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Positive periodic solution for a two-species ratio-dependent predator-prey system with time delay and impulse. (English) Zbl 1111.34051
The paper deals with a predator-prey model with delay and impulses of the form $$\align \dot{x}(t)&=x(t)\left(b_1(t)-a_1(t)x(t)-{{c(t)y(t)}\over{m_1(t)y(t)+x(t)}}\right),\\ \dot{y}(t)&=y(t)\left(-b_2(t)+{{a_2(t)x(t-\tau)}\over{m_2(t)y(t-\tau)+x(t-\tau)}}\right), \quad\text{for }t\ne t_k,\endalign$$ $$\align x(t^+_k)-x(t^-_k)& =c_kx(t_k), \qquad y(t^+_k)-y(t^-_k)= d_ky(t_k), \quad (x(0+), y(0+))=(x_0,y_0),\\ (x(t), y(t))&=(\varphi_1(t), \varphi_2(t)),\text{ for }-\tau\leq t\leq 0.\endalign$$ The authors apply the continuation fixed-point theorem of coincindence degree theory to provide sufficient conditions for the existence of a periodic solution of the problem.

34K13Periodic solutions of functional differential equations
34K45Functional-differential equations with impulses
34K60Qualitative investigation and simulation of models
92D25Population dynamics (general)
Full Text: DOI
[1] Tang, S.; Chen, L.: The periodic predator -- prey Lotka -- Volterra model with impulsive effect. J. mech. Med. biol. 2, 267-296 (2002)
[2] Lu, S.; Ge, W.: Periodic solutions for a kind of second order differential equation with multiple deviating arguments. Appl. math. Comput. 146, 195-209 (2003) · Zbl 1037.34065
[3] Chen, Y.: Multiple periodic solutions of delayed predator -- prey systems with type IV functional responses. Nonlinear anal. Real world appl. 5, 45-53 (2004) · Zbl 1066.92050
[4] Fan, Y. -H.; Li, W. -T.; Wang, L. -L.: Periodic solutions of delayed ratio-dependent predator -- prey models with monotonic or nonmonotonic functional responses. Nonlinear anal. Real world appl. 5, 247-263 (2004) · Zbl 1069.34098
[5] Li, Y.: Periodic solutions of periodic delay Lotka -- Volterra equations and systems. J. math. Anal. appl. 255, 260-280 (2001) · Zbl 1024.34062
[6] Zhao, C.; Debnath, L.; Wang, K.: Positive periodic solutions of a delayed model in population. Appl. math. Lett. 16, 561-565 (2003) · Zbl 1058.34088
[7] Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications. (1993) · Zbl 0815.34001
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