Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy

In this paper, we establish a unilateral diffusion Gompertz model of a single population in two patches in a theoretical way. Firstly, we prove the existence and uniqueness of an order-one periodic solution by the geometry theory of differential equations and the method of successor function. Secondly, we prove the stability of the order-one periodic solution by imitating the theory of the limit cycle of an ordinary differential equation. Finally, we verify the theoretical results by numerical simulations.

[1]  Wanbiao Ma,et al.  Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input , 2017 .

[2]  Lansun Chen,et al.  An impulsive state feedback control model for releasing white-headed langurs in captive to the wild , 2016, Commun. Nonlinear Sci. Numer. Simul..

[3]  Xinyu Song,et al.  Modelling and analysis of impulsive releases of sterile mosquitoes , 2017, Journal of biological dynamics.

[4]  Xinzhu Meng,et al.  Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment , 2016 .

[5]  Zuxiong Li,et al.  Periodic solution of a chemostat model with variable yield and impulsive state feedback control , 2012 .

[6]  Tongqian Zhang,et al.  A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey , 2011, Appl. Math. Comput..

[7]  Xinzhu Meng,et al.  Dynamical Analysis of SIR Epidemic Model with Nonlinear Pulse Vaccination and Lifelong Immunity , 2015 .

[8]  Xinzhu Meng,et al.  Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse , 2017, Advances in Difference Equations.

[9]  Tongqian Zhang,et al.  Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay , 2017 .

[10]  Bing Liu,et al.  DYNAMICS ON A HOLLING II PREDATOR–PREY MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL , 2012 .

[11]  Tonghua Zhang,et al.  Periodic solution of a prey-predator model with nonlinear state feedback control , 2015, Appl. Math. Comput..

[12]  Xiangyun Shi,et al.  Study on a giant panda reintroduction state feedback control pulse model with diffusion between two patches , 2016 .

[13]  Hui Zhang,et al.  Evaluation of Bus Networks in China: From Topology and Transfer Perspectives , 2015 .

[14]  Bing Liu,et al.  Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control , 2004 .

[15]  Rui Liu,et al.  Periodic solution of a pest management Gompertz model with impulsive state feedback control , 2014 .

[16]  Fang Wang,et al.  Existence and Attractiveness of Order One Periodic Solution of a Holling I Predator-Prey Model , 2012 .

[17]  Qingling Zhang,et al.  The geometrical analysis of a predator-prey model with two state impulses. , 2012, Mathematical biosciences.

[18]  Lansun Chen,et al.  A pest management model with state feedback control , 2016 .

[19]  Tonghua Zhang,et al.  Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects , 2017 .

[20]  Jian Zhang,et al.  Threshold Dynamics of a Stochastic SIR Model with Vertical Transmission and Vaccination , 2017, Comput. Math. Methods Medicine.

[21]  Jianjun Jiao,et al.  Pest management through continuous and impulsive control strategies , 2007, Biosyst..

[22]  Elizabeth Pennisi,et al.  When to Go, Where to Stop , 2006, Science.

[23]  Lansun Chen,et al.  Persistence and periodic orbits for two-species nonautonomous diffusion lotka-volterra models , 1994 .

[24]  Sanyi Tang,et al.  Modelling the regulatory system of a chemostat model with a threshold window , 2017, Math. Comput. Simul..

[25]  Lansun Chen,et al.  Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects , 2011, Math. Comput. Simul..

[26]  Sanyi Tang,et al.  Global attractivity in a “food-limited” population model with impulsive effects☆ , 2004 .

[27]  Elena Braverman,et al.  Global stabilization of periodic orbits using a proportional feedback control with pulses , 2012 .

[28]  Guodong Liu,et al.  Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps , 2017, Complex..

[29]  Wei Wang,et al.  Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis , 2017 .

[30]  Guoping Pang,et al.  Periodic solution of the system with impulsive state feedback control , 2014 .

[31]  R. Paine,et al.  Disturbance, patch formation, and community structure. , 1974, Proceedings of the National Academy of Sciences of the United States of America.

[32]  J. B. Shukla,et al.  Population diffusion in a two-patch environment. , 1989, Mathematical biosciences.

[33]  Xinzhu Meng,et al.  Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps , 2017, Journal of Inequalities and Applications.

[34]  Lansun Chen,et al.  Existence and global stability of periodic solution for impulsive predator-prey model with diffusion and distributed delay , 2010 .

[35]  Guirong Jiang,et al.  Impulsive state feedback control of a predator-prey model , 2007 .

[36]  Xinyu Song,et al.  Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting , 2013 .

[37]  Jianjun Jiao,et al.  Dynamical analysis of a delayed predator-prey model with impulsive diffusion between two patches , 2009, Math. Comput. Simul..

[38]  P. Sattayatham,et al.  Dynamics of an SIR epidemic model with stage structure and pulse vaccination , 2016 .

[39]  Meng Zhang,et al.  A state feedback impulse model for computer worm control , 2016 .

[40]  M. Reid,et al.  Intrinsic and extrinsic determinants of mountain pine beetle population growth , 2009 .

[41]  Xinzhu Meng,et al.  Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance , 2016 .

[42]  Tonghua Zhang,et al.  A Stage-Structured Predator-Prey SI Model with Disease in the Prey and Impulsive Effects , 2013 .

[43]  Lin Zou,et al.  The dynamics of an eco-epidemic model with distributed time delay and impulsive control strategy , 2011, J. Frankl. Inst..

[44]  Lansun Chen,et al.  A novel method for analyzing the stability of periodic solution of impulsive state feedback model , 2016, Appl. Math. Comput..

[45]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[46]  Xinzhu Meng,et al.  Optimal harvesting control and dynamics of two-species stochastic model with delays , 2017 .