Figure 1.
A water-plant interaction system with the infiltration feedback
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Minimization of carbon abatement cost: Modeling, analysis and simulation
September
2017, 22(7): 2971-3006.
doi: 10.3934/dcdsb.2017159
Interaction between water and plants: Rich dynamics in a simple model
| 1. | School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
| 2. | Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA |
An ordinary differential equation model describing interaction of water and plants in ecosystem is proposed. Despite its simple looking, it is shown that the model possesses surprisingly rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov-Takens bifurcation. We classify bifurcation diagrams of the system using the rain-fall rate as bifurcation parameter. In the transition from global stability of bare-soil state for low rain-fall to the global stability of high vegetation state for high rain-fall rate, oscillatory states or multiple equilibrium states can occur, which can be viewed as a new indicator of catastrophic environmental shift.
Keywords:
Water-plant model,
bistability,
Hopf bifurcations,
Bogdanov-Takens bifurcations,
homoclinic bifurcations,
closed loop of cycles.
Mathematics Subject Classification:
Primary:34C23, 34C60, 37G15, 92D40.
Citation:
Xiaoli Wang, Junping Shi, Guohong Zhang. Interaction between water and plants: Rich dynamics in a simple model. Discrete and Continuous Dynamical Systems - B,
2017, 22
(7)
: 2971-3006.
doi: 10.3934/dcdsb.2017159
![]()
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Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
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Rising variance: A leading indicator of ecological transition, Ecol. Lett., 9 (2006), 311-318.
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S.-N. Chow and J. Mallet-Paret,
The Fuller index and global Hopf bifurcation, J. Differential Equations, 29 (1978), 66-85.
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E. Gilad, M. Shachak and E. Meron,
Dynamics and spatial organization of plant communities in water-limited systems, Theor. popul. biol., 72 (2007), 214-230.
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E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: From pattern formation to habitat creation Phys. Rev. Lett. 93 (2004), 098105.
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K.P. Hadeler and P. van den Driessche,
Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.
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S.-B. Hsu and T.-W. Huang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
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S.-B. Hsu and T.-W. Huang,
Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.
|
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C.G. Jones, J.H. Lawton and M. Shachak,
Organisms as ecosystem engineers, Ecosystem Management, (1996), 130-147.
doi: 10.1007/978-1-4612-4018-1_14. |
| [13] |
C.G. Jones, J.H. Lawton and M. Shachak,
Positive and negative effects of organisms as physical ecosystem engineers, Ecology, 78 (1997), 1946-1957.
|
| [14] |
C.A. Klausmeier,
Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.
doi: 10.1126/science.284.5421.1826. |
| [15] |
A.J. Koch and H. Meinhardt,
Biological pattern formation: From basic mechanisms to complex structures, Rev. Mod. Phys., 66 (1994), 1481-1507.
doi: 10.1103/RevModPhys.66.1481. |
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Y. A. Kuznetsov, Elements of Applied Bifurcation Theory volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998. |
| [17] |
R. Lefever and O. Lejeune,
On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.
doi: 10.1016/S0092-8240(96)00072-9. |
| [18] |
R.-S. Liu, Z.-L. Feng, H.-P. Zhu and D.L. DeAngelis,
Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 245 (2008), 442-467.
doi: 10.1016/j.jde.2007.10.034. |
| [19] |
M.-X. Liu, E. Liz and G. Röst,
Endemic bubbles generated by delayed behavioral response: Global stability and bifurcation switches in an SIS model, SIAM J. Appl. Math., 75 (2015), 75-91.
doi: 10.1137/140972652. |
| [20] |
A. Manor and N.M. Shnerb,
Dynamical failure of Turing patterns, Europhys. Lett., 74 (2006), 837-843.
doi: 10.1209/epl/i2005-10580-5. |
| [21] |
R.M. May,
Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
doi: 10.1038/269471a0. |
| [22] |
H. Meinhardt,
Pattern formation in biology: A comparison of models and experiments, Rep. Prog. Phys., 55 (1992), 797-849.
doi: 10.1088/0034-4885/55/6/003. |
| [23] |
E. Meron, E. Gilad and J. von Hardenberg,
Vegetation patterns along a rainfall gradient, Chaos, Solitons Fract., 19 (2004), 367-376.
doi: 10.1016/S0960-0779(03)00049-3. |
| [24] |
L. Perko, Differential Equations and Dynamical Systems volume 7 of Texts in Applied Mathematics, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
| [25] |
M. Rietkerk, S.C. Dekker, P.C. De Ruiter and J. van de Koppel,
Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929.
doi: 10.1126/science.1101867. |
| [26] |
M. Scheffer, J. Bascompte and W.A. Brock,
Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.
doi: 10.1038/nature08227. |
| [27] |
M. Scheffer, S. Carpenter and J.A. Foley,
Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.
doi: 10.1038/35098000. |
| [28] |
N. M. Shnerb, P. Sarah, H. Lavee and S. Solomon, Reactive Glass and Vegetation Patterns Phys. Rev. Lett. 90 (2003), 038101.
doi: 10.1103/PhysRevLett. 90. 038101. |
| [29] |
H.-Y. Shu, L. Wang and J.-H. Wu,
Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.
doi: 10.1016/j.jde.2013.06.020. |
| [30] |
A.M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
| [31] |
J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification Phys. Rev. Lett. 87 (2001), 198101.
doi: 10.1103/PhysRevLett. 87. 198101. |
| [32] |
J.-L. Wang, S.-Q. Liu, B.-W. Zheng and Y. Takeuchi,
Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modelling, 55 (2012), 710-722.
doi: 10.1016/j.mcm.2011.08.045. |
| [33] |
J.-F. Wang, J.-P. Shi and J.-J. Wei,
Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
| [34] |
W.-D. Wang,
Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
| [35] |
W.-D. Wang and S.-G. Ruan,
Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793.
doi: 10.1016/j.jmaa.2003.11.043. |
| [36] |
X.-L. Wang, W.-D. Wang and G.-H. Zhang,
Global analysis of predator-prey system with hawk and dove tactics, Stud. Appl. Math., 124 (2010), 151-178.
doi: 10.1111/j.1467-9590.2009.00466.x. |
| [37] |
A.S. Watt,
Pattern and process in the plant community, J. Ecol., 35 (1947), 1-22.
doi: 10.2307/2256497. |
| [38] |
X. Zhang and X.-N. Liu,
Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.
doi: 10.1016/j.jmaa.2008.07.042. |
| [39] |
L.-H. Zhou and M. Fan,
Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), 312-324.
doi: 10.1016/j.nonrwa.2011.07.036. |
show all references
References:
| [1] |
J. C. Alexander and J. A. Yorke,
Global bifurcations of periodic orbits, Amer. J. Math., 100 (1978), 263-292.
doi: 10.2307/2373851. |
| [2] |
J. Arino, C.C. McCluskey and P. van den Driessche,
Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
| [3] |
F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology volume 40 of Texts in Applied Mathematics, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
| [4] |
S.R. Carpenter and W.A. Brock,
Rising variance: A leading indicator of ecological transition, Ecol. Lett., 9 (2006), 311-318.
doi: 10.1111/j.1461-0248.2005.00877.x. |
| [5] |
S.-N. Chow and J. Mallet-Paret,
The Fuller index and global Hopf bifurcation, J. Differential Equations, 29 (1978), 66-85.
doi: 10.1016/0022-0396(78)90041-4. |
| [6] |
E. Gilad, M. Shachak and E. Meron,
Dynamics and spatial organization of plant communities in water-limited systems, Theor. popul. biol., 72 (2007), 214-230.
doi: 10.1016/j.tpb.2007.05.002. |
| [7] |
E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: From pattern formation to habitat creation Phys. Rev. Lett. 93 (2004), 098105.
doi: 10.1103/PhysRevLett. 93. 098105. |
| [8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields volume 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
| [9] |
K.P. Hadeler and P. van den Driessche,
Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
| [10] |
S.-B. Hsu and T.-W. Huang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
| [11] |
S.-B. Hsu and T.-W. Huang,
Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.
|
| [12] |
C.G. Jones, J.H. Lawton and M. Shachak,
Organisms as ecosystem engineers, Ecosystem Management, (1996), 130-147.
doi: 10.1007/978-1-4612-4018-1_14. |
| [13] |
C.G. Jones, J.H. Lawton and M. Shachak,
Positive and negative effects of organisms as physical ecosystem engineers, Ecology, 78 (1997), 1946-1957.
|
| [14] |
C.A. Klausmeier,
Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.
doi: 10.1126/science.284.5421.1826. |
| [15] |
A.J. Koch and H. Meinhardt,
Biological pattern formation: From basic mechanisms to complex structures, Rev. Mod. Phys., 66 (1994), 1481-1507.
doi: 10.1103/RevModPhys.66.1481. |
| [16] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998. |
| [17] |
R. Lefever and O. Lejeune,
On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.
doi: 10.1016/S0092-8240(96)00072-9. |
| [18] |
R.-S. Liu, Z.-L. Feng, H.-P. Zhu and D.L. DeAngelis,
Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 245 (2008), 442-467.
doi: 10.1016/j.jde.2007.10.034. |
| [19] |
M.-X. Liu, E. Liz and G. Röst,
Endemic bubbles generated by delayed behavioral response: Global stability and bifurcation switches in an SIS model, SIAM J. Appl. Math., 75 (2015), 75-91.
doi: 10.1137/140972652. |
| [20] |
A. Manor and N.M. Shnerb,
Dynamical failure of Turing patterns, Europhys. Lett., 74 (2006), 837-843.
doi: 10.1209/epl/i2005-10580-5. |
| [21] |
R.M. May,
Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
doi: 10.1038/269471a0. |
| [22] |
H. Meinhardt,
Pattern formation in biology: A comparison of models and experiments, Rep. Prog. Phys., 55 (1992), 797-849.
doi: 10.1088/0034-4885/55/6/003. |
| [23] |
E. Meron, E. Gilad and J. von Hardenberg,
Vegetation patterns along a rainfall gradient, Chaos, Solitons Fract., 19 (2004), 367-376.
doi: 10.1016/S0960-0779(03)00049-3. |
| [24] |
L. Perko, Differential Equations and Dynamical Systems volume 7 of Texts in Applied Mathematics, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
| [25] |
M. Rietkerk, S.C. Dekker, P.C. De Ruiter and J. van de Koppel,
Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929.
doi: 10.1126/science.1101867. |
| [26] |
M. Scheffer, J. Bascompte and W.A. Brock,
Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.
doi: 10.1038/nature08227. |
| [27] |
M. Scheffer, S. Carpenter and J.A. Foley,
Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.
doi: 10.1038/35098000. |
| [28] |
N. M. Shnerb, P. Sarah, H. Lavee and S. Solomon, Reactive Glass and Vegetation Patterns Phys. Rev. Lett. 90 (2003), 038101.
doi: 10.1103/PhysRevLett. 90. 038101. |
| [29] |
H.-Y. Shu, L. Wang and J.-H. Wu,
Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.
doi: 10.1016/j.jde.2013.06.020. |
| [30] |
A.M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
| [31] |
J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification Phys. Rev. Lett. 87 (2001), 198101.
doi: 10.1103/PhysRevLett. 87. 198101. |
| [32] |
J.-L. Wang, S.-Q. Liu, B.-W. Zheng and Y. Takeuchi,
Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modelling, 55 (2012), 710-722.
doi: 10.1016/j.mcm.2011.08.045. |
| [33] |
J.-F. Wang, J.-P. Shi and J.-J. Wei,
Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
| [34] |
W.-D. Wang,
Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
| [35] |
W.-D. Wang and S.-G. Ruan,
Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793.
doi: 10.1016/j.jmaa.2003.11.043. |
| [36] |
X.-L. Wang, W.-D. Wang and G.-H. Zhang,
Global analysis of predator-prey system with hawk and dove tactics, Stud. Appl. Math., 124 (2010), 151-178.
doi: 10.1111/j.1467-9590.2009.00466.x. |
| [37] |
A.S. Watt,
Pattern and process in the plant community, J. Ecol., 35 (1947), 1-22.
doi: 10.2307/2256497. |
| [38] |
X. Zhang and X.-N. Liu,
Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.
doi: 10.1016/j.jmaa.2008.07.042. |
| [39] |
L.-H. Zhou and M. Fan,
Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), 312-324.
doi: 10.1016/j.nonrwa.2011.07.036. |
Figure 2.
The graph of biomass per capita death rate $\mu(b)=\mu_0+\frac{\mu_1}{b+1}$ . Here $\mu_0=0.5$ , $\mu_1=2$ , and $0\le b\le 10$
Figure 3.
Possible bifurcation diagrams for (1) - (2) for different $(\lambda ,{\mu _0},{\mu _1})$ . In all diagrams, the horizontal axis is the rain-fall rate $R$ , and the vertical axis is the biomass $b$ . (a) forward bifurcation and no cycle; (b) forward bifurcation and bubble loop of cycles; (c) backward bifurcation and no cycle; (d) backward bifurcation and bubble loop of cycles; (e) backward bifurcation, and bubble loop of cycles with one subcritical and one supercritical Hopf bifurcations and a saddle-node bifurcation of cycles; (f) backward bifurcation, subcritical Hopf bifurcation, and branch of cycles into a homoclinic bifurcation; (g) backward bifurcation, supercritical Hopf bifurcation, and branch of cycles into a homoclinic bifurcation; (h) backward bifurcation, subcritical Hopf bifurcation, saddle-node bifurcation of cycles and branch of cycles into a homoclinic bifurcation
Figure 4.
Illustration of the stability parameter subregions in the $\mu_0-\mu_1$ plane. (Left) $0< \lambda<1$ ; (Right) $\lambda\ge 1$
Figure 5.
Bifurcation diagrams (bubble branch of cycles) and phase portraits of limit cycles when there are two Hopf bifurcation points $R=R_2$ and $R=R_3$ . In (a), (c) and (e), the blue curve, the cyan curve and the purple curve represent the biomass $b$ , the minimum value and the maximum value of $b$ of the limit cycles versus the rain-fall rate $R$ , respectively; corresponding phase portraits are shown in (b), (d) and (f). In (a) and (b) $\lambda=0.2$ , $(\mu_0, \mu_1)=(7, 4.7)\in \textrm{Ⅲ}$ , and $R=11.14$ in (b); in (c) and (d) $\lambda=0.2$ , $(\mu_0, \mu_1)=(20, 4.8)\in \textrm{V}$ , and $R=26$ in (d); and in (e) and (f) $\lambda=1.2$ , $(\mu_0, \mu_1)=(5, 8)\in \textrm{Ⅶ}$ , and $R=16$ in (f)
Figure 6.
Hopf bifurcation and multiple limit cycles when $\lambda=0.2$ and $(\mu_0, \mu_1)\in \textrm{Ⅲ}$ . (a): The bifurcation diagram when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ . The blue curve, the cyan curve and the purple curve represent the biomass $b$ , the minimum value and the maximum value of $b$ of the limit cycles vs. the rain-fall rate $R$ , respectively. (b): Two limit cycles when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ and $R=31.4$ . (c): Period of limit cycles versus $R$ when $(\mu_0, \mu_1)=(7, 4.7)\in \textrm{Ⅲ}$ , and the period is monotonically decreasing in $R$ . (d): Period of limit cycles versus $R$ when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ , and the period is not monotone in $R$ and is not single-valued (indicating multiple limit cycles). (e): Time series of the small amplitude periodic orbit when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ and $R=31.4$ . (f): Time series of the large amplitude periodic orbit when $(\mu_0, \mu_1)=(20, 6.5)\in \textrm{Ⅲ}$ and $R=31.4$
Figure 7.
Homoclinic bifurcation, saddle-node bifurcation, Hopf bifurcation and the existence of one limit cycle when $\lambda=0.2$ and $(\mu_0,\mu_1)\in \textrm{Ⅱ}$ . In (a), (c), the blue curve, the cyan curve and the purple curve represent the biomass $b$ , the minimum value and the maximum value of $b$ of the limit cycles vs. the rain-fall rate $R$ , respectively. (a): Bifurcation diagram when $(\mu_0,\mu_1)=(5,5.2)\in \textrm{Ⅱ}$ . (b): Phase portrait with periodic orbit when $(\mu_0,\mu_1)=(5,5.2)$ and $R=9.17$ . (c): Bifurcation diagram when $(\mu_0,\mu_1)=(7,4.9)\in \textrm{Ⅱ}$ . (d): Phase portrait with periodic orbit when $(\mu_0, \mu_1)=(7,4.9)$ and $R=11.38$ . (e): Period of the stable periodic orbits when $(\mu_0,\mu_1)=(5,5.2)$ . (f): Period of the unstable limit cycles when $(\mu_0,\mu_1)=(7,4.9)$
Figure 8.
Evolution of phase portraits of (3) for $\lambda=0.2$ , $(\mu_0, \mu_1)=(5,5.2)\in \textrm{Ⅱ}$ and $R>R_1$ . (a) $R=9.14$ ; (b) $R=9.17$ ; (c) $R=9.1830408$ ; (d) $R=9.19$
Figure 9.
Evolution of phase portraits of (3) for $\lambda=0.2$ and $(\mu_0, \mu_1)=(7,4.9)\in \textrm{Ⅱ}$ and $R>R_1$ . (a) $R=11.3$ ; (b) $R=11.363757$ ; (c) $R=11.38$ ; (d) $R=11.4$
Figure 10.
Homoclinic bifurcation, saddle-node bifurcation, Hopf bifurcation and multiple limit cycles for $\lambda=0.2$ and $(\mu_0,\mu_1)=(7,5)\in \textrm{Ⅱ}$ . (a): The bifurcation diagram. The blue curve, the cyan curve and the purple curve represent the biomass $b$ bifurcation diagram, the minimum value and the maximum value of $b$ of the limit cycles vs. the rain-fall rate $R$ , respectively. (b): Phase portrait with one stable limit cycle when $R=11.48$ . (c): Period of limit cycles versus $R$ . (d): Phase portrait with two limit cycles when $R=11.498$ . (e): Time series of the small amplitude periodic orbit when $R=11.498$ . (f): Time series of the large amplitude periodic orbit when $R=11.498$
Figure 11.
Evolution of phase portraits of (3) for $\lambda=0.2$ , $(\mu_0, \mu_1)=(7, 5)\in \textrm{Ⅱ}$ and $R>R_1$ . (a) $R=11.4$ ; (b) $R=11.47377$ ; (c) $R=11.48$ ; (d) $R=11.498$ ; (e) $R=11.5$ ; (f) $R=12$
Figure 12.
The cyan curve represents the Hopf bifurcation curve, the blue curve represents the saddle-node bifurcation curve and the black line is $\mu_1=R-\mu_0$ . The "BT" mark indicates a Bogdanov-Takens bifurcation point; the "CP" mark indicates a cusp bifurcation point; and the "GH" mark indicates a generalized Hopf point where the first Lyapunov coefficient vanishes while the second Lyapunov coefficient does not vanish, which indicates that it is nondegenerate, i.e. Hopf bifurcation changes from subcritical to supercritical. (a): $\lambda=0.2, \mu_0=5$ . (b): $\lambda=0.2, \mu_0=7$ . (c): $\lambda=0.2, \mu_0=20$ . (d): $\lambda=1.2, \mu_0=5$
Table 1.
Results of bifurcation points
| λ | (µ0, µ1) | Transcritical at R = R0 | Saddle-node bifurcation | Hopf bifurcation points | Homoclinic bifurcation | B-T bifurcation |
| 0 < λ < 1 | Ⅰ | backward | R1 | none | none | R1 |
| 0 < λ < 1 | Ⅱ | backward | R1 | R3 | exist | R1 |
| 0 < λ < 1 | Ⅲ | backward | R1 | R2, R3 | none | R1 |
| 0 < λ < 1 | Ⅳ | forward | none | none | none | none |
| 0 < λ < 1 | Ⅴ | forward | none | R2, R3 | none | none |
| λ ≥ 1 | Ⅵ | forward | none | none | none | none |
| λ ≥ 1 | Ⅶ | forward | none | R2, R3 | none | none |
| λ | (µ0, µ1) | Transcritical at R = R0 | Saddle-node bifurcation | Hopf bifurcation points | Homoclinic bifurcation | B-T bifurcation |
| 0 < λ < 1 | Ⅰ | backward | R1 | none | none | R1 |
| 0 < λ < 1 | Ⅱ | backward | R1 | R3 | exist | R1 |
| 0 < λ < 1 | Ⅲ | backward | R1 | R2, R3 | none | R1 |
| 0 < λ < 1 | Ⅳ | forward | none | none | none | none |
| 0 < λ < 1 | Ⅴ | forward | none | R2, R3 | none | none |
| λ ≥ 1 | Ⅵ | forward | none | none | none | none |
| λ ≥ 1 | Ⅶ | forward | none | R2, R3 | none | none |
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