Theory of rotated equations and applications to a population model

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April 2018, 38(4): 2171-2185. doi: 10.3934/dcds.2018089

Theory of rotated equations and applications to a population model

Maoan Han ^1,2, , Xiaoyan Hou ^3, , Lijuan Sheng ^3,, and Chaoyang Wang ^4,

1.	Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2.	School of Mathematics Sciences, Qufu Normal University, Qufu 273165, China
3.	Department of Mathematics, Shanghai Normal University Shanghai, Shanghai 200234, China
4.	College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

* Corresponding author: Lijuan Sheng

Received June 2017 Revised July 2017 Published January 2018

Fund Project: The first author is supported by National Natural Science Foundation of China (11431008 and 11771296)

We consider a family of scalar periodic equations with a parameter and establish theory of rotated equations, studying the behavior of periodic solutions with the change of the parameter. It is shown that a stable (completely unstable) periodic solution of a rotated equation varies monotonically with respect to the parameter and a semi-stable periodic solution splits into two periodic solutions or disappears as the parameter changes in one direction or another. As an application of the obtained results, we give a further study of a piecewise smooth population model verifying the existence of saddle-node bifurcation.

Keywords: Periodic solution, rotated equation, saddle-node bifurcation.

Mathematics Subject Classification: Primary: 37C10; Secondary: 34K18.

Citation: Maoan Han, Xiaoyan Hou, Lijuan Sheng, Chaoyang Wang. Theory of rotated equations and applications to a population model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2171-2185. doi: 10.3934/dcds.2018089

References:

[1]	D. Batenkov and G. Binyamini, Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation, J. Differential Equations, 259 (2015), 5769-5781. doi: 10.1016/j.jde.2015.07.009.
[2]	F. Brauer and D. A. Sanchez, Periodic environments and periodic harvesting, Natural Resource Modeling, 16 (2003), 233-244.
[3]	F. Brauer and D. A. Sanchez, Constant rate population harvesting: Equilibrium and stability, Theoret. Pop. biol., 8 (1975), 12-30. doi: 10.1016/0040-5809(75)90036-2.
[4]	D. Campbell and S. R. Kaplan, A bifurcation problem in differential equations, Math. Mag., 73 (2000), 194-203. doi: 10.2307/2691522.
[5]	G. F. D. Duff, Limit cycles and rotated vector fields, Ann.of Math., 57 (1953), 15-31. doi: 10.2307/1969724.
[6]	J. Giné, J. Llibre, K. Wu and X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations, 260 (2016), 4130-4156. doi: 10.1016/j.jde.2015.11.005.
[7]	M. Han and D. Zhu, Bifurcation Theory of Differential Equation, Coal Industry Publishing House, Beijing, 1994.
[8]	M. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Alpha Science International Ltd., Oxford, 2017.
[9]	M. Han, Global behavior of limit cycles in rotated vector fields, Journal of Differential Equations, 151 (1999), 20-35. doi: 10.1006/jdeq.1998.3508.
[10]	P. Liu, J. Shi and Y. Wang, Periodic solutions of a logistic type population model with harvesting, J. Math. Anal. Appl., 369 (2010), 730-735. doi: 10.1016/j.jmaa.2010.04.027.
[11]	S. Oruganti, J. Shi and R. Shivaji, Diffusive logistic equaiton with constant yield harvesting, I. Steady States, Trans. Amer. Math. Soc., 354 (2002), 3601-3619. doi: 10.1090/S0002-9947-02-03005-2.
[12]	D. Xiao, Dynamics and bifurcation on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst., 21 (2016), 699-719.

show all references

References:

[1]	D. Batenkov and G. Binyamini, Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation, J. Differential Equations, 259 (2015), 5769-5781. doi: 10.1016/j.jde.2015.07.009.
[2]	F. Brauer and D. A. Sanchez, Periodic environments and periodic harvesting, Natural Resource Modeling, 16 (2003), 233-244.
[3]	F. Brauer and D. A. Sanchez, Constant rate population harvesting: Equilibrium and stability, Theoret. Pop. biol., 8 (1975), 12-30. doi: 10.1016/0040-5809(75)90036-2.
[4]	D. Campbell and S. R. Kaplan, A bifurcation problem in differential equations, Math. Mag., 73 (2000), 194-203. doi: 10.2307/2691522.
[5]	G. F. D. Duff, Limit cycles and rotated vector fields, Ann.of Math., 57 (1953), 15-31. doi: 10.2307/1969724.
[6]	J. Giné, J. Llibre, K. Wu and X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations, 260 (2016), 4130-4156. doi: 10.1016/j.jde.2015.11.005.
[7]	M. Han and D. Zhu, Bifurcation Theory of Differential Equation, Coal Industry Publishing House, Beijing, 1994.
[8]	M. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Alpha Science International Ltd., Oxford, 2017.
[9]	M. Han, Global behavior of limit cycles in rotated vector fields, Journal of Differential Equations, 151 (1999), 20-35. doi: 10.1006/jdeq.1998.3508.
[10]	P. Liu, J. Shi and Y. Wang, Periodic solutions of a logistic type population model with harvesting, J. Math. Anal. Appl., 369 (2010), 730-735. doi: 10.1016/j.jmaa.2010.04.027.
[11]	S. Oruganti, J. Shi and R. Shivaji, Diffusive logistic equaiton with constant yield harvesting, I. Steady States, Trans. Amer. Math. Soc., 354 (2002), 3601-3619. doi: 10.1090/S0002-9947-02-03005-2.
[12]	D. Xiao, Dynamics and bifurcation on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst., 21 (2016), 699-719.

Figure 1. Local behavior of stable or completely unstable periodic solution

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Figure 2. Local behavior of semi-stable periodic solution

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Figure 3. Asymptotic behavior of solutions for $\lambda>\lambda_0$

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Figure 4. Local behavior near a semi-stable periodic solution

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Figure 5. Behavior of solutions as $\lambda>\lambda^*$ and $x_0>K$ or $x_0<0$

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Figure 6. Behavior of $\varphi_s(t,\lambda)$ and $\varphi_u(t,\lambda)$

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Figure 7. Behavior of solutions as $\lambda<0$

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Table 1. Behavior of periodic solutions as $\lambda$ varies

behavior of periodic solution	stable	completely unstable	upper-stable lower-unstable	upper-unstable lower-stable
$\frac{\partial f}{\partial \lambda}\geq 0$	increasing with $\lambda$ increasing	increasing with $\lambda$ decreasing	split with $\lambda$ increasing	disappears with $\lambda$ increasing
$\frac{\partial f}{\partial \lambda}\leq 0$	increasing with $\lambda$ decreasing	increasing with $\lambda$ increasing	disappears with $\lambda$ increasing	split with $\lambda$ increasing

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behavior of periodic solution	stable	completely unstable	upper-stable lower-unstable	upper-unstable lower-stable
$\frac{\partial f}{\partial \lambda}\geq 0$	increasing with $\lambda$ increasing	increasing with $\lambda$ decreasing	split with $\lambda$ increasing	disappears with $\lambda$ increasing
$\frac{\partial f}{\partial \lambda}\leq 0$	increasing with $\lambda$ decreasing	increasing with $\lambda$ increasing	disappears with $\lambda$ increasing	split with $\lambda$ increasing

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