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  January 2021 Early access  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 and 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. 

[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. 

[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.

[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.

[5]

T. Ando, Totally positive matrices, Linear Algebra Appl., 90 (1987), 165-219.  doi: 10.1016/0024-3795(87)90313-2.

[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.

[7]

K. Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Numer. Anal., 12 (1975), 213-222.  doi: 10.1137/0712020.

[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.

[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.
[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.

[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.

[12]

W.-J. BeynT. 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.

[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.

[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.

[15] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation: An Introduction, University Press, Princeton NJ, 2003.  doi: 10.1515/9781400884339.
[16]

D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980.

[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.

[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.

[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.

[20]

————, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.

[21]

S. DayO. 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.

[22]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin etc., 1985. doi: 10.1007/978-3-662-00547-7.

[23]

G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Berlin etc., 1996.

[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.

[25]

D. P. HardinP. 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.

[26]

————, Dispersion population models discrete in time and continuous in space, J. Math. Biol., 28 (1990), 1-20.  doi: 10.1007/BF00171515.

[27]

G. Iooss, Bifurcation of Maps and Applications, Mathematics Studies 36, North-Holland, Amsterdam etc., 1979.

[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.

[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.

[30]

T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin etc., 1980.

[31]

C. Kelley, Solving Nonlinear Equations with Newton's Method, Fundamentals of Algorithms 1, SIAM, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718898.

[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.

[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.

[34]

U. Krause, Positive Dynamical Systems in Discrete Time, Studies in Mathematics 62, de Gruyter, Berlin etc., 2015. doi: 10.1515/9783110365696.

[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.

[36]

P. LiuJ. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.  doi: 10.1016/j.jfa.2007.06.015.

[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.

[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.

[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.

[40]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics 49, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.

[41]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics 11, John Wiley & Sons, Chichester etc., 1976.

[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.

[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.

[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.

[45]

C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, Preprint, 2020.

[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.

[47]

J. R. ReimerM. 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.

[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.

[49]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.

[50]

M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756. 

[51]

H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84.  doi: 10.1007/BF01309313.

[52]

————, Discrete time population dynamics on the state space of measures, Math. Biosci. ngin., 17 (2020), 1168-1217.

[53]

R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bull. Math. Biol., 59 (1997), 107-137. 

[54]

D. S. Watkins, The Matrix Eigenvalue Problem –- GR and Krylov Subspace Methods, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898717808.

[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.

[56]

E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Applied Mathematical Sciences 109, Springer, Heidelberg, 1995.

[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.

[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.

show all references

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. 

[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. 

[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.

[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.

[5]

T. Ando, Totally positive matrices, Linear Algebra Appl., 90 (1987), 165-219.  doi: 10.1016/0024-3795(87)90313-2.

[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.

[7]

K. Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Numer. Anal., 12 (1975), 213-222.  doi: 10.1137/0712020.

[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.

[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.
[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.

[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.

[12]

W.-J. BeynT. 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.

[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.

[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.

[15] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation: An Introduction, University Press, Princeton NJ, 2003.  doi: 10.1515/9781400884339.
[16]

D. Cohn, Measure Theory, Birkhäuser, Boston etc., 1980.

[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.

[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.

[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.

[20]

————, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal., 52 (1973), 161-180.  doi: 10.1007/BF00282325.

[21]

S. DayO. 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.

[22]

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin etc., 1985. doi: 10.1007/978-3-662-00547-7.

[23]

G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Berlin etc., 1996.

[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.

[25]

D. P. HardinP. 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.

[26]

————, Dispersion population models discrete in time and continuous in space, J. Math. Biol., 28 (1990), 1-20.  doi: 10.1007/BF00171515.

[27]

G. Iooss, Bifurcation of Maps and Applications, Mathematics Studies 36, North-Holland, Amsterdam etc., 1979.

[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.

[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.

[30]

T. Kato, Perturbation Theory for Linear Operators (corrected 2nd ed.), Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin etc., 1980.

[31]

C. Kelley, Solving Nonlinear Equations with Newton's Method, Fundamentals of Algorithms 1, SIAM, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718898.

[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.

[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.

[34]

U. Krause, Positive Dynamical Systems in Discrete Time, Studies in Mathematics 62, de Gruyter, Berlin etc., 2015. doi: 10.1515/9783110365696.

[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.

[36]

P. LiuJ. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.  doi: 10.1016/j.jfa.2007.06.015.

[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.

[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.

[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.

[40]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics 49, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.

[41]

R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics 11, John Wiley & Sons, Chichester etc., 1976.

[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.

[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.

[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.

[45]

C. Pötzsche and E. Ruß, Reduction principle for nonautonomous integrodifference equations at work, Preprint, 2020.

[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.

[47]

J. R. ReimerM. 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.

[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.

[49]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.

[50]

M. Slatkin, Gene flow and selection in a cline, Genetics, 75 (1973), 733-756. 

[51]

H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84.  doi: 10.1007/BF01309313.

[52]

————, Discrete time population dynamics on the state space of measures, Math. Biosci. ngin., 17 (2020), 1168-1217.

[53]

R. W. Van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats, Bull. Math. Biol., 59 (1997), 107-137. 

[54]

D. S. Watkins, The Matrix Eigenvalue Problem –- GR and Krylov Subspace Methods, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898717808.

[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.

[56]

E. Zeidler, Applied Functional Analysis: Main Principles and their Applications, Applied Mathematical Sciences 109, Springer, Heidelberg, 1995.

[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.

[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.

Figure 7.  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)
Figure 10.  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)
Figure 15.  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)
Figure 1.  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)
Figure 2.  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
Figure 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)
Figure 4.  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)
Figure 5.  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)
Figure 6.  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) $
Figure 8.  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)
Figure 9.  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)
Figure 11.  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)
Figure 12.  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)
Figure 13.  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) $
Figure 14.  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)
Table 1.  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 $
Table 2.  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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