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Periodicity and attractivity of a ratio-dependent Leslie system with impulses. (English) Zbl 1216.34045
The authors study the ratio-dependent Leslie predator-pray model with impulses $$\aligned \dot x_1&= x_1(t)\Biggl[b(t)- a(t)x_1(t)- {c(t) x_1(t)x_2(t)\over h^2 x^2_2(t)+ x^2_1(t)}\Biggr],\\ \dot x_2(t)&= x_2(t)\Biggl[e(t)- f(t){x_2(t)\over x_1(t)}\Biggr],\qquad t\ne t_k,\\ x_i(t^+_k)&= (1+ h^i_k) x_i(t_k),\qquad x_i(0)> 0,\quad i= 1,2,\endaligned\tag1$$ where $x_i(t)$, $i= 1,2$ denote the density of prey and predator at time $t$, respectively. $b$, $a$, $c$, $d$, $e$, $f$, $p$, $\alpha_i,\beta_i,\gamma_i\in C(\bbfR,\bbfR_+)$, $i= 1,2$, are $\omega$-periodic functions. Sufficient conditions for the existence of positive periodic solutions of (1) are derived. Sufficient conditions for the existence of a unique positive $\omega$-periodic solution which is globally attractive are found.

34C60Qualitative investigation and simulation of models (ODE)
34A37Differential equations with impulses
34C25Periodic solutions of ODE
34D20Stability of ODE
92D25Population dynamics (general)
Full Text: DOI
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