# American Institute of Mathematical Sciences

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

## Theory of rotated equations and applications to a population model

 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.

Citation: Maoan Han, Xiaoyan Hou, Lijuan Sheng, Chaoyang Wang. Theory of rotated equations and applications to a population model. Discrete and Continuous Dynamical Systems, 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.
Local behavior of stable or completely unstable periodic solution
Local behavior of semi-stable periodic solution
Asymptotic behavior of solutions for $\lambda>\lambda_0$
Local behavior near a semi-stable periodic solution
Behavior of solutions as $\lambda>\lambda^*$ and $x_0>K$ or $x_0<0$
Behavior of $\varphi_s(t,\lambda)$ and $\varphi_u(t,\lambda)$
Behavior of solutions as $\lambda<0$
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
 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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