition. Texts in Applied Mathematics, 40. Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar [15] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation: An Introduction, University Press, Princeton NJ, 2003.  doi: 10.1515/9781400884339.  Google Scholar [16] D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980.  Google Scholar [17] J. M. Cushing and A. S. Ackleh, A net reproductive number for periodic matrix models, J. Biol. Dyn., 6 (2012), 166-188.  doi: 10.1080/17513758.2010.544410.  Google Scholar [18] J. M. Cushing and S. M. Henson, Periodic matrix models for seasonal dynamics of structured populations with application to a seabird population, J. Math. Biol., 77 (2018), 1689-1720.  doi: 10.1007/s00285-018-1211-4.  Google Scholar [19] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [20] ————, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar [21] S. Day, O. Junge and K. Mischaikow, A rigerous numerical method for the global dynamics of infinite-dimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3(2) (2004), 117-160.  doi: 10.1137/030600210.  Google Scholar [22] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin etc., 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [23] G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Berlin etc., 1996.  Google Scholar [24] I. Győri and M. Pituk, The converse of the theorem on stability by the first approximation for difference equations, Nonlin. Analysis (TMA), 47 (2001), 4635-4640.  doi: 10.1016/S0362-546X(01)00576-4.  Google Scholar [25] D. P. Hardin, P. Takáč and G. F. Webb, A comparison of dispersal strategies for survival of spatially heterogeneous populations, SIAM J. Appl. Math., 48 (1988), 1396-1423.  doi: 10.1137/0148086.  Google Scholar [26] ————, Dispersion population models discrete in time and continuous in space, J. Math. Biol., 28 (1990), 1-20.  doi: 10.1007/BF00171515.  Google Scholar [27] G. Iooss, Bifurcation of Maps and Applications, Mathematics Studies 36, North-Holland, Amsterdam etc., 1979.  Google Scholar [28] J. Jacobsen and T. McAdam, A boundary value problem for integrodifference population models with cyclic kernels, Discrete Contin. Dyn. Syst. (Series B), 19 (2014), 3191-3207.  doi: 10.3934/dcdsb.2014.19.3191.  Google Scholar [29] W. Jin and H. R. Thieme, An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Discrete Contin. Dyn. Syst. (Series B), 21 (2016), 447-470.  doi: 10.3934/dcdsb.2016.21.447.  Google Scholar [30] T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin etc., 1980. Google Scholar [31] C. Kelley, Solving Nonlinear Equations with Newton's Method, Fundamentals of Algorithms 1, SIAM, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718898.  Google Scholar [32] H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs (2nd ed.), Applied Mathematical Sciences 156, Springer, New York etc., 2012. doi: 10.1007/978-1-4614-0502-3.  Google Scholar [33] M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar [34] U. Krause, Positive Dynamical Systems in Discrete Time, Studies in Mathematics 62, de Gruyter, Berlin etc., 2015. doi: 10.1515/9783110365696.  Google Scholar [35] R. Kress, Linear Integral Equations ($3$rd ed.), Applied Mathematical Sciences 82, Springer, New York etc., 2014. doi: 10.1007/978-1-4614-9593-2.  Google Scholar [36] P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.  doi: 10.1016/j.jfa.2007.06.015.  Google Scholar [37] R. Luís, S. Elaydi and H. Oliveira, Local bifurcation in one-dimensional nonautonomous periodic difference equations, Int. J. Bifurcation Chaos, 23 (2013), 1350049, 18 pp. doi: 10.1142/S0218127413500491.  Google Scholar [38] F. Lutscher and M. A. Lewis, Spatially-explicit matrix models, J. Math. Biol., 48 (2004), 293-324.  doi: 10.1007/s00285-003-0234-6.  Google Scholar [39] F. Lutscher and S. Petrovskii, The importance of census times in discrete-dime growth-dispersal models, J. Biol. Dynamics, 2 (2008), 55-63.  doi: 10.1080/17513750701769899.  Google Scholar [40] F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics 49, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.  Google Scholar [41] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics 11, John Wiley & Sons, Chichester etc., 1976.  Google Scholar [42] A. Pinkus, Spectral properties of totally positive kernels and matrices, in Total Positivity and Its Applications (M. Gasca et al., eds.), Mathematics and Its Applications, 359, Kluwer, Dordrecht (1996), 477–511. doi: 10.1007/978-94-015-8674-0_23.  Google Scholar [43] C. Pötzsche, Bifurcations in a periodic discrete-time environment, Nonlinear Analysis: Real World Applications, 14 (2013), 53-82.  doi: 10.1016/j.nonrwa.2012.05.002.  Google Scholar [44] ————, Numerical dynamics of integrodifference equations: Basics and discretization errors in a $C^0$-setting, Appl. Math. Comput., 354 (2019), 422-443.  doi: 10.1016/j.amc.2019.02.033.  Google Scholar [45] C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, Preprint, 2020. Google Scholar [46] I. K. Rana, An Introduction to Measure and Integration ($2$nd ed.), Graduate Studies in Mathematics 45, American Mathematical Society, Providence RI, 2002. doi: 10.1090/gsm/045.  Google Scholar [47] J. R. Reimer, M. B. Bonsall and P. K. Maini, Approximating the critical domain size of integrodifference equations, Bull. Math. Biol., 78 (2016), 72-109.  doi: 10.1007/s11538-015-0129-x.  Google Scholar [48] S. L. Robertson and J. M. Cushing, A bifurcation analysis of stage-structured density dependent integrodifference equations, J. Math. Anal. Appl., 388 (2012), 490-499.  doi: 10.1016/j.jmaa.2011.09.064.  Google Scholar [49] J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar [50] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.   Google Scholar [51] H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84.  doi: 10.1007/BF01309313.  Google Scholar [52] ————, Discrete time population dynamics on the state space of measures, Math. Biosci. ngin., 17 (2020), 1168-1217. Google Scholar [53] R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bull. Math. Biol., 59 (1997), 107-137.   Google Scholar [54] D. S. Watkins, The Matrix Eigenvalue Problem –- GR and Krylov Subspace Methods, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898717808.  Google Scholar [55] R. Weiss, On the approximation of fixed points of nonlinear compact operators, SIAM J. Numer. Anal., 11 (1974), 550-553.  doi: 10.1137/0711046.  Google Scholar [56] E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Applied Mathematical Sciences 109, Springer, Heidelberg, 1995.  Google Scholar [57] X.-Q. Zhao, Dynamical Systems in Population Biology (2nd ed.), CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar [58] Y. Zhou and W. F. Fagan, A discrete-time model for population persistence in habitats with time-varying sizes, J. Math. Biol., 75 (2017), 649-704.  doi: 10.1007/s00285-017-1095-8.  Google Scholar

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##### References:
 [1] C. Aarset and C. Pötzsche, Bifurcations in periodic integrodifference equations in $C(\Omega)$ II: Discrete torus bifurcations, Commun. Pure Appl. Anal., 19 (2020), 1847-1874.   Google Scholar [2] M. Y. M. Alzoubi, The net reproductive number and bifurcation in an integro-difference system of equations, Appl. Math. Sci., 4 (2010), 191-200.   Google Scholar [3] H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, Studies in Mathematics 13, Walter de Gruyter, Berlin-New York, 1990. doi: 10.1515/9783110853698.  Google Scholar [4] M. Andersen, Properties of some density-dependent integrodifference equation population models, Math. Biosci., 104 (1991), 135-157.  doi: 10.1016/0025-5564(91)90034-G.  Google Scholar [5] T. Ando, Totally positive matrices, Linear Algebra Appl., 90 (1987), 165-219.  doi: 10.1016/0024-3795(87)90313-2.  Google Scholar [6] P. Anselone and J. Lee, Spectral properties of integral operators with nonnegative kernels, Linear Algebra Appl., 9 (1974), 67-87.  doi: 10.1016/0024-3795(74)90027-5.  Google Scholar [7] K. Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Numer. Anal., 12 (1975), 213-222.  doi: 10.1137/0712020.  Google Scholar [8] K. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. Integr. Equat. Appl., 4 (1992), 15-46.  doi: 10.1216/jiea/1181075664.  Google Scholar [9] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Monographs on Applied and Comp. Mathematics 4, University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511626340.  Google Scholar [10] N. Bacaër, Periodic matrix population models: Growth rate, basic reproduction number, and entropy, Bull. Math. Biol., 71 (2009), 1781-1792.  doi: 10.1007/s11538-009-9426-6.  Google Scholar [11] N. Bacaër and E. H. Ait Dads, On the biological interpretation of a definition for the parameter $R_0$ in periodic population models, J. Math. Biol., 65 (2012), 601-621.  doi: 10.1007/s00285-011-0479-4.  Google Scholar [12] W.-J. Beyn, T. Hüls and M.-C. Samtenschnieder., On $r$-periodic orbits of $k$-periodic maps, J. Difference Equ. Appl., 14 (2008), 865-887.  doi: 10.1080/10236190801940010.  Google Scholar [13] J. Bramburger and F. Lutscher, Analysis of integrodifference equations with a separable dispersal kernel, Acta Applicandae Mathematicae, 161 (2019), 127-151.  doi: 10.1007/s10440-018-0207-9.  Google Scholar [14] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second edition. Texts in Applied Mathematics, 40. Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar [15] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation: An Introduction, University Press, Princeton NJ, 2003.  doi: 10.1515/9781400884339.  Google Scholar [16] D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980.  Google Scholar [17] J. M. Cushing and A. S. Ackleh, A net reproductive number for periodic matrix models, J. Biol. Dyn., 6 (2012), 166-188.  doi: 10.1080/17513758.2010.544410.  Google Scholar [18] J. M. Cushing and S. M. Henson, Periodic matrix models for seasonal dynamics of structured populations with application to a seabird population, J. Math. Biol., 77 (2018), 1689-1720.  doi: 10.1007/s00285-018-1211-4.  Google Scholar [19] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar [20] ————, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.  Google Scholar [21] S. Day, O. Junge and K. Mischaikow, A rigerous numerical method for the global dynamics of infinite-dimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3(2) (2004), 117-160.  doi: 10.1137/030600210.  Google Scholar [22] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin etc., 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [23] G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Berlin etc., 1996.  Google Scholar [24] I. Győri and M. Pituk, The converse of the theorem on stability by the first approximation for difference equations, Nonlin. Analysis (TMA), 47 (2001), 4635-4640.  doi: 10.1016/S0362-546X(01)00576-4.  Google Scholar [25] D. P. Hardin, P. Takáč and G. F. Webb, A comparison of dispersal strategies for survival of spatially heterogeneous populations, SIAM J. Appl. Math., 48 (1988), 1396-1423.  doi: 10.1137/0148086.  Google Scholar [26] ————, Dispersion population models discrete in time and continuous in space, J. Math. Biol., 28 (1990), 1-20.  doi: 10.1007/BF00171515.  Google Scholar [27] G. Iooss, Bifurcation of Maps and Applications, Mathematics Studies 36, North-Holland, Amsterdam etc., 1979.  Google Scholar [28] J. Jacobsen and T. McAdam, A boundary value problem for integrodifference population models with cyclic kernels, Discrete Contin. Dyn. Syst. (Series B), 19 (2014), 3191-3207.  doi: 10.3934/dcdsb.2014.19.3191.  Google Scholar [29] W. Jin and H. R. Thieme, An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Discrete Contin. Dyn. Syst. (Series B), 21 (2016), 447-470.  doi: 10.3934/dcdsb.2016.21.447.  Google Scholar [30] T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin etc., 1980. Google Scholar [31] C. Kelley, Solving Nonlinear Equations with Newton's Method, Fundamentals of Algorithms 1, SIAM, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718898.  Google Scholar [32] H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs (2nd ed.), Applied Mathematical Sciences 156, Springer, New York etc., 2012. doi: 10.1007/978-1-4614-0502-3.  Google Scholar [33] M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar [34] U. Krause, Positive Dynamical Systems in Discrete Time, Studies in Mathematics 62, de Gruyter, Berlin etc., 2015. doi: 10.1515/9783110365696.  Google Scholar [35] R. Kress, Linear Integral Equations ($3$rd ed.), Applied Mathematical Sciences 82, Springer, New York etc., 2014. doi: 10.1007/978-1-4614-9593-2.  Google Scholar [36] P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.  doi: 10.1016/j.jfa.2007.06.015.  Google Scholar [37] R. Luís, S. Elaydi and H. Oliveira, Local bifurcation in one-dimensional nonautonomous periodic difference equations, Int. J. Bifurcation Chaos, 23 (2013), 1350049, 18 pp. doi: 10.1142/S0218127413500491.  Google Scholar [38] F. Lutscher and M. A. Lewis, Spatially-explicit matrix models, J. Math. Biol., 48 (2004), 293-324.  doi: 10.1007/s00285-003-0234-6.  Google Scholar [39] F. Lutscher and S. Petrovskii, The importance of census times in discrete-dime growth-dispersal models, J. Biol. Dynamics, 2 (2008), 55-63.  doi: 10.1080/17513750701769899.  Google Scholar [40] F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics 49, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.  Google Scholar [41] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics 11, John Wiley & Sons, Chichester etc., 1976.  Google Scholar [42] A. Pinkus, Spectral properties of totally positive kernels and matrices, in Total Positivity and Its Applications (M. Gasca et al., eds.), Mathematics and Its Applications, 359, Kluwer, Dordrecht (1996), 477–511. doi: 10.1007/978-94-015-8674-0_23.  Google Scholar [43] C. Pötzsche, Bifurcations in a periodic discrete-time environment, Nonlinear Analysis: Real World Applications, 14 (2013), 53-82.  doi: 10.1016/j.nonrwa.2012.05.002.  Google Scholar [44] ————, Numerical dynamics of integrodifference equations: Basics and discretization errors in a $C^0$-setting, Appl. Math. Comput., 354 (2019), 422-443.  doi: 10.1016/j.amc.2019.02.033.  Google Scholar [45] C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, Preprint, 2020. Google Scholar [46] I. K. Rana, An Introduction to Measure and Integration ($2$nd ed.), Graduate Studies in Mathematics 45, American Mathematical Society, Providence RI, 2002. doi: 10.1090/gsm/045.  Google Scholar [47] J. R. Reimer, M. B. Bonsall and P. K. Maini, Approximating the critical domain size of integrodifference equations, Bull. Math. Biol., 78 (2016), 72-109.  doi: 10.1007/s11538-015-0129-x.  Google Scholar [48] S. L. Robertson and J. M. Cushing, A bifurcation analysis of stage-structured density dependent integrodifference equations, J. Math. Anal. Appl., 388 (2012), 490-499.  doi: 10.1016/j.jmaa.2011.09.064.  Google Scholar [49] J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar [50] M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756.   Google Scholar [51] H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84.  doi: 10.1007/BF01309313.  Google Scholar [52] ————, Discrete time population dynamics on the state space of measures, Math. Biosci. ngin., 17 (2020), 1168-1217. Google Scholar [53] R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bull. Math. Biol., 59 (1997), 107-137.   Google Scholar [54] D. S. Watkins, The Matrix Eigenvalue Problem –- GR and Krylov Subspace Methods, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898717808.  Google Scholar [55] R. Weiss, On the approximation of fixed points of nonlinear compact operators, SIAM J. Numer. Anal., 11 (1974), 550-553.  doi: 10.1137/0711046.  Google Scholar [56] E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Applied Mathematical Sciences 109, Springer, Heidelberg, 1995.  Google Scholar [57] X.-Q. Zhao, Dynamical Systems in Population Biology (2nd ed.), CMS Books in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar [58] Y. Zhou and W. F. Fagan, A discrete-time model for population persistence in habitats with time-varying sizes, J. Math. Biol., 75 (2017), 649-704.  doi: 10.1007/s00285-017-1095-8.  Google Scholar
First critical values $\alpha_i^0$ for bifurcations from the trivial branch (left). Nontrivial branch $\phi_0^0$ of the primary transcritical bifurcation at $\alpha_0^0\approx 1.74$ for the Beverton-Holt growth-dispersal IDE with right-hand side (5.12), Laplace kernel (3.10) and $a = 1$, $L = 2$ (center). Total population over $\alpha\in[0,6]$ along $\phi_0^0$ (right)
First critical values $\alpha_i^0$ for bifurcations from the trivial branch (left). Nontrivial branch $\phi_0^0$ of the primary transcritical bifurcation at $\alpha_0^0\approx 1.74$ for the Beverton-Holt dispersal-growth IDE with right-hand side (5.17), Laplace kernel (3.10) and $a = 1$, $L = 1$ (center). Total population over $\alpha\in[0,6]$ along $\phi_0^0$ (right)
First critical values $\alpha_i^1$ for bifurcations along the primary branch $\phi_0^0$ (left). The $\alpha$-dependence of the corresponding eigenvalues $\lambda^1_{i-1}(\alpha)$ for $\alpha\geq 0.1$ (center). Period doubling cascade for the Ricker IDE with right-hand side (5.18) and $\alpha\in[0,40]$ for the Gauß kernel (5.19) with $a=1$, $L=2$ (right)
Equilibrium branches $\phi^\pm(aL)$ as functions of the dispersal parameter $\alpha=aL$ (left). Four largest eigenvalues $\lambda^\pm(aL)$ along these two branches of nontrivial solutions to (3.11) (right)
Subcritical ($\tfrac{g_{20}}{g_{11}}>0$) and supercritical ($\tfrac{g_{20}}{g_{11}}<0$) fold bifurcation of $\theta$-periodic solutions to $\left(\Delta_{\alpha}\right)$ described in Thm. 4.2, as well as the exchange of stability between the branches $\Gamma^+$ and $\Gamma^-$ from unstable (dashed line) to exponentially stable (solid) covered in Cor. 4.3
Transcritical bifurcation of $\theta$-periodic solutions to $\left(\Delta_{\alpha}\right)$ from a branch $\Gamma_1$ into $\Gamma_2$ described in Prop. 4.6, as well as the exchange of stability from unstable (dashed line) to exponentially stable (solid)
Subcritical $(\bar g/g_{11}>0)$ and supercritical $(\bar g/g_{11}<0)$ pitchfork bifurcation of $\theta$-periodic solutions to $\left(\Delta_{\alpha}\right)$ from a branch $\Gamma_1$ into $\Gamma_2$ described in Prop. 4.7, as well as the exchange of stability from unstable (dashed line) to exponentially stable (solid)
Branch of the subcritical fold bifurcation for $\left(\Delta_{\alpha}\right)$ with right-hand side (5.2) and kernel (5.1) (left). Total population over $\alpha\in[0.0,0.3]$ with $a = \tfrac{1}{4}$, $L = 2$ along the branch (right)
Schematic bifurcation diagrams for the cosine kernel (5.1) with $aL\leq\tfrac{1}{2}$ illustrating branches of $\theta$-periodic solutions: Ex. 4 has a subcritical fold bifurcation of fixed points at $(\phi^\ast,\alpha^\ast)$ (left). After a supercritical period doubling at $(0,\alpha_0^0)$, Ex. 5 has a supercritical pitchfork bifurcation of $2$-periodic solutions at $(\phi_\pm(\alpha_1^0),\alpha_1^0)$ (right). Fixed point branches are solid $(\theta = 1)$, branches of $2$-periodic solutions are dashed $(\theta = 2)$ and $4$-periodic solutions are indicated by dotted lines $(\theta = 4)$
Branch $\phi_1^0$ of the supercritical pitchfork bifurcation at $\alpha_1^0\approx 5.12$ for the Beverton-Holt growth-dispersal IDE with right-hand side (5.12), Laplace kernel (3.10) and $a = 1$, $L = 2$ (left). Total population over $\alpha\in[\alpha_1^0,7]$ along $\phi_1^0$ (right)
Branch $\phi_2^0$ of the transcritical bifurcation at $\alpha_2^0\approx 12.73$ for the Beverton-Holt growth-dispersal IDE with right-hand side (5.12), Laplace kernel (3.10) and $a = 1$, $L = 2$ as part of a fold (left). Total population over parameters $\alpha\in[12.4, 15]$ reflecting a fold in $\phi_2^0$ (right)
Branch $\phi_1^0$ of the supercritical pitchfork bifurcation at $\alpha_1^0\approx 5.12$ for the Beverton-Holt dispersal-growth IDE with right-hand side (5.17), Laplace kernel (3.10) and $a = 1$, $L = 2$ (left). Total population over $\alpha\in[5,6]$ along $\phi_1^0$ (right)
Branch $\phi_2^0$ of the transcritical bifurcation at $\alpha_2^0\approx 12.73$ for the Beverton-Holt dispersal-growth IDE with right-hand side (5.17), Laplace kernel (3.10) and $a=1$, $L=2$ as part of a fold (left). Total population over parameters $\alpha\in[12,13]$ reflecting a fold in $\phi_2^0$ (right)
Schematic bifurcation diagrams: Branches of $\theta$-periodic solutions in the Beverton-Holt IDE having the right-hand sides (5.12)/(5.17), $\alpha\in[0,50]$ and the Laplace kernel (5.19) with $a=1$, $L=2$ (left), as well as IDE with right-hand side (5.18) and $\alpha\in[0,3000]$ (logarithmic axis) for the Gauß kernel (5.19) with $a=1$, $L=2$ (right). Fixed point branches are solid $(\theta=1)$, branches of $2$-periodic solutions are dashed $(\theta=2)$ and $4$-periodic solutions are indicated by dotted lines $(\theta=4)$
First critical values $\alpha_i^0$ for bifurcations from the trivial branch (left). The $\alpha$-dependence of the corresponding eigenvalues $\lambda_{i-1}^0(\alpha)$ (center). Primary transcritical bifurcation of the branch $\phi_0^0$ at $\alpha^0_0\approx 1.36$ for the Gauß kernel (5.19) with $a=1$, $L=2$ (right)
Coefficients in commonly used growth functions $\hat g$, $c>0$
 growth function $\hat g(z)$ $c_2$ $d_2$ $c_3$ $d_3$ logistic $z(1-z)$ $-2$ $-2$ $0$ $0$ Hassell $\tfrac{z}{(1+z)^c}$ $-2c$ $-2c$ $3(1+c)c$ $3(1+c)c$ Ricker $ze^{-z}$ $-2$ $-2$ $3$ $3$
 growth function $\hat g(z)$ $c_2$ $d_2$ $c_3$ $d_3$ logistic $z(1-z)$ $-2$ $-2$ $0$ $0$ Hassell $\tfrac{z}{(1+z)^c}$ $-2c$ $-2c$ $3(1+c)c$ $3(1+c)c$ Ricker $ze^{-z}$ $-2$ $-2$ $3$ $3$
Quadrature rules (B.1) with $h: = \tfrac{b-a}{n}$
 rule nodes $\eta_j$ weights $w_j$ $r$ $N_n$ midpoint $a+h(j-\tfrac{1}{2})$ $h$ $2$ $n$ trapezoidal $a+h(j-1)$ $\tfrac{h}{2}\text{ for }j\in\left\{{1,n+1}\right\}$ $2$ $n+1$ $h$ else Chebyshev $a+(j-\tfrac{\sqrt{3}+1}{2\sqrt{3}})h$ for $j\leq n$ $\tfrac{h}{2}$ $4$ $2n$ $a+(j-n+\tfrac{\sqrt{3}-1}{2\sqrt{3}})h$ for n < j
 rule nodes $\eta_j$ weights $w_j$ $r$ $N_n$ midpoint $a+h(j-\tfrac{1}{2})$ $h$ $2$ $n$ trapezoidal $a+h(j-1)$ $\tfrac{h}{2}\text{ for }j\in\left\{{1,n+1}\right\}$ $2$ $n+1$ $h$ else Chebyshev $a+(j-\tfrac{\sqrt{3}+1}{2\sqrt{3}})h$ for $j\leq n$ $\tfrac{h}{2}$ $4$ $2n$ $a+(j-n+\tfrac{\sqrt{3}-1}{2\sqrt{3}})h$ for n < j
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