Dynamical analysis in a hybrid bioeconomic system with multiple time delays and strong Allee effect

In this paper, a multiple delayed differentialalgebraic preypredator system is established, where commercial harvesting on predator and strong Allee effect in prey growth are considered. Three time delays are introduced to represent maturation delay for prey (1), reaction delay to changes of prey surviving environment due to strong Allee effect (2) and gestation delay for predator (3), respectively. Positivity of solutions and uniform persistence of system are discussed. In absence of time delay, existence of singularity induced bifurcation and local stability analysis are investigated due to variation of economic interest of commercial harvesting. In presence of multiple time delays, existence of Hopf bifurcation and local stability analysis are discussed by analyzing associated characteristic equation. By using new normal form of multiple delayed differentialalgebraic system and center manifold theorem, properties of Hopf bifurcation are studied. Furthermore, existence of global continuation of periodic solutions bifurcating from interior equilibrium is discussed by using a global Hopf bifurcation theorem. A bioeconomic system with strong Allee effect and multiple time delays is proposed.Positivity of solutions and uniform persistence of hybrid system are investigated.Combined dynamic effects of multiple time delays and economic interest are studied.Properties of singularity induced bifurcation and Hopf bifurcation are discussed.

[1]  Wei Liu,et al.  Hopf bifurcation for a predator-prey biological economic system with Holling type II functional response , 2011, J. Frankl. Inst..

[2]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[3]  P. D. N. Srinivasu,et al.  Bio-economics of a renewable resource subjected to strong Allee effect , 2014, Commun. Nonlinear Sci. Numer. Simul..

[4]  Prashanta Kumar Mandal,et al.  Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect , 2014, Math. Comput. Simul..

[5]  Gang Wang,et al.  The competitive dynamics of populations subject to an Allee effect , 1999 .

[6]  W. C. Allee.  Animal Aggregations: A Study in General Sociology , 1931 .

[7]  B Chen,et al.  NORMAL FORMS AND BIFURCATIONS FOR THE DIFFERENTIAL-ALGEBRAIC SYSTEMS , 2000 .

[8]  Hüseyin Merdan,et al.  Stability analysis of a general discrete-time population model involving delay and Allee effects , 2012, Appl. Math. Comput..

[9]  Eduardo González-Olivares,et al.  Consequences of double Allee effect on the number of limit cycles in a predator-prey model , 2011, Comput. Math. Appl..

[10]  D. Garcelon,et al.  Pathogen exposure in endangered island fox (Urocyon littoralis) populations: Implications for conservation management , 2006, Biological Conservation.

[11]  Andrew M. Liebhold,et al.  The role of Allee effects in gypsy moth, Lymantria dispar (L.), invasions , 2009, Population Ecology.

[12]  Boshan Chen,et al.  Hopf bifurcation in a delayed differential–algebraic biological economic system ☆ , 2011 .

[13]  H. Schättler,et al.  Local bifurcations and feasibility regions in differential-algebraic systems , 1995, IEEE Trans. Autom. Control..

[14]  R. Shivaji,et al.  Population models with diffusion, strong Allee effect and constant yield harvesting , 2009 .

[15]  Bai-lian Li,et al.  The maximum sustainable yield of Allee dynamic system , 2002 .

[16]  Grenfell,et al.  Inverse density dependence and the Allee effect. , 1999, Trends in ecology & evolution.

[17]  Ludek Berec,et al.  Multiple Allee effects and population management. , 2007, Trends in ecology & evolution.

[18]  Yi Shen,et al.  Hopf bifurcation of a predator–prey system with predator harvesting and two delays , 2013 .

[19]  G. Livadiotis,et al.  General Allee effect in two-species population biology , 2012, Journal of biological dynamics.

[20]  William J. Sutherland,et al.  What Is the Allee Effect , 1999 .

[21]  Pablo Aguirre,et al.  Bifurcations and global dynamics in a predator-prey model with a strong Allee effect on the prey, and a ratio-dependent functional response , 2014 .

[22]  Guodong Zhang,et al.  Hopf bifurcation for a differential-algebraic biological economic system with time delay , 2012, Appl. Math. Comput..

[23]  Abdul-Aziz Yakubu,et al.  Predator-induced and mating limitation-induced Allee effects in a discrete-time SIMS epidemic model , 2013, Comput. Math. Appl..

[24]  Richard T. Roush,et al.  Mate finding, dispersal, number released, and the success of biological control introductions , 1993 .

[25]  Eduardo González-Olivares,et al.  Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey , 2011 .

[26]  Horst R. Thieme,et al.  Mathematics in Population Biology , 2003 .

[27]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[28]  Louis J. Gross,et al.  Applied Mathematical Ecology , 1990 .

[29]  Corrigendum to “Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey” [Nonlinear Anal. RWA 12 (2011) 2931–2942] , 2013 .

[30]  Brian Dennis,et al.  ALLEE EFFECTS: POPULATION GROWTH, CRITICAL DENSITY, AND THE CHANCE OF EXTINCTION , 1989 .

[31]  Eduardo González-Olivares,et al.  Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response , 2012 .

[32]  Qingling Zhang,et al.  Modeling and analysis in a prey-predator system with commercial harvesting and double time delays , 2016, Appl. Math. Comput..

[33]  Rodrigo Ramos-Jiliberto,et al.  Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey , 2011 .

[34]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[35]  H. I. Freedman,et al.  The trade-off between mutual interference and time lags in predator-prey systems , 1983 .

[36]  H. Gordon.  The Economic Theory of a Common-Property Resource: The Fishery , 1954, Journal of Political Economy.

[37]  Eugenio Aulisa,et al.  Continuous-time predator-prey systems with Allee effects in the prey , 2014, Math. Comput. Simul..

[38]  Xue Zhang,et al.  Bifurcation Analysis and Control of a Class of Hybrid Biological Economic Models , 2009 .

[39]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[40]  N. Macdonald,et al.  Time delay in prey-predator models— II. Bifurcation theory , 1977 .

[41]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[42]  Peiyong Liu,et al.  Dynamical Behavior and Stability Analysis in a Hybrid Epidemiological-Economic Model with Incubation , 2014 .

[43]  Michael A. McCarthy,et al.  The Allee effect, finding mates and theoretical models , 1997 .

[44]  Qingling Zhang,et al.  Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting , 2009, J. Comput. Appl. Math..

[45]  Yongli Cai,et al.  Dynamical complexity induced by Allee effect in a predator–prey model , 2014 .

[46]  Kunal Chakraborty,et al.  Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay , 2011 .

[47]  Eduardo Sáez,et al.  Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect , 2009 .

[48]  Wansheng Tang,et al.  DYNAMICAL ANALYSIS AND CONTROL IN A DELAYED DIFFERENTIAL-ALGEBRAIC BIO-ECONOMIC MODEL WITH STAGE STRUCTURE AND DIFFUSION , 2012 .